Spectrum,bifurcation and hypersurfaces of prescribed k -th mean curvature in Minkowski space

Abstract

By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k-th mean curvature problem in Minkowski spacetime $\begin{matrix} \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k} (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ.

Keywords

Spectrum Bifurcation Nodal solutions A priori bounds Asymptotic behaviors One-sign solution

1. Introduction and the main results

It is well known that the Minkowski space $L^{N + 1}$ is the space $R^{N} \times R$ equipped with the metric $\begin{matrix} d s^{2} = d x_{1}^{2} + \dots + d x_{n}^{2} - d x_{N + 1}^{2} . \end{matrix}$ A hypersurface in the Minkowski space $L^{N + 1}$ is said to be spacelike if its induced metric is a Riemannian one. Clearly, if a function u is defined over a subdomain Ω of $R^{N}$ ( $N ⩾ 1$ ), then $M = {(x, u) : x \in Ω}$ is a spacelike hypersurface in $L^{N + 1}$ if and only if $| \nabla u | < 1$ in Ω. The induced metric on $M$ is given by $\begin{matrix} g_{i j} = δ_{i j} - D_{i} u D_{j} u, 1 ⩽ i, j ⩽ N, \end{matrix}$ where $D_{i} u = \partial u / \partial x_{i}$ , $δ_{i j} = 1$ (0) if $j = i$ ( $j \neq i$ ). The second fundamental form of $M$ is $\begin{matrix} h_{i j} = \frac{D_{i j} u}{\sqrt{1 - | \nabla u |^{2}}}, 1 ⩽ i, j ⩽ N, \end{matrix}$ where $D_{i j} u = \partial^{2} u / \partial x_{i} \partial x_{j}$ . The principal curvatures $κ_{1}$ , …, $κ_{N}$ of $M$ are the eigenvalues of matrix $[h_{i j}]$ relative to matrix $[g_{i j}]$ . The k-th mean curvature $H_{k}$ of $M$ is defined to be the k-th elementary symmetric function of the principal curvatures, $\begin{matrix} H_{k} = S_{k} (κ_{1}, \dots, κ_{N}) = \sum_{1 ⩽ i_{1} < \dots < i_{k} ⩽ N} κ_{i_{1}} \dots κ_{i_{k}}, 1 ⩽ k ⩽ N . \end{matrix}$ In particular, $S_{1}$ is the usual mean curvature, $S_{2}$ is the scalar curvature and $S_{N}$ is the Gauss–Kronecker curvature. So the k-th mean curvature equation can be regarded as an extension of the mean curvature equation and the Gauss curvature equation. The k-th mean curvature is also related to General Relativity. A spacelike hypersurface with given k-th mean curvature is a suitable subset of the spacetime where the initial value problems for the different field equations are naturally stated. Each k-th mean curvature function intuitively measures the time evolution towards the future or the past of the spatial universe.

For $k = 1$ , Calabi [13] showed that for $N ⩽ 4$ the equation $S_{1} = 0$ has the Bernstein property that the only entire solutions are linear. This was later extended to all dimensions by Cheng–Yau [14]. Then, Treibergs [43] constructed and classified the entire solutions of the constant mean curvature spacelike hypersurface equation. An important universal existence result was proved by Bartnik–Simon [5]. Further, Bartnik [3] proved the existence of maximal surfaces in asymptotically flat spacetimes. The Dirichlet problem in a more general spacetime was solved by Gerhardt [29] and Bartnik [4]. Recently, there are more contributions, and the interest is many times focused on the existence and multiplicity of solutions, by using a combination of variational techniques, critical point theory, sub-supersolutions, topological degree and bifurcation (see, for example, [2,9–12,15–17,19,21,25] and the references therein).

When $k = 2$ , Bayard [6,7] established the classical solvability of the Dirichlet problem with $N = 3, 4$ and the constant mean curvature, which was extended to higher dimensions by Urbas [44]. Later, Bayard–Delanoë [8] proved the existence and uniqueness of entire spacelike radial hypersurfaces contained in the future of the origin O and asymptotic to the light-cone. And Gerhardt [30] obtained the existence of closed hypersurfaces of prescribed scalar curvature in Lorentzian manifolds provided there are barriers.

The Gauss–Kronecker curvature has also been quite well studied. Here we highlight the work of Li [35] on spacelike hypersurfaces with constant Gauss–Kronecker curvature. And Delanoë [27] proved that there exists a unique smooth solution of the Dirichlet problem with a given Lorentz–Gauss curvature function defined on a bounded convex domain, which was extended to general (non-convex) domain by Guan [31]. Further, Guan, Jian and Schoen [32] studied the entire spacelike convex hypersurfaces of positive constant or prescribed Gauss curvature in Minkowski space $R^{N, 1}$ . More recently, Huang, Jian and Su [33] proved an existence theorem for spacelike convex hypersurfaces of prescribed Gauss curvature in exterior domain.

However, little attention has been paid to hypersurfaces with prescribed k-th mean curvature with $3 ⩽ k < N$ . In 2003, Urbas [45] derived interior curvature bounds for strictly spacelike hypersurfaces of prescribed k-th mean curvature in Minkowski space.

If $Ω = B_{R} : = {x \in R^{N} : | x | < R}$ with $R > 0$ and u is radially symmetric, very recently, Fuente, Romero and Torres [28] have shown that $\begin{matrix} S_{k} = \frac{C_{N}^{k}}{N r^{N - 1}} {(r^{N - k} ϕ^{k} (v^{'}))}^{'}, r \in (0, R), \end{matrix}$ where $v (r) = u (x)$ with $r = | x |$ , $ϕ (s) = s / \sqrt{1 - s^{2}}$ , and $C_{N}^{k} = N! / ((N - k)! k!)$ is the combinatorial constant. Then, by the Schauder Point Fixed Theorem, under some suitable assumptions on $H_{k}$ , they proved the existence of rotationally symmetric solution of the associated Dirichlet problem. Furthermore, they showed that such graph can be extended to the whole space and may be unique.

To the best of our knowledge, there is no result on nonexistence or multiplicity spacelike hypersurfaces of prescribed k-th mean curvature in Minkowski space except the case of $k = 1$ . It is clear that a good understanding of the nonexistence/existence and multiplicity of such surfaces is needed.

Motivated by the above interesting and important studies, in this paper we continue the investigations on the nonexistence/existence and multiplicity of one-sign or nodal solutions of the following problem $\begin{matrix} (1.1) & \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k} (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0, \end{array} \end{matrix}$ where λ is a nonnegative parameter which may indicate the strength of $H_{k}$ . It follows from $v (r) = \int_{R}^{r} v^{'} (τ) d τ$ that any spacelike solution v of problem (1.1) is uniformly bounded by R and $‖ v ‖_{\infty} ⩽ R ‖ v^{'} ‖_{\infty}$ . Hence, the image of v must lie in $[- R, R] : = I_{R}$ . So, from now on, we assume that $H_{k}$ is a real continuous function defined on $[0, R] \times I_{R}$ . The solution of problem (1.1) is understood in the sense that $v \in C^{2} (0, R] \cap C^{1} [0, R]$ and satisfies problem (1.1).

As a previous step, we investigate the spectral structure of the following eigenvalue problem $\begin{matrix} (1.2) & \{\begin{array}{l} - {(r^{N + 1 - p} | v^{'} |^{p - 2} v^{'})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} | v |^{p - 2} v, r \in (0, R), \\ v^{'} (0) = v (R) = 0 \end{array} \end{matrix}$ for any $p \in [2, N + 1]$ , which has its own interest. In [20, Lemma 4.5], we have shown that problem (1.2) possesses a principal eigenvalue $λ_{1} (p)$ . Here we further obtain the spectral structure as follows.

Theorem 1.1.
Problem ( 1.2 ) has a sequence of simple eigenvalues $\begin{matrix} 0 < λ_{1} (p) < λ_{2} (p) < \dots < λ_{i} (p) < \dots, lim_{i \to + \infty} λ_{i} (p) = + \infty, \end{matrix}$ and no other eigenvalue. Moreover, for each $i \in N$ , $λ_{i} (p)$ is continuous with respect to p and any eigenfunction $φ_{i, p}$ corresponding to $λ_{i} (p)$ has exactly $i - 1$ simple zeros in $(0, R)$ .

When k is odd and $p = k + 1$ , we can easily check that problem (1.2) is equivalent to $\begin{matrix} (1.3) & \{\begin{array}{ll} {(r^{N - k} {(- v^{'})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} v^{k}, & r \in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ According to Theorem 1.1, problem (1.3) possesses a sequence of simple eigenvalues ${λ_{i}}$ such that $0 < λ_{1} < λ_{2} < \dots < λ_{i} < \dots$ , ${lim}_{i \to + \infty} λ_{i} = + \infty$ , and any eigenfunction $φ_{i}$ corresponding to $λ_{i}$ has exactly $i - 1$ simple zeros in $(0, R)$ .

When k is even, applying the results of [20, Lemmas 4.5–4.9] with $p = k + 1$ , we know that problem (1.3) possesses a unique principal eigenvalue $λ_{1}$ with a positive eigenfunction $φ_{1}$ , which is minimal, simple and isolated. So, we can choose δ small enough such that there is no eigenvalue of problem (1.3) in $(λ_{1}, λ_{1} + δ)$ .

Since problem (1.1) has only one-sign nontrivial solutions when k is even (see [28]), we study the existence and multiplicity of one-sign nontrivial solutions when k is even. While, we investigate the existence and multiplicity of nodal solutions when k is odd.

If k is odd (even) and ${lim}_{s \to 0} H_{k} (r, s) / s^{k} = - 1$ uniformly in $r \in (0, R)$ , we shall show that the necessary condition for $(μ, 0)$ being a bifurcation point of problem (1.1) is that $μ = λ_{i}$ for some $i \in N$ ( $i = 1$ ) (see Sections 4–5). To realize our goal, the key is to show the sufficiency of $(λ_{i}, 0)$ being a bifurcation point for every $i \in N$ ( $i = 1$ ). It is well known that the index formula of an isolated zero (see [26, Theorem 8.10]) plays a key role in the applications of bifurcation theory. While, this index formula is only involving linear map which cannot be directly applied to problem (1.3). Naturally, we can think of using the invariance of the topological degree under a compact homotopy to obtain an index jumping result. Since k is an integer, it is difficult to obtain a compact homotopy via k, which is why we introduce problem (1.2).

Let $\begin{matrix} X = {v \in C^{1} [0, R] : v^{'} (0) = v (R) = 0} \end{matrix}$ with the norm $‖ v ‖ : = ‖ v^{'} ‖_{\infty}$ . Since $‖ v ‖_{\infty} ⩽ ‖ v^{'} ‖_{\infty} R$ , the norm $‖ v ‖$ is equivalent to the usual norm $‖ v ‖_{\infty} + ‖ v^{'} ‖_{\infty}$ . Set $X^{+} : = {v \in X : v (0) > 0 on [0, R)}$ and $X^{-} : = - X^{+}$ . Let $S_{i}^{+}$ denote the set of functions in $X^{+}$ which have exactly $i - 1$ interior nodal (i.e. non-degenerate) zeros in $(0, R)$ and $S_{i}^{-} = - S_{i}^{+}$ . It is clear that $S_{i}^{+}$ ( $X^{+}$ ) and $S_{i}^{-}$ ( $X^{-}$ ) are disjoint and open in X. Let $Φ_{i}^{\pm} = R \times S_{i}^{\pm}$ under the product topology. Following [41], we add the point ∞ to our space $R \times X$ .

Then, we have the following three theorems, which are the main results of this paper. The first one is global bifurcation result for k being odd.
Theorem 1.2.
Let k be odd. Assume that $H_{k} : [0, R] \times [- R, R] \to R$ is continuous, $H_{k} (r, s) s < 0$ for any $r \in [0, R]$ and $s \in [- R, R]$ and there exists $H_{0} \in [0, + \infty]$ such that $\begin{matrix} lim_{s \to 0} \frac{- H_{k} (r, s)}{s^{k}} = H_{0} \end{matrix}$ uniformly in $r \in (0, R)$ . For any $i \in N$ ,
if $H_{0} = 1$ , there are two unbounded continua, $C_{i}^{+}$ and $C_{i}^{-}$ , of the set of nontrivial solutions of problem ( 1.1 ) bifurcating from $(λ_{i}, 0)$ such that $C_{i}^{ν} \subseteq (Φ_{i}^{ν} \cup \{(λ_{i}, 0)\})$ , $(λ_{i}, + \infty) \subseteq {pr}_{R} (C_{i}^{ν})$ , $‖ v_{λ} ‖ < 1$ and ${lim}_{λ \to + \infty} ‖ v_{λ} ‖ = 1$ for $(λ, v_{λ}) \in (C_{i}^{+} \cup C_{i}^{-}) ∖ \{(λ_{i}, 0)\}$ , where $ν \in {+, -}$ and ${pr}_{R} (C_{i}^{ν})$ denotes the projection of $C_{i}^{ν}$ on $R$ ,

if $H_{0} = + \infty$ , there are two unbounded continua, $C_{i}^{+}$ and $C_{i}^{-}$ , of the set of nontrivial solutions of problem ( 1.1 ) emanating from $(0, 0)$ and joining to $(+ \infty, 1)$ such that $C_{i}^{ν} \subseteq (R_{+} \times S_{i}^{ν} \cup {(0, 0)})$ and $‖ v_{λ} ‖ < 1$ for $(λ, v_{λ}) \in (C_{i}^{+} \cup C_{i}^{-}) ∖ {(0, 0)}$ with $λ < + \infty$ , where $R_{+} = [0, + \infty)$ ,

if $H_{0} = 0$ , there are two unbounded continua, $C_{i}^{+}$ and $C_{i}^{-}$ , of the set of nontrivial solutions of problem ( 1.1 ) such that $C_{k}^{ν} \subseteq R_{+} \times S_{i}^{ν}$ , joins $(+ \infty, 1)$ to $(+ \infty, 0)$ , $C_{k}^{ν} ∖ {\infty} \neq \emptyset$ and $‖ v_{λ} ‖ < 1$ for any $(λ, v_{λ}) \in C_{k}^{ν}$ with $λ < + \infty$ .

Fig. 1.
Bifurcation diagrams of Theorem 1.2.

The existence and multiplicity of nodal solutions of problem (1.1) can be easily seen from these bifurcation diagrams in Fig. 1. In particular, if $H_{0} = 1$ , for $λ \in (λ_{i}, λ_{i + 1}]$ , problem (1.1) possesses at least i pairs solutions $v_{j}^{\pm} \in S_{j}^{\pm}$ with $j = 1, 2, \dots, i$ ; if $H_{0} = + \infty$ , problem (1.1) has at least one pairs solutions $v_{i}^{\pm} \in S_{i}^{\pm}$ ; if $H_{0} = 0$ , there exits $λ_{} > 0$ such that problem (1.1) possesses at least two pairs solutions $v_{i, j}^{\pm} \in S_{i}^{\pm}$ with $j = 1, 2$ .

For $i > 1$ , solutions obtained in Theorem 1.2 must change sign, which is different from the case of $i = 1$ . So, it is very difficult to study the asymptotic behavior* of solutions with respect to λ. We shall introduce some new techniques to overcome this difficulty. Theorem 1.1 extends the corresponding results of [17, Theorems 1.1–1.3] where only the case of $k = 1$ and $i = 1$ is considered. Up to the authors’ knowledge, Theorem 1.2 is the first result involving the existence of sign-changing spacelike hypersurfaces of prescribed k-th mean curvature in Minkowski space with $k > 1$ .

When k is even, we have that
Theorem 1.3.
Let k be even. Assume that $H_{k} (r, s) > 0$ for any $r \in [0, R]$ , $s \in I_{R} ∖ {0}$ , and there exists $H_{0} \in [0, + \infty]$ such that $\begin{matrix} lim_{s \to 0} \frac{H_{k} (r, s)}{s^{k}} = H_{0} \end{matrix}$ uniformly in $r \in (0, R)$ . Then,
if $H_{0} = 1$ , there are two unbounded continua $C^{+}$ and $C^{-}$ of the set of nontrivial solutions of problem ( 1.1 ) bifurcating from $(λ_{1}, 0)$ such that $C^{\pm} \subseteq (R_{+} \times S_{1}^{\mp}) \cup \{(λ_{1}, 0)\}$ , $(λ_{1}, + \infty) \subseteq {pr}_{R} (C^{\pm})$ , $‖ v_{λ} ‖ < 1$ and ${lim}_{λ \to + \infty} ‖ v_{λ} ‖ = 1$ for $(λ, v_{λ}) \in C^{\pm} ∖ \{(λ_{1}, 0)\}$ ,

if $H_{0} = + \infty$ , there are two unbounded continua $C^{+}$ and $C^{-}$ of the set of nontrivial solutions of problem ( 1.1 ) emanating from $(0, 0)$ such that $C^{\pm} \subseteq (R_{+} \times S_{1}^{\mp}) \cup {(0, 0)}$ , joins to $(+ \infty, 1)$ and $‖ v_{λ} ‖ < 1$ for any $(λ, v_{λ}) \in C^{\pm} ∖ {(0, 0)}$ with $λ < + \infty$ ,

if $H_{0} = 0$ , there are two unbounded continua $C^{+}$ and $C^{-}$ of the set of nontrivial solutions of problem ( 1.1 ) such that $C^{\pm} \subseteq R_{+} \times S_{1}^{\mp}$ , joins $(+ \infty, 1)$ to $(+ \infty, 0)$ and $‖ v_{λ} ‖ < 1$ for any $(λ, v_{λ}) \in C^{\pm}$ with $λ < + \infty$ .

Finally, we present a result concerning the nonexistence of one-sign solution.
Theorem 1.4.
Assume that there exists a positive constant ϱ such that $\begin{matrix} \frac{{(- 1)}^{k} H_{k} (r, s)}{s^{k}} ⩽ ϱ \end{matrix}$ for any $s \in I_{R} ∖ {0}$ and $r \in (0, R)$ . Then there exists $ϱ_{} > 0$ such that problem (* 1.1 ) has not one-sign solution for $λ \in (0, ϱ_{*})$ .

The outline of the rest of this article is as follows. In Section 2, we show the existence and uniqueness result for problem (1.1), give a priori bounds estimate of derivatives for any solution of problem (1.1) and two Whyburn type topological lemmas. The proof of Theorem 1.1 will be given in Section 3. The proof of Theorem 1.2 will be given in Section 4. In the last Section, we present the proofs of Theorems 1.3–1.4.
2. Preliminaries

Firstly, we have the following existence and uniqueness result for problem (1.1), which will be used later.

Lemma 2.1.
Assume that there exists a constant $M > 0$ such that $\begin{matrix} | H_{k} (r, s) | ⩽ M | s |^{k} \end{matrix}$ for any $r \in [0, R]$ and $s \in I_{R}$ . For any solution v of problem ( 1.1 ) having a double zero, then $v \equiv 0$ .
Proof.
Let v be a solution of problem (1.1) and $r_{} \in [0, R]$ be a double zero. If k is even, we must have $\begin{matrix} \int_{r_{}}^{r} τ^{N - 1} H_{k} (τ, v (τ)) d τ ⩾ 0 \end{matrix}$ for any $r \in [0, R]$ . Indeed, otherwise, we easily find that $\begin{matrix} 0 ⩽ {r_{0}}^{N - k} {(\frac{v^{'} (r_{0})}{\sqrt{1 - v^{' 2} (r_{0})}})}^{k} = λ \frac{N}{C_{N}^{k}} \int_{r_{}}^{r_{0}} τ^{N - 1} H_{k} (τ, v (τ)) d τ < 0 \end{matrix}$ for some $r_{0} \in [0, R]$ , which is a contradiction. So, we have that $\begin{matrix} v (r) = κ \int_{r_{}}^{r} ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{r_{}}^{s} τ^{N - 1} H_{k} (τ, v (τ)) d τ)}^{1 / k}] d s . \end{matrix}$ where $ϕ^{- 1} (s)$ denotes the inverse function of $ϕ (s)$ , $κ = +$ ( $κ = \pm$ ) if k is odd (even). It is easy to show that $\begin{matrix} ϕ^{- 1} (s) = \frac{s}{\sqrt{1 + s^{2}}} . \end{matrix}$ Hence, we have that $\begin{matrix} v (r) = κ \int_{r_{}}^{r} \frac{{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{r_{}}^{s} τ^{N - 1} H_{k} (τ, v (τ)) d τ)}^{1 / k}}{\sqrt{1 + {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{r_{}}^{s} τ^{N - 1} H_{k} (τ, v (τ)) d τ)}^{2 / k}}} d s . \end{matrix}$ If $r_{} = 0$ , we have that $\begin{array}{rcl} {| v (r) |}^{k} & ⩽ & \frac{λ N R^{k}}{C_{N}^{k}} \int_{0}^{r} | H_{k} (τ, v (τ)) | d τ . \end{array}$ Hence, we have that $\begin{matrix} {| v (r) |}^{k} ⩽ \frac{λ N R^{k} M}{C_{N}^{k}} \int_{0}^{r} {| v (τ) |}^{k} d τ . \end{matrix}$ By the Gronwall–Bellman inequality [20, Lemma 2.1], we have that $v \equiv 0$ on $[0, R]$ .

Next, we assume that $r_{} > 0$ . We first consider $r \in [0, r_{}]$ . Reasoning as the above, we obtain that $\begin{array}{r} r^{N - k} {| v (r) |}^{k} ⩽ \frac{λ N R^{k} M}{C_{N}^{k}} \int_{r}^{r_{}} τ^{N - k} {| v (τ) |}^{k} d τ . \end{array}$ It follows from the Gronwall–Bellman inequality [20, Lemma 2.2] that $r^{N - k} | v (r) |^{k} \equiv 0$ on $[0, r_{}]$ . It follows that $v \equiv 0$ on $[0, r_{}]$ . On the other hand, for $r \in [r_{}, R]$ , in the same manner, we have that $\begin{array}{r} {| v (r) |}^{k} ⩽ \frac{λ N R^{N} M}{r_{}^{N - k} C_{N}^{k}} \int_{r_{}}^{r} {| v (τ) |}^{k} d τ . \end{array}$ Again by the Gronwall–Bellman inequality [20, Lemma 2.1], we have that $v \equiv 0$ on $[r_{}, 1]$ . Therefore, we have that $v \equiv 0$ on $[0, R]$ . □

Clearly, if v is nonnegative, we only need that $| H_{k} (r, s) | ⩽ M s^{k}$ for any $r \in [0, R]$ and $s \in [0, R]$ . To prove our main results, we need the following global derivatives estimate.
Lemma 2.2.
For any solution v of problem ( 1.1 ) with any fixed λ, there exists a positive constant $θ = θ (Λ, λ, R, k, N)$ such that $‖ v ‖ ⩽ 1 - θ$ , where $Λ = {max}_{[0, R] \times I_{R}} | H_{k} |$ .
Proof.
We assume that $‖ v ‖ = | v^{'} (ρ) | : = γ$ . If $ρ = 0$ , the conclusion is done since $γ = 0$ . Next, we assume that $ρ > 0$ . By Lemma 6.1 of [28], we know that $γ < 1$ . Integrating the first equation of problem (1.1) from 0 to ρ, we obtain that $\begin{matrix} ρ^{N - k} {(\frac{v^{'} (ρ)}{\sqrt{1 - v^{' 2} (ρ)}})}^{k} = λ \frac{N}{C_{N}^{k}} \int_{0}^{ρ} r^{N - 1} H_{k} (r, v (r)) d r . \end{matrix}$ It follows that $\begin{array}{r} \frac{γ^{k}}{{(1 - γ^{2})}^{k / 2}} ⩽ \frac{λ Λ ρ^{k}}{C_{N}^{k}}, \end{array}$ which implies that $\begin{array}{r} \frac{γ}{\sqrt{1 - γ^{2}}} ⩽ {(\frac{λ Λ}{C_{N}^{k}})}^{1 / k} ρ ⩽ {(\frac{λ Λ}{C_{N}^{k}})}^{1 / k} R . \end{array}$ Then, by some elementary calculations, we obtain that $\begin{array}{rcl} γ & ⩽ & \frac{{(\frac{λ Λ}{C_{N}^{k}})}^{1 / k} R}{\sqrt{1 + {(\frac{λ Λ}{C_{N}^{k}})}^{2 / k} R^{2}}} : = γ^{} . \end{array}$ Clearly, we can see that $γ^{} < 1$ . The desired conclusion is immediately obtained by taking $θ = 1 - γ^{*}$ . □

Note that if v is nonnegative in Lemma 2.2, we only need that $H_{k}$ is continuous on $[0, R] \times [0, R]$ , which is also valid for Theorems 1.2–1.3. We end this section by presenting two Whyburn type topological lemmas, which will be used to prove Theorems 1.2–1.3. Lemma 2.3 ([18, Theorem 2.1]).

Let X be a normal space and let $\{C_{n}\}$ be a sequence of unbounded connected subsets of X. Assume that:

there exists $z^{*} \in {lim inf}_{n \to + \infty} C_{n}$ with $‖ z^{*} ‖ < + \infty$ ;

for every $R > 0$ , $(⋃_{n = 1}^{+ \infty} C_{n}) \cap B_{R}$ is a relatively compact set of X, where $\begin{matrix} B_{R} = {x \in X : ‖ x ‖ ⩽ R} . \end{matrix}$

Then

D = {lim sup}_{n \to + \infty} C_{n}

is unbounded, closed and connected.

Lemma 2.4 ([22, Theorem 1.1]).

If $[A_{i}]$ is an infinite sequence of sets in X such that

for every $R > 0$ , $(⋃_{i = 1}^{+ \infty} A_{i}) \cap {\overline{B}}_{R}$ (if nonempty) is a relatively compact,

for each i, any pair of points of $A_{i}$ can be joined in $A_{i}$ by an $ϵ_{i}$ -chain and $ϵ_{i} \to 0$ as $i \to + \infty$ ,

${lim sup}_{i \to + \infty} [A_{i}] ∖ {\infty} \neq \emptyset$ ,

then

{lim sup}_{i \to + \infty} [A_{i}] ∖ {\infty}

is connected.

3. Proof of Theorem 1.1

Let $I = [0, R]$ , $E = \{v \in C^{1} [0, R] : v (R) = 0\}$ and $W_{p} (I)$ be the real Banach space obtained by completing E under the following norm $\begin{matrix} ‖ v ‖_{p} = {(\int_{0}^{R} r^{N + 1 - p} {| v^{'} |}^{p} d r)}^{1 / p} . \end{matrix}$ From Proposition 4.3 [20], we know that the embedding of $W_{p} (I) ↪ L^{p} (r^{N - 1} d r)$ is continuous and compact, where $L^{p} (r^{N - 1} d r) = \{v : \int_{0}^{R} r^{N - 1} | v |^{p} d r < + \infty\}$ with the norm $\begin{matrix} ‖ v ‖_{L^{p}} = {(\int_{0}^{R} r^{N - 1} | v |^{p} d r)}^{1 / p} . \end{matrix}$ Since $L^{p} (B_{R})$ is separable, using Theorem 1.21 of [1], one has that $L^{p} (r^{N - 1} d r)$ is separable. Further, we have that $W_{p} (I)$ is separable.

For any $v \in W_{p} (I)$ , define $\begin{matrix} Φ (v) = \frac{1}{p} \int_{0}^{R} r^{N + 1 - p} {| v^{'} |}^{p} d r, Ψ (v) = \frac{N}{p C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p} d r . \end{matrix}$ Use $W_{p}^{*} (I)$ to denote the dual space of $W_{p} (I)$ . It is obvious that the functional Φ is continuously Gâteaux differentiable whose Gâteaux derivative at the point $v \in W_{p} (I)$ is the functional $Φ^{'} (v) \in W_{p}^{*} (I)$ , given by $\begin{matrix} ⟨ Φ^{'} (v), ϕ ⟩ = \int_{0}^{R} r^{N + 1 - p} {| v^{'} |}^{p - 2} v^{'} ϕ^{'} d r \end{matrix}$ for any $ϕ \in W_{p} (I)$ , where $⟨ \cdot, \cdot ⟩$ denotes the pairing between $W_{p} (I)$ and $W_{p}^{*} (I)$ . Firstly, we have the following properties about the operator $Φ^{'}$ .

Proposition 3.1.
Let $L = Φ^{'}$ . Then, we have that
$L : W_{p} (I) \to W_{p}^{} (I)$ is a continuous and strictly monotone operator;*

if $v_{n} ⇀ v$ in $W_{p} (I)$ and $\underset{n \to + \infty}{lim^{‾}} ⟨L (v_{n}) - L (v), v_{n} - v⟩ ⩽ 0$ , then $v_{n} \to v$ in $W_{p} (I)$ .

Proof.
(i) Let $v_{n} \to v$ in $W_{p} (I)$ , i.e. ${lim}_{n \to + \infty} ‖ v_{n} - v ‖_{p} = 0$ . For any $ϕ \in X$ , by using of the Hölder inequality, we have that $\begin{array}{r} ⟨ Φ^{'} (v_{n} - v), ϕ ⟩ ⩽ ‖ ϕ ‖_{p} ‖ v_{n} - v ‖_{p}^{p / q} \to 0 as n \to + \infty, \end{array}$ where $q = p / (p - 1)$ . It follows that L is a continuous operator. The monotonicity of L can be easily derived from the following inequalities (see [34,36]) $\begin{array}{c} (| ξ |^{p - 2} ξ - | η |^{p - 2} η) \cdot (ξ - η) {(| ξ |^{p} + | η |^{p})}^{\frac{2 - p}{p}} ⩾ (p - 1) | ξ - η |^{p} if 1 < p < 2, \\ (| ξ |^{p - 2} ξ - | η |^{p - 2} η) \cdot (ξ - η) ⩾ {(\frac{1}{2})}^{p} | ξ - η |^{p} if p ⩾ 2, \end{array}$ where $ξ, η \in R^{N}$ .

(ii) From (i), if $v_{n} ⇀ v$ and $\underset{n \to + \infty}{lim^{‾}} ⟨ L (v_{n}) - L (v), v_{n} - v ⟩ ⩽ 0$ , then we have that $\begin{matrix} lim_{n \to + \infty} ⟨ L (v_{n}) - L (v), v_{n} - v ⟩ = 0 . \end{matrix}$ It follows that $v_{n}^{'}$ converges in measure to $v^{'}$ in $(0, R)$ . Further, by Riesz’s Theorem, we get a subsequence (which we still denote by $v_{n}$ ) satisfying $v_{n}^{'} (r) \to v^{'} (r)$ , a.e. $r \in (0, R)$ . Applying the Lebesgue Dominated Convergence Theorem, we find that $\begin{matrix} (3.1) & lim_{n \to + \infty} \int_{0}^{R} r^{N + 1 - p} {| v_{n}^{'} |}^{p} d r = \int_{0}^{R} r^{N + 1 - p} {| v^{'} |}^{p} d r . \end{matrix}$ From (3.1) it follows that the integrals of the functions family $\{r^{N + 1 - p} | v_{n}^{'} |^{p}\}$ possess absolutely equi-continuity on $[0, R]$ (see [37, Chapter 6, Section 3]). Since $\begin{matrix} | ξ - η |^{p} ⩽ 2^{p - 1} (| ξ |^{p} + | η |^{p}), \end{matrix}$ the integrals of the family $\{r^{N + 1 - p} | v_{n}^{'} - v |^{p}\}$ are also absolutely equi-continuous on $[0, R]$ . By the Vitali convergence theorem, we obtain that $\begin{matrix} lim_{n \to + \infty} \int_{0}^{R} r^{N + 1 - p} {| v_{n}^{'} - v |}^{p} d r = 0 . \end{matrix}$ Therefore, $v_{n} \to v$ in $W_{p} (I)$ . □

Then, by the Ljusternik–Schnirelmann theory, we can show the existence of a sequence eigenvalues of problem (1.2) as follows.
Proposition 3.2.
The eigenvalue problem ( 1.2 ) has a sequence of eigenvalues $\begin{matrix} 0 < μ_{1} ⩽ μ_{2} ⩽ \dots ⩽ μ_{k} ⩽ \dots, lim_{k \to + \infty} μ_{k} = + \infty . \end{matrix}$ Moreover, one has that $μ_{1} = inf \{\int_{0}^{R} r^{N + 1 - p} | v^{'} |^{p} d r : \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p} d r = 1\}$ .
Proof.
Set $M : = {v \in W_{p} (I) : p Ψ (v) = 1}$ . It is obvious that $M$ is a closed symmetric $C^{1}$ -submanifold of $W_{p} (I)$ with $0 \notin M$ , and $Φ \in C^{1} (M, R)$ is even, bounded from below. It follows from the Lagrange multiplier rule that $v \in M$ is a critical point of Φ if and only if $\begin{matrix} \int_{0}^{R} r^{N + 1 - p} {| v^{'} |}^{p - 2} v^{'} ϕ^{'} d r = λ \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p - 2} v ϕ d r, \forall ϕ \in W_{p} (I) . \end{matrix}$ Taking $ϕ = v$ , we see that the Lagrange multiplier λ equals the corresponding critical value $p Φ (v)$ and the eigenvalues of problem (1.2) are precisely the critical values of the functional $p Φ$ on $M$ .

Now, we check that Φ satisfies the Palais–Smale condition at any level set $c \in R$ . Suppose that ${v_{n}} \subset W_{p} (I)$ , $| Φ (v_{n}) | ⩽ c$ and $Φ^{'} (v_{n}) \to 0$ . For any constant $θ > p$ , we have that $\begin{array}{r} c + ‖ v_{n} ‖_{p} ⩾ Φ (v_{n}) - \frac{1}{θ} Φ^{'} (v_{n}) v_{n} ⩾ (\frac{1}{p} - \frac{1}{θ}) ‖ v_{n} ‖_{p}^{p} . \end{array}$ Since $p ⩾ 2$ , $\{‖ v_{n} ‖_{p}\}$ is bounded. Without loss of generality, we may assume that $v_{n} ⇀ v$ in $W_{p} (I)$ . Then, one has that $Φ^{'} (v_{n}) (v_{n} - v) \to 0$ . Hence, we find that $\begin{matrix} - \int_{0}^{R} r^{N + 1 - p} {| v_{n}^{'} |}^{p - 2} v_{n}^{'} (v_{n}^{'} - v^{'}) d r \to 0 . \end{matrix}$ By Proposition 3.1, we conclude that $v_{n} \to v$ .

Define $\begin{matrix} Γ_{i} = {K \subset M : K is symmetric, compact and γ (K) ⩾ i}, \end{matrix}$ where $γ (K)$ is the genus of K. We claim that $Γ_{i} \neq \emptyset$ for any $i ⩾ 1$ . Set $N = {v \in W_{p} (I) : p Ψ (v) ⩽ 1}$ . Assume that ${e_{n}}_{n \in N}$ be the standard orthogonal basis of $W_{p} (I)$ and let $X_{i} : = span {e_{1}, \dots, e_{i}}$ . Clearly, $dim X_{i} = i$ and $X_{i} \cap N$ is a bounded and symmetric neighbourhood of $0 \in X_{i}$ . According to (f) of [42, Proposition 2.3], we have that $γ (X_{i} \cap M) = i$ . Since $X_{i} \cap M$ is symmetric and compact, we have that $Γ_{i} \neq \emptyset$ , which verified the desired claim.

Applying Corollary 4.1 of [42], we obtain that problem (1.2) possesses a sequence of positive eigenvalues $\begin{matrix} μ_{1} ⩽ μ_{2} ⩽ \dots ⩽ μ_{i} ⩽ \dots, lim_{i \to + \infty} μ_{i} = + \infty \end{matrix}$ with $\begin{matrix} μ_{i} = inf_{K \in Γ_{i}} sup_{v \in K} p Φ (v), i \in N . \end{matrix}$ For $i = 1$ , taking $K = {v, - v : v \in M}$ , we have that $\begin{array}{rcl} μ_{1} & = & inf_{v \in M} p Φ (u) \\ = & inf {\int_{0}^{R} r^{N + 1 - p} {| v^{'} |}^{p} d r : \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p} d r = 1} \\ = & inf_{v \in W_{p} (I) ∖ {0}} \frac{\int_{0}^{R} r^{N + 1 - p} | v^{'} |^{p} d r}{\frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p} d r} > 0, \end{array}$ which is just the desired conclusion. □

From [20, Lemma 4.5], we know that $μ_{1} = λ_{1} (p)$ . For the consistency of symbols, from now on, we write $λ_{i} (p)$ as $μ_{i}$ . Let S denote the unit sphere of $W_{p} (I)$ . From Proposition 3.2, we see that $\begin{matrix} (3.2) & \frac{1}{λ_{i} (p)} = max_{\underset{dim W = i}{W \subset W_{p} (I)}} min_{u \in W \cap S} \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p} d r . \end{matrix}$ In particular, we have that $\begin{matrix} \frac{1}{λ_{1} (p)} = max_{v \in S} \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p} d r . \end{matrix}$ By [20, Lemmas 4.5–4.9], we know that $λ_{1} (p)$ is simple, isolated, continuous with respect to p and the unique positive principal eigenvalue. Further, Corollary 4.1 of [42] implies that $λ_{1} (p) < λ_{2} (p)$ .

We write $λ (J)$ to denote that λ is depending on some interval J. Any connected component of $I ∖ {r \in I : v (r) = 0}$ is called a nodal domain of v. To prove Theorem 1.1, we first show the following technical lemma.
Lemma 3.1.

The restriction of a nontrivial solution v of problem ( 1.2 ), on a nodal domain ω, is an eigenfunction of problem ( 1.2 ) on ω. And we have $λ (I) = λ_{1} (ω)$ .

$λ_{1} (I)$ verifies the strict monotonicity property with respect to the domain I, i.e, if J is a strict subinterval of I with nonzero measure, then $λ_{1} (I) < λ_{1} (J)$ .

Proof.
(i) Without loss of generality, we assume that $ω = [a, b]$ . Clearly, v satisfies the following problem $\begin{matrix} (3.3) & \{\begin{array}{ll} - {(r^{N + 1 - p} | v^{'} |^{p - 2} v^{'})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} | v |^{p - 2} v, & r \in (a, b), \\ B v (a) = v (b) = 0, \end{array} \end{matrix}$ where $B v (a) = v^{'} (a)$ if $a = 0$ and $B v (a) = v (a)$ if $a > 0$ . By arguments similar to those of Proposition 3.2 and [20, Lemmas 4.5–4.9] with obvious changes, we can show that problem (3.3) has a unique positive, simple, isolated, minimal principal eigenvalue $λ_{1} (ω)$ which is continuous with respect to p. Define $\begin{matrix} \tilde{v} = \{\begin{array}{ll} v, & r \in ω, \\ 0, & r \in I ∖ ω, \end{array} \end{matrix}$ which is called the extension by zero of v on I. It is obvious that $\tilde{v} \in W_{p} (I)$ . Multiplying (3.3) by $\tilde{v}$ , we reach that $\begin{matrix} \int_{a}^{b} r^{N + 1 - p} {| v^{'} |}^{p} d r = λ (I) \frac{N}{C_{N}^{k}} \int_{a}^{b} r^{N - 1} | v |^{p} d r . \end{matrix}$ It follows that the restriction of v on ω is a (weak) solution of problem (3.3) with constant sign. Thus, we have that $λ (I) = λ_{1} (ω)$ .

(ii) Let $v_{1}$ be an eigenfunction corresponding to $λ_{1} (J)$ with $‖ v_{1} ‖_{W_{p} (J)} = 1$ . Then, we see that $\begin{matrix} \frac{1}{λ_{1} (J)} = \frac{N}{C_{N}^{k}} \int_{J} r^{N - 1} | v_{1} |^{p} d r . \end{matrix}$ Define ${\tilde{v}}_{1}$ as that of $\tilde{v}$ . Since ${\tilde{v}}_{1}$ vanishes in $I ∖ J$ , it cannot be an eigenfunction corresponding to the principal eigenvalue $λ_{1} (I)$ . Then, we obtain that $\begin{matrix} \frac{1}{λ_{1} (J)} = \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | {\tilde{v}}_{1} |^{p} d r < \frac{1}{λ_{1} (I)} . \end{matrix}$ So, $λ_{1} (I)$ is strictly decreasing with respect to the domain I. □

Let $Z (v) = {r \in [0, R) : v (r) = 0}$ . From [20], we know that any eigenfunction $φ_{i, p}$ corresponding to $λ_{i} (p)$ is classical solution of problem (1.2). On the basis of Lemma 3.1, we next show that any eigenfunction corresponding to $λ_{2} (I)$ has a unique zero in I.
Lemma 3.2.
For any eigenfunction v corresponding to $λ_{2} (I)$ , there exists a unique real number $c_{2, 1} \in (0, R)$ such that $Z (v) = \{c_{2, 1}\}$ .
Proof.
By [20, Lemma 4.5], v must change its sign. By Lemma 4.1 of [20], we have that $v (0) \neq 0$ and v has a finite number of zeros in $(0, R)$ . Without loss of generality, we assume that $c_{1} > 0$ be the first zero and $c_{2} ⩾ c_{1}$ be the last zero.

It is enough to show that $c_{1} = c_{2}$ . By contradiction, we assume that $c_{1} < c_{2}$ . Choose $d \in (c_{1}, c_{2})$ and put $J_{1} = [0, d)$ , $J_{2} = (d, R)$ , $I_{1} = [0, c_{1})$ , $I_{2} = (c_{2}, R)$ . Clearly, $I_{i} \subset J_{i}$ strictly for $i = 1, 2$ . From the argument of Lemma 3.1, we see that the strict monotonicity property of $λ_{1} (ω)$ with the domain is also valid for problem (3.3). So, using Lemma 3.1, we obtain that $\begin{matrix} (3.4) & λ_{1} (J_{1}) < λ_{1} (I_{1}) = λ_{2} (I) \end{matrix}$ and $\begin{matrix} (3.5) & λ_{1} (J_{2}) < λ_{1} (I_{2}) = λ_{2} (I) . \end{matrix}$ For $i = 1, 2$ , let $ϕ_{i} \in S$ be an eigenfunction corresponding to $λ_{1} (J_{i})$ . By Proposition 3.2 and the argument of Lemma 3.1, we see that $\begin{matrix} (3.6) & \frac{1}{λ_{1} (J_{i})} = \frac{N}{C_{N}^{k}} \int_{J_{i}} r^{N - 1} | ϕ_{i} |^{p} d r . \end{matrix}$ Let ${\tilde{ϕ}}_{i}$ be the extension by zero of $ϕ_{i}$ on I, $i = 1, 2$ . Consider the two dimensional subspace $W : = span \{{\tilde{ϕ}}_{1}, {\tilde{ϕ}}_{2}\}$ and set $K_{2} : = W \cap S$ . For $v \in K_{2}$ , obviously, there exist α, $β \in R$ such that $v = α {\tilde{ϕ}}_{1} + β {\tilde{ϕ}}_{2}$ and $| α |^{p} + | β |^{p} = 1$ . By (3.2) and (3.4)–(3.6), we have that $\begin{array}{rcl} \frac{1}{λ_{2} (I)} & ⩾ & min_{v \in K_{2}} \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} | v |^{p} d r \\ = & \frac{N}{C_{N}^{k}} (min_{v = α {\tilde{ϕ}}_{1} + β {\tilde{ϕ}}_{2} \in K_{2}} (| α |^{p} \int_{J_{1}} r^{N - 1} | ϕ_{1} |^{p} d r + | β |^{p} \int_{J_{2}} r^{N - 1} | ϕ_{2} |^{p} d r)) \\ = & \frac{N}{C_{N}^{k}} (| α_{0} |^{p} \int_{J_{1}} r^{N - 1} | ϕ_{1} |^{p} d r + | β_{0} |^{p} \int_{J_{2}} r^{N - 1} | ϕ_{2} |^{p} d r) \\ = & \frac{| α_{0} |^{p}}{λ_{1} (J_{1})} + \frac{| β_{0} |^{p}}{λ_{1} (J_{2})} > \frac{| α_{0} |^{p} + | β_{0} |^{p}}{λ_{2} (I)} = \frac{1}{λ_{2} (I)} \end{array}$ for some constants $α_{0}$ and $β_{0}$ with $| α_{0} |^{p} + | β_{0} |^{p} = 1$ , which is a contradiction. Therefore, every eigenfunction corresponding to $λ_{2} (I)$ has one and only one zero in $(0, R)$ , which is denoted by $c_{2, 1}$ .

Let w be another eigenfunction corresponding to $λ_{2} (I)$ . Let d be the unique zero in $(0, R)$ . It suffices to show that $d = c_{2, 1}$ . Suppose, on the contrary, that $c_{2, 1} \neq d$ . Without loss of generality, we assume that $c_{2, 1} < d$ . By Lemma 3.1, we have that $\begin{matrix} λ_{2} (I) = λ_{1} ([0, d)) < λ_{1} ([0, c_{2, 1})) = λ_{2} (I), \end{matrix}$ which is absurd. □

Further, we can show that $λ_{2} (I)$ is simple, isolated and strictly decreasing with respect to the domain I.
Lemma 3.3.
$λ_{2} (I)$ is simple and $λ_{2} (I) < λ_{3} (I)$ . Moreover, $λ_{2} (I)$ verifies the strict monotonicity property with respect to the domain I.
Proof.
Let u and v be any two eigenfunctions corresponding to $λ_{2} (I)$ . Lemma 3.1 shows that the restrictions of u and v on $[0, c_{2, 1})$ are eigenfunctions corresponding to $λ_{1} ([0, c_{2, 1}))$ . So, we have $u = α v$ on $[0, c_{2, 1})$ for some constant $α \neq 0$ . Similarly, we have $u = β v$ on $(c_{2, 1}, R)$ for some constant $β \neq 0$ . It follows that $\begin{matrix} α v^{'} (c_{2, 1}) = u^{'} (c_{2, 1}) = β v^{'} (c_{2, 1}) . \end{matrix}$ By Lemma 4.1 of [20], we have that $v^{'} (c_{2, 1}) \neq 0$ . So, we must have $α = β$ , that is to say $λ_{2} (I)$ is simple. If $λ_{2} (I) = λ_{3} (I)$ , we have that $φ_{3} = c φ_{2}$ for some constant $c \neq 0$ , which contradicts Corollary 4.1 of [42]. Therefore, we must have $λ_{2} (I) < λ_{3} (I)$ .

Let $J = (c, d)$ be a strict subinterval of I. Analogously to that of Lemma 3.2, we can show that any eigenfunctions corresponding to $λ_{2} (J)$ has a unique zero in J, which is denoted by $c_{2, 1}^{'}$ . If $c_{2, 1} = c_{2, 1}^{'} : = e$ , we must have that $(c, e)$ is a strict subinterval of $[0, e)$ . Note that the conclusions of Lemma 3.1 are also valid for problem (3.3). By Lemma 3.1, we have that $\begin{matrix} λ_{2} (I) = λ_{1} ([0, e)) < λ_{1} ((c, e)) = λ_{2} (J) . \end{matrix}$ If $c_{2, 1} \neq c_{2, 1}^{'}$ , without loss of generality, letting $c_{2, 1} > c_{2, 1}^{'}$ and using Lemma 3.1, we reach that $\begin{matrix} λ_{2} (I) = λ_{1} ([0, c_{2, 1})) < λ_{1} ((c, c_{2, 1}^{'})) = λ_{2} (J) . \end{matrix}$ Therefore, $λ_{2} (I)$ is strictly monotonous with respect to the domain I. □

Moreover, we shall show that problem (1.2) has no other eigenvalue except $λ_{2} (I)$ such that the corresponding functions have exactly one zero.
Lemma 3.4.
If any eigenfunction u corresponding to some positive eigenvalue $λ (I)$ is such that $Z (u) = {c}$ for some real number c, then $λ (I) = λ_{2} (I)$ .
Proof.
If $c = c_{2, 1}$ , we have that $λ (I) = λ_{1} ([0, c)) = λ_{2} (I)$ . If $c \neq c_{2, 1}$ , without loss of generality, assume that $c < c_{2, 1}$ . [20, Lemma 4.5] implies $c > 0$ . Then, it follows from Lemma 3.1 that $\begin{array}{r} λ (I) = λ_{1} ((c, R)) < λ_{1} ((c_{2, 1}, R)) = λ_{2} (I) = λ_{1} ([0, c_{2, 1})) < λ_{1} ([0, c)) = λ (I), \end{array}$ which is a contradiction. □

For $λ_{i}$ with $i > 2$ , we can use a recurrence argument. For any $n ⩾ 2$ and $i \in N$ with $2 ⩽ i ⩽ n$ , we assume that the following conclusions hold:

For any eigenfunction v corresponding to the i-th eigenvalue $λ_{i} (I)$ , there exists a unique $c_{i, j}$ , $1 ⩽ j ⩽ i - 1$ , such that $Z (v) = \{c_{i, j}, 1 ⩽ j ⩽ i - 1\}$ .

$λ_{i} (I)$ is simple such that $λ_{1} (I) < λ_{2} (I) < \dots < λ_{n + 1} (I)$ .

$λ_{i} (I)$ verifies the strict monotonicity property with respect to the domain I.

If $(v, λ (I))$ is a solution of problem (1.2) such that $Z (v) = \{c_{j}, 1 ⩽ j ⩽ i - 1\}$ , then $λ (I) = λ_{i} (I)$ .

Then, we shall show that the above conclusions are also valid for $i = n + 1$ .
Lemma 3.5.
For any eigenfunction v corresponding to $λ_{n + 1} (I)$ , there exists a unique family $\{c_{n + 1, j} : 1 ⩽ j ⩽ n\}$ such that $Z (v) = \{c_{n + 1, j}, 1 ⩽ j ⩽ n\}$ .
Proof.
Let v be an eigenfunction corresponding to $λ_{n + 1} (I)$ . By (A1), (A2) and (A4), v has at least n zeros. By Lemma 4.1 of [20], we know that v has a finite number of zeros in $[0, R)$ and the first zero $c_{1}$ is positive. Let c be the last zero of v in $[0, R)$ . Then, we consider the $n + 1$ nodal domains $I_{1} = [0, c_{1})$ , $I_{2} = (c_{1}, c_{2})$ , …, $I_{n} = (c_{n - 1}, c_{n})$ and $I_{n + 1} = (c, R)$ . Clearly, the restrictions of u on $[0, c_{i + 1})$ , $1 ⩽ i ⩽ n - 1$ , has i zeros. Then, by (A4), we have that $λ_{n + 1} (I) = λ_{i + 1} ([0, c_{i + 1}))$ .

We claim that $c = c_{n}$ . Otherwise, $c_{n} < c$ . Choose d in $(c_{n}, c)$ and set $J_{1} = [0, d)$ , $J_{2} = (d, R)$ . Then $[0, c_{n})$ is a strict subinterval of $J_{1}$ and $(c, R)$ is a strict subinterval of $J_{2}$ . It follows from (A3) that $\begin{matrix} λ_{n} (J_{1}) < λ_{n} ([0, c_{n})) = λ_{n + 1} (I) \end{matrix}$ and $\begin{matrix} λ_{1} (J_{2}) < λ_{1} ((c, R)) = λ_{n + 1} (I) . \end{matrix}$ Let $ϕ_{n + 1}$ be an eigenfunction corresponding to $λ_{1} (J_{2})$ and $φ_{n + 1}$ be an eigenfunction corresponding to $λ_{n} (J_{1})$ . Denote by $ϕ_{i}$ the restrictions of $φ_{n + 1}$ on $I_{i}$ for $1 ⩽ i ⩽ n$ and ${\tilde{ϕ}}_{i}$ the extensions of $ϕ_{i}$ on I for $1 ⩽ i ⩽ n + 1$ . Letting $W = span \{{\tilde{ϕ}}_{1}, {\tilde{ϕ}}_{2}, \dots, {\tilde{ϕ}}_{n + 1}\}$ and $K_{2} = W \cap S$ , by an argument similar to that of Lemma 3.2 with obvious changes, we can derive a contradiction.

Let w be any eigenfunction corresponding to $λ_{n + 1} (I)$ . Denote by $d_{1}$ , $d_{2}$ , …, $d_{n}$ the zeros of w. If $d_{1} \neq c_{1}$ , then $λ_{n + 1} (I) = λ_{1} ([0, d_{1})) \neq λ_{1} ([0, c_{1})) = λ_{n + 1} (I)$ , an absurd. Hence, one has $d_{1} = c_{1}$ . Similarly, we can check that $d_{i} = c_{i}$ for all $2 ⩽ i ⩽ n$ . □

By an argument similar to that of Lemma 3.3 and in view of our assumptions, it is not difficult to establish the following lemma.
Lemma 3.6.
$λ_{n + 1} (I)$ is simple such that $λ_{n + 1} (I) < λ_{n + 2} (I)$ and verifies the strict monotonicity property with respect to the domain I.

In addition, we can show the following lemma. Lemma 3.7.
If $(v, λ (I))$ is a solution of ( 1.2 ) such that $Z (v) = \{d_{1}, d_{2}, \dots, d_{n}\}$ , then $λ (I) = λ_{n + 1} (I)$ .
Proof.
It is sufficient to prove that $d_{i} = c_{n + 1, i}$ for all $1 ⩽ i ⩽ n$ . If $c_{n + 1, 1} \neq d_{1}$ , without loss of generality, assuming $c_{n + 1, 1} < d_{1}$ , by Lemma 3.1, (A3) and (A4), we obtain that $\begin{array}{r} λ (I) = λ_{1} ([0, d_{1})) < λ_{1} ([0, c_{n + 1, 1})) = λ_{n + 1} (I) = λ_{n} ((c_{n + 1, 1}, R)) < λ_{n} ((d_{1}, R)) = λ (I), \end{array}$ which is a contradiction. □
Proof of Theorem 1.1.
In view of Lemmas 3.1–3.7, it is enough to show that $λ_{i} (p)$ ( $i \in N$ ) is continuous with respect to p. From Lemma 3.1, we have that $λ_{i} (p) = λ_{1} ((0, c_{i, 1}))$ . It follows from Lemma 4.8 of [20] that $λ_{1} ((0, c_{i, 1}))$ is continuous with respect to p. Therefore, $λ_{i} (p)$ is continuous with respect to p. □

4. Proof of Theorem 1.2

We first give a Sturm-type comparison result, which will be used later.

Lemma 4.1.
Let $b_{2} (r) ⩾ b_{1} (r) > 0$ for $r \in (α, β)$ with $0 < α < β < + \infty$ and $b_{i} \in C [α, β]$ , $i = 1, 2$ . Also let $u_{i}$ be solution of the following equations $\begin{matrix} {(r^{N - k} {(- u^{'})}^{k})}^{'} = b_{i} u^{k}, i = 1, 2, \end{matrix}$ respectively, where $u_{i}$ satisfies $u_{1}^{'} (α) = u_{1} (β) = 0$ , $u_{2} (α) \neq 0$ and $u_{2}^{'} (α) u_{2} (α) < 0$ . If $u_{1} \neq 0$ in $(α, β)$ , then either there exists $τ \in (α, β)$ such that $u_{2} (τ) = 0$ , or $b_{2} \equiv b_{1}$ and $u_{2} \equiv μ u_{1}$ on $[α, β]$ for some constant $μ \neq 0$ .
Proof.
If $u_{2} \neq 0$ in $(α, β)$ , without loss of generality, we assume that $u_{1} > 0$ , $u_{2} > 0$ in $(α, β)$ . By the Picone–Hessian identity (see [20]), we obtain that $\begin{array}{c} \int_{α}^{β} {(\frac{u_{1}^{k + 1} r^{N - k} {(- u_{2}^{'})}^{k}}{u_{2}^{k}} - u_{1} r^{N - k} {(- u_{1}^{'})}^{k})}^{'} d r \\ (4.1) & = \int_{α}^{β} (w u_{1}^{k + 1} + r^{N - k} ({(- u_{1}^{'})}^{k + 1} + k {(\frac{- u_{1} u_{2}^{'}}{u_{2}})}^{k + 1} + (k + 1) u_{1}^{k} u_{1}^{'} {(\frac{- u_{2}^{'}}{u_{2}})}^{k})) d r . \end{array}$ The left-hand side of (4.1) equals $\begin{matrix} lim_{r \to β^{-}} \frac{u_{1}^{k + 1} r^{N - k} {(- u_{2}^{'})}^{k}}{u_{2}^{k}} - lim_{r \to α^{+}} \frac{u_{1}^{k + 1} r^{N - k} {(- u_{2}^{'})}^{k}}{u_{2}^{k}} : = L . \end{matrix}$ Let $\begin{matrix} L_{β} = lim_{r \to β^{-}} \frac{u_{1}^{k + 1} r^{N - k} {(- u_{2}^{'})}^{k}}{u_{2}^{k}} and L_{α} = lim_{r \to α^{+}} \frac{u_{1}^{k + 1} r^{N - k} {(- u_{2}^{'})}^{k}}{u_{2}^{k}} . \end{matrix}$ As that of [20, Lemma 2.3], we can show that $L_{β} = 0$ . Since $u_{2} > 0$ in $(α, β)$ , $u_{2} (α) \neq 0$ and $u_{2}^{'} (α) u_{2} (α) < 0$ , one has that $u_{2} (α) > 0$ and $u_{2}^{'} (α) < 0$ . It follows that $L_{α} ⩾ 0$ . Then, reasoning as that of [20, Lemma 2.3], we can obtain that $b_{2} \equiv b_{1}$ and $u_{2} \equiv μ u_{1}$ on $[α, β]$ for some constant $μ \neq 0$ . □

Then, for any α, $β \in (0, + \infty)$ with $α < β$ , we consider the following eigenvalue problem $\begin{matrix} (4.2) & \{\begin{array}{ll} {(r^{N - k} {(- v^{'})}^{k})}^{'} = μ \frac{N}{C_{N}^{k}} r^{N - 1} v^{k}, & r \in (α, β), \\ v^{'} (α) = v (β) = 0 . \end{array} \end{matrix}$ We shall use the following Krein–Rutman theorem to obtain a principal eigenvalue of problem (4.2).
Proposition 4.1 (see [39] or [38]).

Let $(E, ‖ \cdot ‖)$ be a Banach space and K be a closed cone in E with a vertex at 0. Let $T : R_{+} \times K ⟶ K$ be a compact operator such that $T (0, u) = 0$ for all $u \in E$ ; then there exists an unbounded, connected component $C$ of $R_{+} \times K$ of solutions $u = T (μ, u)$ and starting from $(0, 0)$ .

Then, we have that

Lemma 4.2.
Problem ( 4.2 ) possesses a principal eigenvalue $μ_{1}^{+} > 0$ with a positive eigenfunction $ψ_{1}^{+}$ .
Proof.
Let $E = {v \in C^{1} [α, β] : v^{'} (α) = v (β) = 0}$ and $K = {v \in E : v ⩾ 0}$ . Define $\begin{matrix} T (μ, v) = \int_{r}^{β} {(\frac{μ N}{C_{N}^{k} s^{N - k}} \int_{α}^{s} τ^{N - 1} v^{k} (τ) d τ)}^{1 / k} d s : = T (μ, v) . \end{matrix}$ for any $v \in E$ . Then, it is not difficult to check that problem (4.2) is equivalent to $v = T (μ, v)$ and $T : R_{+} \times K ⟶ K$ is compact such that $T (0, u) = 0$ for all $u \in E$ .

Taking $v_{0} \in K ∖ {0}$ , we claim that there exists $M > 0$ such that $T (1, v_{0}) ⩾ v_{0} / M$ . Suppose, by contradiction, that $T (1, v_{0}) - v_{0} / M \notin K$ for all $M > 0$ . Letting $M \to + \infty$ , we have that $T (1, v_{0}) \notin Int (K)$ , which is a contradiction.

For any $ε > 0$ , consider $T_{ε} (μ, v) : = T (μ, v) + T (μ ϵ, v_{0})$ . Applying Proposition 4.1, we obtain a component $C_{ε}$ of the solution to $v = T_{ε} (μ, v)$ . For any $(μ, v) \in C_{ε}$ , we have that $\begin{matrix} v = T (μ, v) + T (μ ϵ, v_{0}) ⩾ T (μ ϵ, v_{0}) = {(μ ϵ)}^{1 / k} T (1, v_{0}) ⩾ \frac{{(μ ϵ)}^{1 / k}}{M} v_{0} . \end{matrix}$ Acting $T (1, \cdot)$ on the both sides of the above inequality, we have that $\begin{matrix} T (1, v) ⩾ \frac{{(μ ϵ)}^{1 / k}}{M} T (1, v_{0}) ⩾ \frac{{(μ ϵ)}^{1 / k}}{M^{2}} v_{0} . \end{matrix}$ Since $v ⩾ T (μ, v)$ , we obtain that $\begin{matrix} v ⩾ {(\frac{μ^{1 / k}}{M})}^{2} ϵ v_{0} . \end{matrix}$ By recurrence, we find that $\begin{matrix} v ⩾ {(\frac{μ^{1 / k}}{M})}^{n} ϵ v_{0} for all n ⩾ 2 . \end{matrix}$ It follows that $μ ⩽ M^{k}$ . So, $C_{ε}$ is unbounded in the direction of K. In particular, there exists $(μ_{ε}, v_{ε}) \in C_{ε}$ such that $‖v_{ε}‖ = 1$ . By the compact of $T (1, \cdot)$ , up to a subsequence, we find $μ_{1}^{+} \in [0, M^{k}]$ and $ψ_{1}^{+} \in K$ with $‖ ψ_{1}^{+} ‖ = 1$ such that $ψ_{1}^{+} = T_{ε} (μ_{1}^{+}, ψ_{1}^{+})$ . Clearly, $μ_{1}^{+} > 0$ and $ψ_{1}^{+}$ is positive in $[α, β)$ . □

For any $v \in X$ and k being odd, define $\begin{matrix} A (λ, v) (r) = - \int_{r}^{R} ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} H_{k} (τ, v (τ)) d τ)}^{1 / k}] d s . \end{matrix}$ From the construct of $A$ (see [28]), we know that $A (λ, \cdot) : X ⟶ X$ is completely continuous. It is not difficult to check that v is a solution of problem (1.1) if and only if v is a fixed point of $A$ . Proof of Theorem 1.2.
(a) Since $H_{0} = 1$ , one has that $H_{k} (r, 0) \equiv 0$ in $(0, R)$ . For any $w \in X ∖ {0}$ , we have that $\begin{array}{rcl} lim_{t \to 0} \frac{A (λ, t w) (r) - A (λ, 0) (r)}{t} & = & - lim_{t \to 0^{+}} \frac{\int_{r}^{R} ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} H_{k} (τ, t w (τ)) d τ)}^{1 / k}] d s}{t} \\ = & - lim_{t \to 0} \int_{r}^{R} \frac{{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} H_{k} (τ, t w (τ)) d τ)}^{1 / k}}{t \sqrt{1 + {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} H_{k} (τ, t w) d τ)}^{2 / k}}} d s \\ = & - lim_{t \to 0} \int_{r}^{R} \frac{{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} H_{k} (τ, t w (τ)) d τ)}^{1 / k}}{t} d s \\ = & - lim_{t \to 0} \int_{r}^{R} {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} \frac{H_{k} (τ, t w (τ)) d τ}{t^{k}})}^{1 / k} d s \\ = & \int_{r}^{R} {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} w^{k} (τ) d τ)}^{1 / k} d s . \end{array}$ Then, by the Implicit Function Theorem, the necessary condition for $(μ, 0)$ being a bifurcation point of problem (1.1) is that $\begin{matrix} w = \int_{r}^{R} {(\frac{μ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} w^{k} (τ) d τ)}^{1 / k} d s : = T (μ, w) . \end{matrix}$ By Theorem 1.1, $μ = λ_{i}$ for some $i \in N$ .

Obviously, $I - T (λ, \cdot)$ is a completely continuous vector field. So, the Leray–Schauder degree $deg (I - T (λ, \cdot), B_{ρ} (0), 0)$ is well defined for arbitrary ρ-ball $B_{ρ} (0) : = {v \in X : ‖ v ‖ < ρ}$ and $λ \neq λ_{i}$ for any $i \in N$ . For any $v \in X$ , define $\begin{matrix} \int_{r}^{R} φ_{p}^{- 1} (\frac{λ N}{C_{N}^{k} s^{N + 1 - p}} \int_{0}^{s} τ^{N - 1} φ_{p} (v) d τ) d s : = T_{p} (λ, v) . \end{matrix}$ By the Arzelè-Ascoli Theorem, it is not difficult to check that $T_{p} (λ, \cdot) : X ⟶ X$ is completely continuous. Clearly, the equation $v = T_{p} (λ, v)$ is equivalent to problem (1.2). Then, by an argument similar to that of [23, Lemma 3.2] and in view of Theorem 1.1, we can obtain the following index formula $\begin{matrix} deg (I - T_{p} (λ, \cdot), B_{ρ} (0), 0) = \{\begin{array}{ll} 1 & if λ \in (0, λ_{1} (p)), \\ {(- 1)}^{i} & if λ \in (λ_{i} (p), λ_{i + 1} (p)), i \in N . \end{array} \end{matrix}$ In particular, taking $p = k + 1$ , we have that $\begin{matrix} deg (I - T (λ, \cdot), B_{ρ} (0), 0) = \{\begin{array}{ll} 1 & if λ \in (0, λ_{1}), \\ {(- 1)}^{i} & if λ \in (λ_{i}, λ_{i + 1}), i \in N . \end{array} \end{matrix}$ Letting $ξ (r, s) = - H_{k} (r, s) - s^{k}$ for any $(r, s) \in [0, R] \times I_{R}$ , we have that $\begin{matrix} lim_{s \to 0} \frac{ξ (x, s)}{s^{k}} = 0 \end{matrix}$ uniformly in $r \in (0, R)$ . Then, problem (1.1) is equivalent to $\begin{matrix} \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = - λ \frac{N}{C_{N}^{k}} r^{N - 1} v^{k} - λ \frac{N}{C_{N}^{k}} r^{N - 1} ξ (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ For any $t \in [0, 1]$ , we consider the following problem $\begin{matrix} (4.3) & \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = - λ \frac{N}{C_{N}^{k}} r^{N - 1} v^{k} - λ \frac{N}{C_{N}^{k}} r^{N - 1} t ξ (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ Obviously, problem (4.3) is equivalent to $\begin{matrix} v = - \int_{r}^{R} ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (- t ξ (τ, v (τ)) - v^{k} (τ)) d τ)}^{1 / k}] d s : = A_{λ} (t, v) . \end{matrix}$ Clearly, $A_{λ} : [0, 1] \times X ⟶ X$ is completely continuous.

Let $\begin{matrix} \tilde{ξ} (r, v) = max_{| s | ⩽ v} | ξ (r, s) | for any r \in (0, R) . \end{matrix}$ Then, we have that $\tilde{ξ}$ is nondecreasing with respect to v and $\begin{matrix} (4.4) & lim_{v \to 0^{+}} \frac{\tilde{ξ} (r, v)}{v^{k}} = 0 . \end{matrix}$ Further, it follows from (4.4) that $\begin{matrix} (4.5) & | \frac{ξ (r, v)}{‖ v ‖} | ⩽ \frac{\tilde{ξ} (r, v)}{‖ v ‖} ⩽ \frac{\tilde{ξ} (r, ‖ v ‖_{\infty})}{‖ v ‖} ⩽ R \frac{\tilde{ξ} (r, R ‖ v ‖)}{R ‖ v ‖} \to 0 as ‖ v ‖ \to 0 \end{matrix}$ uniformly in $r \in (0, R)$ .

We claim that the Leray–Schauder degree $deg (I - A_{λ} (t, \cdot), B_{ρ} (0), 0)$ is well defined for $λ \neq λ_{i}$ and ρ small enough. Suppose, on the contrary, that there exists a sequence $\{v_{n}\}$ such that $v_{n} = A_{λ} (t, v_{n})$ and $‖ v_{n} ‖ \to 0$ as $n \to + \infty$ . Letting $w_{n} = v_{n} / ‖ v_{n} ‖$ , we have that $w_{n}$ satisfies $\begin{array}{rcl} w_{n} & = & - \frac{\int_{r}^{R} ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (- t ξ (τ, v_{n} (τ)) - v_{n}^{k} (τ)) d τ)}^{1 / k}] d s}{‖ v_{n} ‖} \\ = & - \int_{r}^{R} \frac{{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (- t ξ (τ, v_{n} (τ)) - v_{n}^{k} (τ)) d τ)}^{1 / k} d s}{‖ v_{n} ‖ \sqrt{1 + {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (- t ξ (τ, v_{n} (τ)) - v_{n}^{k} (τ)) d τ)}^{2 / k}}} \\ = & \int_{r}^{R} \frac{{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (\frac{t ξ (τ, v_{n} (τ))}{‖ v_{n} ‖^{k}} + w_{n}^{k} (τ)) d τ)}^{1 / k} d s}{\sqrt{1 + {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (- t ξ (τ, v_{n} (τ)) - v_{n}^{k} (τ)) d τ)}^{2 / k}}} . \end{array}$ From the compactness of $A_{λ}$ , up to a subsequence, we have that $w_{n} \to w$ in X as $n \to + \infty$ . Letting $n \to + \infty$ in the above equality, using (4.5) and the continuity of $A_{λ}$ , we obtain that $\begin{matrix} w = \int_{r}^{R} {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} w^{k} (τ) d τ)}^{1 / k} d s . \end{matrix}$ It follows that $\begin{matrix} \{\begin{array}{ll} {(r^{N - k} {(- w^{'})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} w^{k}, & r \in (0, R), \\ w^{'} (0) = w (R) = 0 . \end{array} \end{matrix}$ Clearly, one has that $‖ w ‖ = 1$ . So, λ is an eigenvalue of problem (1.3), which is absurd. By the invariance of the degree under homotopies, we find that $\begin{array}{r} deg (I - A (λ, \cdot), B_{ρ} (0), 0) = deg (I - A_{λ} (1, \cdot), B_{ρ} (0), 0) = deg (I - A_{λ} (0, \cdot), B_{ρ} (0), 0) . \end{array}$ Furthermore, for any $t \in [0, 1]$ , we consider $\begin{matrix} A_{λ}^{} (0, t, v) : = \int_{r}^{R} \frac{{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (v^{k} (τ)) d τ)}^{1 / k}}{\sqrt{1 + t {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (v^{k} (τ)) d τ)}^{2 / k}}} d s . \end{matrix}$ Then, clearly, one has that $A_{λ}^{} (0, 1, v) = A_{λ} (0, v)$ and $A_{λ}^{} (0, 0, v) = T (λ, v)$ . Obviously, $A_{λ}^{} (0, t, \cdot) : [0, 1] \times X ⟶ X$ is completely continuous. Reasoning as the above, we can show that the Leray–Schauder degree $deg (I - A_{λ}^{} (0, t, \cdot), B_{ρ} (0), 0)$ is well defined for $λ \neq λ_{i}$ and ρ small enough. So, using the invariance of the degree under compact homotopies, we obtain that $\begin{array}{rcl} (4.6) & deg (I - A (λ, \cdot), B_{ρ} (0), 0) & = & deg (I - A_{λ} (0, \cdot), B_{ρ} (0), 0) \\ = & deg (I - A_{λ}^{} (0, 1, \cdot), B_{ρ} (0), 0) \\ = & deg (I - A_{λ}^{} (0, 0, \cdot), B_{ρ} (0), 0) \\ = & deg (I - T (λ, \cdot), B_{ρ} (0), 0) \\ = & \{\begin{array}{ll} 1 & if λ \in (0, λ_{1}), \\ {(- 1)}^{i} & if λ \in (λ_{i}, λ_{i + 1}), i \in N . \end{array} \end{array}$ By Lemma 2.2, we know that if $(λ, v)$ is a nontrivial solution of (1.1) with any fixed $λ < + \infty$ , $‖ v ‖ < 1$ . So, applying Proposition 2.1 of [24] with $O = R \times B_{1} (0)$ for each $i \in N$ , there exists a maximum continuum $C_{i}^{+}$ of nontrivial solutions of problem (1.1) bifurcating from $(λ_{i}, 0)$ which satisfies at least one of the following three properties:
$C_{i}^{+}$ is unbounded in $O$ ,

meets $R \times \partial B_{1} (0)$ ,

$C_{i}^{+} \cap (R ∖ \{λ_{i}\} \times {0}) \neq \emptyset$ .

By virtue of Lemma 2.2, the case (ii) does not occur. Similarly as Lemma 1.24 of [40], we can show that if $(λ, v) \in C_{i}^{+}$ and is near $(λ_{i}, 0)$ , then $v = α φ_{i} + w$ with $w = o (| α |)$ for α near $0^{+}$ . Thus, there exists an open neighborhood $N$ of $(λ_{i}, 0)$ such that $\begin{matrix} (C_{i}^{+} ∖ {(λ_{i}, 0)}) \cap N \subset R \times S_{i}^{+} . \end{matrix}$ Furthermore, we claim* that $C_{i}^{+} ∖ \{(λ_{i}, 0)\}$ cannot touch $R \times \partial S_{i}^{+}$ . Otherwise, there exists $(μ, v) \in (C_{i}^{+} ∖ \{(λ_{i}, 0)\}) \cap (R \times \partial S_{i}^{+})$ such that $(μ, v)$ is the limit of $(λ_{n}, v_{n}) \in R \times S_{i}^{+}$ in $R \times X$ . Hence, v has a double zero in $[0, R]$ . In view of $H_{0} = 1$ and our signum condition, we can see that there exists a constant $M > 0$ such that $| H_{k} (r, s) | ⩽ M | s |^{k}$ for any $r \in [0, R]$ and $s \in [- R, R]$ . It follows from Lemma 2.1 that $v \equiv 0$ . Thus, $μ = λ_{i}$ , which is a contradiction. Therefore, we have that $C_{i}^{+} \subseteq (Φ_{i}^{+} \cup \{(λ_{i}, 0)\})$ . Clearly, if (iii) happens, $C_{i}^{+} ∖ \{(λ_{i}, 0)\}$ will touch $R \times \partial S_{i}^{+}$ , which is absurd. So, (iii) is also impossible. So, $C_{i}^{+}$ is unbounded in $O$ . Since $v \equiv 0$ is the only solution of problem (1.1) for $λ = 0$ and 0 is not an eigenvalue of problem (1.2), we have that $C_{i}^{+} \cap ({0} \times X) = \emptyset$ . It follows that $(λ_{i}, + \infty) \subseteq {pr}_{R} (C_{i}^{+})$ and $‖ v ‖ < 1$ for any $(λ, v) \in C_{i}^{+} ∖ \{(λ_{i}, 0)\}$ with $λ < + \infty$ .

Finally, we show the asymptotic behavior of $v_{λ}$ with respect to λ. We suppose, by contradiction, that there exist a constant $δ > 0$ and $(λ_{n}, v_{n}) \in C_{i}^{+} ∖ \{(λ_{i}, 0)\}$ with $λ_{n} \to + \infty$ as $n \to + \infty$ such that $‖ v_{n} ‖^{2} ⩽ 1 - δ^{2}$ for any $n \in N$ .

Let $\begin{matrix} τ (1, n) < τ (2, n) < \dots < τ (i - 1, n) \end{matrix}$ denote the zeros of $v_{n}$ in $(0, R)$ . For convenience, let $τ (0, n) = 0$ and $τ (i, n) = R$ . Up to a subsequence, we have that $\begin{matrix} lim_{n \to + \infty} τ (l, n) = τ (l, \infty), l \in {0, 1, \dots, i} \end{matrix}$ for some $τ (l, \infty) \in [0, R]$ . Note that $τ (l + 1, \infty) ⩾ τ (l, \infty)$ and $\begin{matrix} R = \sum_{l = 0}^{i - 1} (τ (l + 1, n) - τ (l, n)) \to \sum_{l = 0}^{i - 1} (τ (l + 1, \infty) - τ (l, \infty)) = R \end{matrix}$ as $n \to + \infty$ . It follows that there exists $l_{0} \in {0, 1, \dots, i - 1}$ such that $\begin{matrix} (4.7) & τ (l_{0}, \infty) < τ (l_{0} + 1, \infty) . \end{matrix}$ Without loss of generality, assume that $v_{n} > 0$ in $(τ (l_{0}, n), τ (l_{0} + 1, n))$ . Clearly, there exists $r_{0}^{n} \in [τ (l_{0}, n), τ (l_{0} + 1, n))$ such that $v_{n}^{'} (r_{0}^{n}) = 0$ .

Integrating the first equation of problem (1.1) from $r_{0}^{n}$ to any $r \in (r_{0}^{n}, τ (l_{0} + 1, n))$ , we have that $\begin{matrix} v_{n}^{'} (r) = ϕ^{- 1} [{(\frac{λ_{n} N}{r^{N - k} C_{N}^{k}} \int_{r_{0}^{n}}^{r} τ^{N - 1} H_{k} (τ, v_{n}) d τ)}^{1 / k}] . \end{matrix}$ Clearly, $ϕ^{- 1}$ is an odd and increasing diffeomorphism. The signum condition on $H_{k}$ implies $v_{n}^{'} (r) < 0$ in $(r_{0}^{n}, τ (l_{0} + 1, n))$ . Analogously, we also have that $v_{n}^{'} (r) ⩾ 0$ on $[τ (l_{0}, n), r_{0}^{n}]$ .

For any fixed $θ \in (0, τ (l_{0} + 1, \infty) - τ (l_{0}, \infty))$ , (4.7) implies that there exists $N_{1} \in N$ such that $\begin{matrix} τ (l_{0} + 1, n) - τ (l_{0}, n) > θ \end{matrix}$ for any $n > N_{1}$ . Letting $I_{n, 0}^{1} = [τ (l_{0}, n), r_{0}^{n}]$ and $I_{n, 0}^{2} = (r_{0}^{n}, τ (l_{0} + 1, n))$ , we must have that either (i) $| I_{n, 0}^{1} | > θ / 2$ or (ii) $| I_{n, 0}^{2} | > θ / 2$ .

We first consider the latter case. Taking a subsequence if necessary, we have ${lim}_{n \to + \infty} r_{0}^{n} = r_{0}$ for some $r_{0} \in [0, τ (l_{0} + 1, \infty)]$ . It follows that $\begin{matrix} τ (l_{0} + 1, \infty) - r_{0} ⩾ \frac{θ}{2} . \end{matrix}$ So, we can choose α, $β \in (r_{0}, τ (l_{0} + 1, \infty))$ with $α < β$ . Hence, there exists $N_{2} \in N$ with $N_{2} ⩾ N_{1}$ such that $\begin{matrix} r_{0}^{n} < α < β < τ (l_{0} + 1, n) \end{matrix}$ for any $n > N_{2}$ .

We expand the left term of the first equation of problem (1.1) as the following $\begin{array}{r} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = (N - k) r^{N - k - 1} ϕ^{k} (v^{'}) + r^{N - k} k ϕ^{k - 1} (v^{'}) \frac{v^{″}}{{(1 - v^{' 2})}^{3 / 2}}, \end{array}$ where we used the fact of $\begin{array}{r} ϕ^{'} (s) = \frac{1}{{(1 - s^{2})}^{3 / 2}} . \end{array}$ Further, by some elementary calculations, we have that $\begin{array}{r} - r^{1 - k} {(v^{' k})}^{'} = - {(1 - v^{' 2})}^{\frac{k + 2}{2}} λ \frac{N}{C_{N}^{k}} H_{k} (r, v) + (N - k) r^{- k} v^{' k} (1 - v^{' 2}) . \end{array}$ It follows that $\begin{array}{rcl} - {(r^{N - k} v^{' k})}^{'} & = & - r^{N - 1} {(1 - v^{' 2})}^{\frac{k + 2}{2}} λ \frac{N}{C_{N}^{k}} H_{k} (r, v) - (N - k) r^{N - k - 1} v^{' k + 2} \\ (4.8) & = & r^{N - 1} ({(1 - v^{' 2})}^{\frac{k + 2}{2}} λ \frac{N}{C_{N}^{k}} \frac{- H_{k} (r, v)}{v^{k}} - \frac{(N - k) r^{- k} v^{' k + 2}}{v^{k}}) v^{k} . \end{array}$ Since $v_{n}$ is positive and decreasing in α, β, we have that $\begin{array}{r} - \frac{(N - k) r^{- k} v_{n}^{' k + 2}}{v_{n}^{k}} ⩾ 0 in (α, β) . \end{array}$ The signum condition on $H_{k}$ combining $H_{0} = 1$ implies that there exists a positive constant $ρ > 0$ such that $\begin{matrix} (4.9) & \frac{- H_{k} (r, v_{n})}{v_{n}^{k}} ⩾ ρ \end{matrix}$ for any $r \in (α, β)$ and $n \in N$ . Thus, we obtain that $\begin{matrix} {(1 - v_{n}^{' 2})}^{\frac{k + 2}{2}} λ_{n} \frac{N}{C_{N}^{k}} \frac{- H_{k} (r, v_{n})}{v_{n}^{k}} - \frac{(N - k) r^{- k} v_{n}^{' k + 2}}{v_{n}^{k}} ⩾ λ_{n} ρ δ^{k + 2} \frac{N}{C_{N}^{k}} on [α, β] . \end{matrix}$ Taking n large enough such that $λ_{n} ρ σ^{k + 2} > μ_{1}^{+}$ and applying Lemma 4.1 on $v_{n}$ and $ψ_{1}^{+}$ , we conclude that $v_{n}$ changes its sign, which contradicts the fact of $v_{n} > 0$ on $[α, β]$ .

Now, we consider the former case. Clearly, we have that $\begin{matrix} r_{0} - τ (l_{0}, \infty) ⩾ \frac{θ}{2} . \end{matrix}$ Then, choose α, β, $σ \in (τ (l_{0}, \infty), r_{0})$ with $σ < α < β$ . Clearly, we have that $\begin{matrix} r_{0}^{n} > β > α > σ > τ (l_{0}, n) \end{matrix}$ for n large enough.

For any $r \in (τ (l_{0}, n), r_{0}^{n})$ , from the first equation of problem (1.1), we get that $\begin{matrix} - ϕ (v_{n}^{'}) = {(\frac{λ_{n} N}{C_{N}^{k} r^{N - k}} \int_{r}^{r_{0}^{n}} τ^{N - 1} H_{k} (τ, v_{n} (τ)) d τ)}^{1 / k} . \end{matrix}$ It follows that $\begin{matrix} - r^{\frac{N - k}{k}} ϕ (v_{n}^{'}) = {(\frac{λ_{n} N}{C_{N}^{k}} \int_{r}^{r_{0}^{n}} τ^{N - 1} H_{k} (τ, v_{n} (τ)) d τ)}^{1 / k} . \end{matrix}$ Furthermore, for any $r \in [α, r_{0}^{n}]$ , noting k being odd, we have that $\begin{array}{rcl} - {(r^{\frac{N - k}{k}} ϕ (v_{n}^{'}))}^{'} & = & - \frac{1}{k} {(\frac{λ_{n} N}{C_{N}^{k}} \int_{r}^{r_{0}^{n}} τ^{N - 1} H_{k} (τ, v_{n} (τ)) d τ)}^{\frac{1}{k} - 1} \frac{λ_{n} N}{C_{N}^{k}} r^{N - 1} H_{k} (r, v_{n}) \\ = & - \frac{λ_{n} N}{k C_{N}^{k}} r^{N - 1} \frac{H_{k} (r, v_{n})}{v_{n}^{k}} v_{n} {(\frac{λ_{n} N}{C_{N}^{k}} \int_{r}^{r_{0}^{n}} τ^{N - 1} \frac{- H_{k} (τ, v_{n} (τ))}{v_{n}^{k} (r)} d τ)}^{\frac{1}{k} - 1} \\ = & {(\frac{λ_{n} N}{k C_{N}^{k}})}^{\frac{1}{k}} r^{N - 1} \frac{- H_{k} (r, v_{n})}{v_{n}^{k}} v_{n} \frac{1}{{(\int_{r}^{r_{0}^{n}} τ^{N - 1} \frac{- H_{k} (τ, v_{n} (τ))}{v_{n}^{k} (r)} d τ)}^{\frac{k - 1}{k}}} . \end{array}$ As that of (4.9), we see that there exist two constants ρ, $ϱ > 0$ such that $\begin{matrix} ρ ⩾ \frac{- H_{k} (r, v_{n})}{v_{n}^{k}} ⩾ ϱ \end{matrix}$ for any $r \in [α, r_{0}^{n}]$ and $n \in N$ . For some calculations, in view of ${(ϕ^{- 1})}^{'} (s) = {(1 + s^{2})}^{- 3 / 2}$ , we find that $\begin{array}{rcl} v_{n}^{″} & = & \frac{{(m_{n} (r))}^{1 - k}}{{(1 + {(m_{n} (r))}^{2})}^{3 / 2}} \frac{λ_{n} N}{k C_{N}^{k}} \frac{H_{k} (r, v_{n}) + (N - k) \int_{r}^{r_{0}^{n}} τ^{N - 1} H_{k} (τ, v_{n} (τ)) d τ}{r^{N - k + 1}} \\ ⩽ & 0 for r \in (τ (l_{0}, n), r_{0}^{n}), \end{array}$ where $\begin{matrix} m_{n} (r) = {(\frac{λ_{n} N}{C_{N}^{k} r^{N - k}} \int_{r}^{r_{0}^{n}} τ^{N - 1} H_{k} (τ, v_{n} (τ)) d τ)}^{1 / k} . \end{matrix}$ Hence, $v_{n}$ is concave on $[τ (l_{0}, n), r_{0}^{n}]$ . It follows that $\begin{matrix} v_{n} (r) ⩾ v_{n} (τ) \frac{τ - τ (l_{0}, n)}{r - τ (l_{0}, n)} \end{matrix}$ for any $τ \in [r, r_{0}^{n}]$ . Clearly, for $r \in [α, r_{0}^{n}]$ and n large enough, one has that $\begin{matrix} \frac{τ - τ (l_{0}, n)}{r - τ (l_{0}, n)} ⩾ \frac{α - σ}{2 (r_{0} - τ (l_{0}, \infty))} . \end{matrix}$ Then, some simple estimates give that $\begin{array}{rcl} \int_{r}^{r_{0}^{n}} τ^{N - 1} \frac{- H_{k} (τ, v_{n} (τ))}{v_{n}^{k} (r)} d τ & ⩽ & ρ \int_{r}^{r_{0}^{n}} τ^{N - 1} \frac{v_{n}^{k} (τ)}{v_{n}^{k} (r)} d τ \\ ⩽ & \frac{2^{k} ρ}{N} {(\frac{r_{0} - τ (l_{0}, \infty)}{α - σ})}^{k} ({(r_{0}^{n})}^{N} - r^{N}) . \end{array}$ For n large sufficiently, we have that $\begin{array}{r} \int_{r}^{r_{0}^{n}} τ^{N - 1} \frac{- H_{k} (τ, v_{n} (τ))}{v_{n}^{k} (r)} d τ ⩽ \frac{2^{k} ρ}{N} {(\frac{r_{0} - τ (l_{0}, \infty)}{α - θ})}^{k} ({(2 r_{0})}^{N} - α^{N}) : = M . \end{array}$ So, we obtain that $\begin{array}{r} (4.10) & - {(r^{\frac{N - k}{k}} ϕ (v_{n}^{'}))}^{'} ⩾ {(\frac{λ_{n} N}{k C_{N}^{k}})}^{\frac{1}{k}} \frac{ϱ}{M^{\frac{k - 1}{k}}} r^{N - 1} v_{n} . \end{array}$ Let $μ_{1, n}$ denote the first eigenvalue of the following problem $\begin{matrix} \{\begin{array}{ll} - {(r^{\frac{N - k}{k}} v^{'})}^{'} = λ r^{N - 1} v, & r \in (α, r_{0}^{n}), \\ v (α) = v^{'} (r_{0}^{n}) = 0 . \end{array} \end{matrix}$ Reasoning as that of Lemma 3.1, we can check that $μ_{1, n}$ is strictly decreasing with respect to the domain $I_{n} = (α, r_{0}^{n})$ . This implies that $μ_{1, n} ⩽ μ_{1}$ , where $μ_{1}$ is the first eigenvalue of the following problem $\begin{matrix} \{\begin{array}{ll} - {(r^{\frac{N - k}{k}} v^{'})}^{'} = λ r^{N - 1} v, & r \in (α, β), \\ v (α) = v^{'} (β) = 0 . \end{array} \end{matrix}$ Let $ψ_{1, n}$ be a positive eigenfunction corresponding to $μ_{1, n}$ . Multiplying equation (4.10) by $ψ_{1, n}$ , we obtain after integrations by parts that $\begin{array}{rcl} \frac{μ_{1}}{δ} \int_{α}^{r_{0}^{n}} r^{N - 1} ψ_{1, n} v_{n} d r & ⩾ & \frac{μ_{1, n}}{δ} \int_{α}^{r_{0}^{n}} r^{N - 1} ψ_{1, n} v_{n} d r \\ = & \frac{1}{δ} \int_{α}^{r_{0}^{n}} r^{\frac{N - k}{k}} v_{n}^{'} ψ_{1, n}^{'} d r \\ ⩾ & \int_{α}^{r_{0}^{n}} r^{\frac{N - k}{k}} ϕ (v_{n}^{'}) ψ_{1, n}^{'} d r \\ = & {(\frac{λ_{n} N}{k C_{N}^{k}})}^{\frac{1}{k}} \frac{ϱ}{M^{\frac{k - 1}{k}}} \int_{α}^{r_{0}^{n}} r^{N - 1} ψ_{1, n} v_{n} d r . \end{array}$ Therefore, we obtain that $\begin{array}{r} \frac{μ_{1}}{δ} ⩾ {(\frac{λ_{n} N}{k C_{N}^{k}})}^{\frac{1}{k}} \frac{ϱ}{M^{\frac{k - 1}{k}}}, \end{array}$ which is impossible because $λ_{n} \to + \infty$ as $n \to + \infty$ .

To obtain another component, we consider the following auxiliary problem $\begin{matrix} \{\begin{array}{ll} {(r^{N - k} {(\frac{w^{'}}{\sqrt{1 - w^{' 2}}})}^{k})}^{'} = - λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k} (r, - w) & in (0, R), \\ | w^{'} | < 1 & in (0, R), \\ w^{'} (0) = w (R) = 0 . \end{array} \end{matrix}$ It is not difficult to verify that $- H_{k} (r, - w) : = {\tilde{H}}_{k} (r, w)$ satisfies the same assumptions as $H_{k}$ . The previous argument shows that an unbounded continuum ${\tilde{C}}_{i}^{+}$ bifurcates from $(λ_{i}, 0)$ , such that ${\tilde{C}}_{i}^{+} \subseteq (Φ_{i}^{+} \cup \{(λ_{i}, 0)\})$ , $(λ_{i}, + \infty) \subseteq {pr}_{R} ({\tilde{C}}_{i}^{+})$ , $‖ w ‖ < 1$ for any $(λ, w) \in {\tilde{C}}_{i}^{+} ∖ \{(λ_{i}, 0)\}$ with $λ < + \infty$ and meets $(+ \infty, 1)$ . Letting $C_{i}^{-} = - {\tilde{C}}_{i}^{+}$ , clearly, $C_{i}^{-}$ is a nontrivial solution component of problem (1.1) emanating from $(λ_{i}, 0)$ , which satisfies the desired conclusions.

(b) For any $n \in N$ , define $\begin{array}{r} H_{k}^{(n)} (r, s) = \{\begin{array}{ll} - n^{k} s^{k}, & s \in [- \frac{1}{n}, \frac{1}{n}], \\ (H_{k} (r, \frac{2}{n}) + 1) n s - H_{k} (r, \frac{2}{n}) - 2, & s \in (\frac{1}{n}, \frac{2}{n}), \\ (1 - H_{k} (r, - \frac{2}{n})) n s + 2 - H_{k} (r, \frac{2}{n}), & s \in (- \frac{2}{n}, - \frac{1}{n}), \\ H_{k} (r, s), & s \in [- R, - \frac{2}{n}] \cup [\frac{2}{n}, R] . \end{array} \end{array}$ Clearly, we have that ${lim}_{n \to + \infty} H_{k}^{(n)} (r, s) = H_{k} (r, s)$ and $H_{0}^{(n)} = n^{k}$ . Consider the following approximation problem $\begin{matrix} (4.11) & \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k}^{(n)} (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ By (a), for each $i \in N$ and $ν \in {+, -}$ , there exists a sequence unbounded continua $C_{i, n}^{ν}$ of the set of nontrivial solutions of problem (4.11) emanating from $(λ_{i} / n^{k}, 0)$ and joining to the point $(+ \infty, 1)$ such that $\begin{matrix} C_{i, n}^{ν} \subseteq (Φ_{i}^{ν} \cup {(λ_{i} / n^{k}, 0)}) . \end{matrix}$ Taking $z^{} = (0, 0)$ , clearly, one has that $z^{} \in {lim inf}_{n \to + \infty} C_{i, n}^{ν}$ . The compactness of $A$ implies that $(⋃_{n = 1}^{+ \infty} C_{i, n}^{ν}) \cap B_{R}$ is pre-compact, where $B_{R} = {z \in R \times X : ‖ z ‖ < R}$ for any $R > 0$ . By Lemma 2.3, $C_{i}^{ν} = {lim sup}_{n \to + \infty} C_{i, n}^{ν}$ is unbounded and connected such that $z^{} \in C_{i}^{ν}$ and $(+ \infty, 1) \in C_{i}^{ν}$ .

From the definition of superior limit (see [46]) and the continuity of $A$ , it is not difficult to see that v is a solution of problem (1.1) for any $(λ, v) \in C_{i}^{ν}$ . Obviously, $v \in \overline{S_{i}^{ν}}$ for any $(λ, v) \in C_{i}^{ν}$ . By the definition of inferior limit (see [46]) and the fact of $C_{i, n}^{ν} \cap (R_{+} \times {0}) = λ_{i} / n^{k}$ , we can derive that $C_{i}^{ν} \cap ((0, + \infty) \times {0}) = \emptyset$ . Therefore, by Lemma 2.1 and the definition of superior limit, we have that $v \in S_{i}^{ν}$ for any $(λ, v) \in C_{i}^{ν} ∖ {(0, 0)}$ . It follows from Lemma 2.2 that $‖ v ‖ < 1$ for any $(λ, v) \in C_{i}^{ν} ∖ {(0, 0)}$ with $λ < + \infty$ .

(c) For any $n \in N$ , define $\begin{array}{r} H_{k, n} (r, s) = \{\begin{array}{ll} - \frac{1}{n^{k}} s^{k}, & s \in [- \frac{1}{n}, \frac{1}{n}], \\ (H_{k} (r, \frac{2}{n}) + \frac{1}{n^{2 k}}) n s - \frac{2}{n^{2 k}} - H_{k} (r, \frac{2}{n}), & s \in (\frac{1}{n}, \frac{2}{n}), \\ (\frac{1}{n^{2 k}} - H_{k} (r, - \frac{2}{n})) n s + \frac{2}{n^{2 k}} - H_{k} (r, - \frac{2}{n}), & s \in (- \frac{2}{n}, - \frac{1}{n}), \\ H_{k} (r, s), & s \in [- R, - \frac{2}{n}] \cup [\frac{2}{n}, R] . \end{array} \end{array}$ Then, we consider the following approximation problem $\begin{matrix} (4.12) & \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k, n} (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ Obviously, we have that ${lim}_{n \to + \infty} H_{k, n} (r, s) = H_{k} (r, s)$ and $\begin{matrix} lim_{s \to 0^{+}} \frac{H_{k, n} (r, s)}{s^{k}} = \frac{1}{n^{k}} uniformly in r \in (0, R) . \end{matrix}$ It follows from the conclusion of (a) that there exists a sequence unbounded continua $C_{i, n}^{ν}$ of nontrivial solutions set of problem (4.12) in $R_{+} \times S_{i}^{ν}$ emanating from $(λ_{i} n^{k}, 0)$ for any $n \in N$ and joining to $(+ \infty, 1) : = z_{}$ .

Taking $z^{} = (+ \infty, 0)$ , clearly, we have that $z^{} \in C_{i}^{ν}$ . We claim that $C_{i}^{ν} ∖ {\infty} \neq \emptyset$ . It suffices to show that the projection of $C_{i}^{ν}$ on $R$ is nonempty. From (a), we have known that $C_{i, n}^{ν}$ has unbounded projection on $R$ for any fixed $n \in N$ . By Proposition 2.3 of [17], for each fixed $ϵ > 0$ there exists an m such that for every $n > m$ , $C_{i, n}^{ν} \subset V_{ϵ} (C_{i}^{ν})$ . It follows that $\begin{matrix} (λ_{i} n^{k}, + \infty) \subseteq {pr}_{R} (C_{i, n}^{ν}) \subseteq {pr}_{R} (V_{ϵ} (C_{i}^{ν})) . \end{matrix}$ It implies that $(n^{k} λ_{i} + ϵ, + \infty) \subseteq {pr}_{R} (C_{i}^{ν})$ . So the projection of $C_{i}^{ν}$ is nonempty on $R$ . The compactness of $A$ implies that $(⋃_{n = 1}^{+ \infty} C_{i, n}^{ν}) \cap B_{R}$ is pre-compact.

Applying Lemma 2.4, we obtain that $C_{i}^{ν} ∖ {\infty}$ is connected. Applying the argument in the proof of (b), we can show that $C_{i}^{ν} \cap ([0, + \infty) \times {0}) = \emptyset$ , $v \in S_{i}^{ν}$ is a solution of problem (1.1) and $‖ v ‖ < 1$ for any $(λ, v) \in C_{i}^{ν}$ with $λ < + \infty$ . □

5. Proofs of Theorems 1.3–1.4

If k is even, we always assume that $\int_{0}^{r} τ^{N - 1} H_{k} (τ, s) d τ ⩾ 0$ for any $r \in [0, R]$ and $s \in I_{R} ∖ {0}$ . For any $v \in X$ and k being even, we define $\begin{matrix} A^{\mp} (λ, v) (r) = \mp \int_{r}^{R} ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} H_{k} (τ, v (τ)) d τ)}^{1 / k}] d s . \end{matrix}$ According to the construct of $A^{\mp}$ (see [28]), we see that $A^{\mp} (λ, \cdot) : X^{\mp} ⟶ X^{\mp}$ is completely continuous. It is easy to see that v is a solution of problem (1.1) if and only if v is a fixed point of $A^{\mp}$ .

Proof of Theorem 1.3.

(a) As that of Theorem 1.2(a), if $(μ, 0)$ is a bifurcation point of one-sign solutions of problem (1.1), one have that $μ = λ_{1}$ . Define $\begin{matrix} T^{\mp} (λ, w) : = \mp \int_{r}^{R} {(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} w^{k} (τ) d τ)}^{1 / k} d s . \end{matrix}$ Reasoning as that of [20, Lemma 5.2] with obvious changes, we obtain that $\begin{matrix} deg (I - T^{\mp} (λ, \cdot), B_{ρ}^{\mp} (0), 0) = \{\begin{array}{ll} 1 & if λ \in (0, λ_{1}), \\ - 1 & if λ \in (λ_{1}, λ_{1} + δ), \end{array} \end{matrix}$ where $B_{ρ}^{\mp} (0)$ is arbitrary ρ-ball of $X^{\mp}$ . Let $ξ (r, s) = H_{k} (r, s) - s^{k}$ for any $(r, s) \in [0, R] \times I_{R}$ . Then, we define $\begin{matrix} A_{λ}^{\mp} (t, v) : = \mp \int_{r}^{R} ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} s^{N - k}} \int_{0}^{s} τ^{N - 1} (t ξ (τ, v (τ)) + v^{k}) d τ)}^{1 / k}] d s . \end{matrix}$ Modifying the proof of Theorem 1.2(a) with small changes, we obtain that $\begin{array}{rcl} deg (I - A^{\mp} (λ, \cdot), B_{ρ}^{\mp} (0), 0) & = & deg (I - A_{λ}^{\mp} (1, \cdot), B_{ρ}^{\mp} (0), 0) \\ = & deg (I - A_{λ}^{\mp} (0, \cdot), B_{ρ}^{\mp} (0), 0) \\ = & deg (I - T^{\mp} (λ, \cdot), B_{ρ}^{\mp} (0), 0) \\ = & \{\begin{array}{ll} 1 & if λ \in (0, λ_{1}), \\ - 1 & if λ \in (λ_{1}, λ_{1} + δ) . \end{array} \end{array}$ Applying Proposition 2.1 of [24] with $O = R \times B_{1}^{\mp} (0)$ , there exists a continuum $C^{\mp}$ of nontrivial solution of problem (1.1) bifurcating from $(λ_{1}, 0)$ which satisfies one of the following three properties:

$C^{\mp}$ is unbounded in $O$ ,

meets $R \times \partial B_{1}^{\mp} (0)$ ,

$C^{\mp} \cap (R ∖ \{λ_{1}\} \times {0}) \neq \emptyset$ .

In view of Lemmas 2.1–2.2, by a similar argument as Theorem 1.2(a), we can show that $C^{\mp} \subseteq (R_{+} \times S_{1}^{\mp}) \cup \{(λ_{1}, 0)\}$ is unbounded in $O$ and $(λ_{1}, + \infty) \subseteq {pr}_{R} (C^{\mp})$ .

From equation (4.8), we have that $\begin{array}{rcl} {(r^{N - k} v^{' k})}^{'} & = & - r^{N - 1} {(1 - v^{' 2})}^{\frac{k + 2}{2}} λ \frac{N}{C_{N}^{k}} H_{k} (r, v) - (N - k) r^{N - k - 1} v^{' k + 2} \\ = & r^{N - 1} ({(1 - v^{' 2})}^{\frac{k + 2}{2}} λ \frac{N}{C_{N}^{k}} \frac{H_{k} (r, v)}{v^{k}} + \frac{(N - k) r^{- k} v^{' k + 2}}{v^{k}}) v^{k} . \end{array}$ Noting that $\begin{array}{r} \frac{(N - k) r^{- k} v^{' k + 2}}{v^{k}} ⩾ 0, \end{array}$ like that of Theorem 1.2(a), we can obtain the desire asymptotic behavior of v with respect to λ.

(b) For any $n \in N$ , define $\begin{array}{r} H_{k}^{(n)} (r, s) = \{\begin{array}{ll} n^{k} s^{k}, & s \in [- \frac{1}{n}, \frac{1}{n}], \\ (H_{k} (r, \frac{2}{n}) - 1) n s + 2 - H_{k} (r, \frac{2}{n}), & s \in (\frac{1}{n}, \frac{2}{n}), \\ - (H_{k} (r, - \frac{2}{n}) - 1) n s + 2 - H_{k} (r, - \frac{2}{n}), & s \in (- \frac{2}{n}, - \frac{1}{n}), \\ H (r, s), & s \in [- R, - \frac{2}{n}] \cup [\frac{2}{n}, R] \end{array} \end{array}$ and consider the following approximation problem $\begin{matrix} \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k}^{(n)} (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ Then, using an argument similar to that of Theorem 1.2(b), we can derive the desired conclusion.

(c) For any $n \in N$ , define $\begin{array}{r} H_{k, n} (r, s) = \{\begin{array}{ll} \frac{1}{n^{k}} s^{k}, & s \in [- \frac{1}{n}, \frac{1}{n}], \\ (H_{k} (r, \frac{2}{n}) - \frac{1}{n^{2 k}}) n s + 2 \frac{1}{n^{2 k}} - H_{k} (r, \frac{2}{n}), & s \in (\frac{1}{n}, \frac{2}{n}), \\ - (H_{k} (r, - \frac{2}{n}) - \frac{1}{n^{2 k}}) n s + 2 \frac{1}{n^{2 k}} - H_{k} (r, - \frac{2}{n}), & s \in (- \frac{2}{n}, - \frac{1}{n}), \\ H_{k} (r, s), & s \in [- R, - \frac{2}{n}] \cup [\frac{2}{n}, R] . \end{array} \end{array}$ Consider the following problem $\begin{matrix} \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k, n} (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ Applying the argument as the proof of Theorem 1.2(c), we can obtain the desired conclusion. □

Proof of Theorem 1.4.

Suppose that v is an one-sign solution of problem (1.1) with some $λ > 0$ . If k is odd, multiplying the first equation of problem (1.1) by v, in view of Lemma 2.2, we obtain after integrations by parts that $\begin{array}{rcl} \int_{0}^{R} r^{N - k} v^{' k + 1} d r & ⩽ & \int_{0}^{R} r^{N - k} \frac{v^{' k + 1}}{{(1 - v^{' 2})}^{k / 2}} d r \\ = & λ \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} \frac{- H_{k} (r, v)}{v^{k}} v^{k + 1} d r \\ ⩽ & λ ϱ \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} v^{k + 1} d r \\ ⩽ & \frac{λ ϱ}{λ_{1}} \int_{0}^{R} r^{N - k} {(- v^{'})}^{k + 1} d r \\ = & \frac{λ ϱ}{λ_{1}} \int_{0}^{R} r^{N - k} v^{' k + 1} d r, \end{array}$ which follows that $λ ⩾ λ_{1} / ϱ : = ϱ_{*}$ .

On the other hand, when k is even, we claim that $\int_{0}^{r} τ^{N - 1} H_{k} (τ, v) d τ ⩾ 0$ for any $r \in [0, R]$ . Otherwise, we get that $\begin{matrix} 0 ⩽ {r_{1}}^{N - k} {(\frac{v^{'} (r_{1})}{\sqrt{1 - v^{' 2} (r_{1})}})}^{k} = λ \frac{N}{C_{N}^{k}} \int_{0}^{r_{1}} τ^{N - 1} H_{k} (τ, v) d τ < 0 \end{matrix}$ for some $r_{1} \in [0, R]$ , which is impossible. Thus, we have that $\begin{array}{r} v^{'} = \pm ϕ^{- 1} [{(\frac{λ N}{C_{N}^{k} r^{N - k}} \int_{0}^{r} τ^{N - 1} H_{k} (τ, v (τ)) d τ)}^{1 / k}] . \end{array}$ It follows that v and $v^{'}$ have the opposite signs. If v is positive, reasoning as the odd case, we can show that $\begin{array}{rcl} \int_{0}^{R} r^{N - k} {(- v^{'})}^{k + 1} d r & ⩽ & \int_{0}^{R} r^{N - k} \frac{{(- v^{'})}^{k + 1}}{{(1 - v^{' 2})}^{k / 2}} d r \\ = & λ \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} \frac{H_{k} (r, v)}{v^{k}} v^{k + 1} d r \\ ⩽ & λ ϱ \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} v^{k + 1} d r \\ ⩽ & \frac{λ ϱ}{λ_{1}} \int_{0}^{R} r^{N - k} {(- v^{'})}^{k + 1} d r, \end{array}$ which implies that $λ ⩾ ϱ_{*}$ . While, if v is negative, analogously, we have that $\begin{array}{rcl} \int_{0}^{R} r^{N - k} v^{' k + 1} d r & ⩽ & \int_{0}^{R} r^{N - k} \frac{v^{' k + 1}}{{(1 - v^{' 2})}^{k / 2}} d r \\ = & λ \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} \frac{H_{k} (r, v)}{v^{k}} {(- v)}^{k + 1} d r \\ ⩽ & λ ϱ \frac{N}{C_{N}^{k}} \int_{0}^{R} r^{N - 1} {(- v)}^{k + 1} d r \\ ⩽ & \frac{λ ϱ}{λ_{1}} \int_{0}^{R} r^{N - k} v^{' k + 1} d r, \end{array}$ which still shows that $λ ⩾ ϱ_{*}$ . □

Footnotes

Acknowledgement

Research supported by NNSF of China (No. 12031015, 12371110) and National Key R&D Program of China (2022YFA1005601).

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