Arbitrarily small nodal solutions for nonhomogeneous Robin problems

Abstract

In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in $C^{1} (\overline{Ω})$ .

Keywords

Nonhomogeneous differential operator locally defined reaction cut-off function extremal constant sign solution nodal solution

1. Introduction

Let $Ω \subseteq R^{N}$ be a bounded domain with a $C^{2}$ -boundary $\partial Ω$ . In this paper we study the following nonlinear nonhomogeneous Robin problem $\begin{matrix} (1) & \{\begin{matrix} - div a (D u) + ξ (z) | u |^{p - 2} u = f (z, u) & in Ω, \\ \frac{\partial u}{\partial n_{a}} + β (z) | u |^{p - 2} u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ with $1 < p < \infty$ . The map $a : R^{N} \to R^{N}$ in the definition of the differential operator is continuous and strictly monotone (thus, maximal monotone too) and satisfies certain other regularity and growth conditions, which are listed in hypotheses $H_{0}$ (see Section 2). These conditions provide a general framework in which we can fit many operators of interest such as the p-Laplacian and the $(p, q)$ -Laplacian (the sum of a p-Laplacian and of a q-Laplacian with $1 < q < p$ ). There is also a potential term $ξ (z) | u |^{p - 2} u$ with $ξ \in L^{\infty} (Ω)$ , $ξ (z) ⩾ 0$ for a.a. $z \in Ω$ . In the reaction (right hand side of (1)), we have a Carathéodory function $f (z, x)$ (that is measurable in z and continuous in x), which is only locally defined in x, namely, all the hypotheses on $f (z, \cdot)$ concern an interval $[- θ, θ]$ with $θ > 0$ . In the boundary condition, $\frac{\partial u}{\partial n_{a}}$ denotes the conormal derivative of u corresponding to the map $a (\cdot)$ . It is interpreted by using the nonlinear Green’s identity (see e.g. [14], p. 34). If $u \in C^{1} (\overline{Ω})$ , then $\begin{matrix} \frac{\partial u}{\partial n_{a}} = {(a (D u), n)}_{R^{N}}, \end{matrix}$ with $n (\cdot)$ being the outward unit normal on $\partial Ω$ . The boundary coefficient $β \in C^{0, α} (\partial Ω)$ with $0 < α < 1$ and $β (z) ⩾ 0$ for all $z \in \partial Ω$ . The case $β \equiv 0$ is also included and corresponds to the Neumann problem.

Using cut-off techniques, variational tools from the critical point theory and an extended version of the symmetric mountain pass theorem due to Kajikiya [5], we prove that problem (1) has a whole sequence ${u_{n}}_{n \in N}$ of distinct nodal (sign changing) solutions such that $u_{n} \to 0$ in $C^{1} (\overline{Ω})$ as $n \to \infty$ . Our work here extends those of Papageorgiou–Vetro–Vetro [15] (semilinear problems) and Papageorgiou–Rǎdulescu [10] (nonlinear problems). In particular in [10], the reaction is globally defined and it is required that $f (z, θ) ⩽ 0 ⩽ f (z, - θ)$ for a.a. $z \in Ω$ . This excludes for example, a “concave” reaction of the form $| x |^{τ - 2} x$ with $x \in [- θ, θ]$ and $τ < p$ , which is admissible for the setting of problem (1), see hypotheses $H_{2}$ below. We also mention the recent works of Liu–Papageorgiou [7], Papageorgiou–Zhang [16], Vetro [21]. In [7] the authors studied a critical double phase problem and produced a sequence of nodal solutions converging to zero in the corresponding Musielak–Orlicz–Sobolev space. In [16] the authors dealt with a parametric $(p, 2)$ -Dirichlet problem and in the reaction they have the competing effects of a globally defined parametric term $λ | u |^{r - 2} u$ ( $p < r ⩽ p^{*}$ ) and of a locally defined Carathéodory perturbation $g (z, x)$ which is $(p - 1)$ -sublinear near zero. They proved multiplicity theorems providing sign information for all the solutions when the parameter $λ > 0$ is big, and studied the asymptotic behavior of these solutions as $λ \to + \infty$ . Finally, Vetro [21] using the Orlicz–Sobolev functional setting studied general nonlinear elliptic problems with a globally defined reaction, and proved vanishing and blow-up phenomena for the solutions. All the aforementioned works deal with Dirichlet problems. As we already mentioned, we use cut-off techniques to take care of the fact that our reaction $f (z, \cdot)$ is locally defined. The first work to do this, is that of Wang [22], who studied Dirichlet equations driven by the p-Laplacian and with a concave contribution in the reaction and produced a sequence ${u_{n}}_{n \in N}$ of weak solutions such that $‖ u_{n} ‖_{\infty} \to 0$ . These solutions need not be nodal.

2. Mathematical background and hypotheses

The main spaces in the study of problem (1), are the Sobolev space $W^{1, p} (Ω)$ , the Banach space $C^{1} (\overline{Ω})$ and the “boundary” Lebesgue space $L^{p} (\partial Ω)$ . By $‖ \cdot ‖$ we denote the norm of the Sobolev space $W^{1, p} (Ω)$ . So, we have $\begin{matrix} ‖ u ‖ = {[‖ u ‖_{p}^{p} + ‖ D u ‖_{p}^{p}]}^{\frac{1}{p}} for all u \in W^{1, p} (Ω) . \end{matrix}$ It is well-known that the Banach space $C^{1} (\overline{Ω})$ is an ordered Banach space with positive (order) cone $C_{+}$ $\begin{matrix} C_{+} = {u \in C^{1} (\overline{Ω}) ∣ u (z) ⩾ 0 for all z \in \overline{Ω}} . \end{matrix}$ This cone has a nonempty interior given by $\begin{matrix} int C_{+} = {u \in C_{+} ∣ u (z) > 0 for all z \in \overline{Ω}} . \end{matrix}$

On the boundary $\partial Ω$ , we consider the $(N - 1)$ -dimensional Hausdorff (surface) measure $σ (\cdot)$ . Using this measure on $\partial Ω$ , we can define, in the usual way, the boundary Lebesgue space $L^{q} (\partial Ω)$ ( $1 ⩽ q ⩽ \infty$ ). From the theory of Sobolev spaces, we know that there exists a unique continuous linear operator ${\hat{γ}}_{0} : W^{1, p} (Ω) \to L^{p} (\partial Ω)$ , known as the trace map such that ${\hat{γ}}_{0} (u) = u |_{\partial Ω}$ for all $u \in W^{1, p} (Ω) \cap C (\overline{Ω})$ . So, the trace operator generalizes the notion of boundary values to all Sobolev functions. The operator is compact into $L^{q} (\partial Ω)$ for all $1 ⩽ q < \frac{(N - 1) p}{N - p}$ if $p < N$ , and into $L^{q} (\partial Ω)$ for all $1 ⩽ q < \infty$ if $p ⩾ N$ . Moreover, we have $\begin{matrix} Im {\hat{γ}}_{0} = W^{\frac{1}{p^{'}}, p} (\partial Ω) (\frac{1}{p} + \frac{1}{p^{'}} = 1), Ker {\hat{γ}}_{0} = W_{0}^{1, p} (Ω) . \end{matrix}$ In what follows, for notational economy we drop the use of the trace operator ${\hat{γ}}_{0}$ . All restrictions of Sobolev functions on $\partial Ω$ are understood in the sense of traces.

Given $u : Ω \to R$ a measurable function, we define $\begin{matrix} u^{\pm} = max {\pm u, 0} . \end{matrix}$ We know that if $u \in W^{1, p} (Ω)$ , then $u^{\pm} \in W^{1, p} (Ω)$ and $u = u^{+} - u^{-}$ , $| u | = u^{+} + u^{-}$ . Also, if $u, v : Ω \to R$ are measurable functions and $u (z) ⩽ v (z)$ for a.a. $z \in Ω$ , then we define the order interval $\begin{matrix} [u, v] = {h \in W^{1, p} (Ω) ∣ u (z) ⩽ h (z) ⩽ v (z) for a.a. z \in Ω} . \end{matrix}$ Also by $| \cdot |_{N}$ we denote the Lebesgue measure on $R^{N}$ .

Next we introduce our hypotheses on the map $a (\cdot)$ . Let $η \in C^{1} (0, \infty)$ be such that $\begin{matrix} 0 < \hat{c} ⩽ \frac{t η^{'} (t)}{η (t)} ⩽ c_{0} and c_{1} t^{p - 1} ⩽ η (t) ⩽ c_{2} [t^{s - 1} + t^{p - 1}] \end{matrix}$ for all $t ⩾ 0$ , some $c_{1}, c_{2} > 0$ and $1 ⩽ s < p$ .

The hypotheses on the map $a (\cdot)$ are the following:

$a (y) = a_{0} (| y |) y$ for all $y \in R^{N}$ with $a_{0} (t) > 0$ for all $t > 0$ and

$a_{0} \in C^{1} (0, \infty)$ , $t \mapsto a_{0} (t) t$ is strictly increasing, $a_{0} (t) t \to 0^{+}$ as $t \to 0^{+}$ and $- 1 < {lim}_{t \to 0^{+}} \frac{a_{0}^{'} (t) t}{a_{0} (t)}$ ;

$| \nabla a (y) | ⩽ c_{3} \frac{η (| y |)}{| y |}$ for all $y \in R^{N} ∖ {0}$ , some $c_{3} > 0$ ;

$\frac{η (| y |)}{| y |} | ξ |^{2} ⩽ {(\nabla a (y) ξ, ξ)}_{R^{N}}$ for all $y \in R^{N} ∖ {0}$ and all $ξ \in R^{N}$ ;

if $G_{0} (t) = \int_{0}^{t} a_{0} (s) s d s$ for all $t ⩾ 0$ , then there exists $τ \in (1, p)$ such that $t \mapsto G_{0} (t^{\frac{1}{τ}})$ is convex and ${lim}_{t \to 0^{+}} \frac{G_{0} (t)}{t^{τ}} = 0$ .

Remark 1.
Hypotheses $H_{0}$ (i)–(iii) are dictated by the nonlinear regularity theory of Lieberman [6] and the nonlinear maximum principle of Pucci–Serrin [18]. Hypothesis $H_{0}$ (iv) serves the particular needs of our problem (1). However, it is a mild condition and it is satisfied in all cases of interest (see the examples below).

These hypotheses imply that $G_{0} (\cdot)$ is strictly increasing and strictly convex. We set $G (y) = G_{0} (| y |)$ for all $y \in R^{N}$ . Then $G (\cdot)$ is convex, differentiable and $G (0) = 0$ . Also we have $\begin{matrix} \nabla G (y) = G_{0}^{'} (y) \frac{y}{| y |} = a_{0} (| y |) y = a (y) for all y \in R^{N} ∖ {0}, and \nabla G (0) = 0 . \end{matrix}$ Thus, $G (\cdot)$ is the primitive of $a (\cdot)$ and then using the properties of convex functions, we have $\begin{matrix} (2) & G (y) ⩽ {(a (y), y)}_{R^{N}} for all y \in R^{N} . \end{matrix}$

The following lemma summarizes the main properties of the map $a (\cdot)$ and it is an easy consequence of hypotheses $H_{0}$ (i)–(iii) (see Papageorgiou–Rǎdulescu [11]).
Lemma 2.
If hypotheses $H_{0}$ (i)–(iii) hold, then we have
$y \mapsto a (y)$ is continuous, strictly monotone hence maximal monotone too;

$| a (y) | ⩽ c_{4} [| y |^{s - 1} + | y |^{p - 1}]$ for all $y \in R^{N}$ , some $c_{4} > 0$ ;

$\frac{c_{1}}{p - 1} | y |^{p} ⩽ {(a (y), y)}_{R^{N}}$ for all $y \in R^{N}$ .

Combining this lemma with the inequality (2), we obtain the following bilateral growth restrictions for the primitive $G (\cdot)$ . In particular we see that $G (\cdot)$ has balanced growth.
Corollary 3.
If hypotheses $H_{0}$ (i)–(iii) hold, then $\begin{array}{rcl} \frac{c_{1}}{p (p - 1)} | y |^{p} ⩽ G (y) ⩽ c_{5} [| y |^{s} + | y |^{p}] for all y \in R^{N} \end{array}$ with some $c_{5} > 0$ .

Below we give some characteristic examples of functions $a (\cdot)$ which satisfy hypotheses $H_{0}$ (see Papageorgiou–Rǎdulescu [11]).
Example 4.

$a (y) = | y |^{p - 2} y$ with $1 < p < \infty$ . This map corresponds to the p-Laplace differential operator defined by $\begin{matrix} Δ_{p} u = div (| D u |^{p - 2} D u) for all u \in W^{1, p} (Ω) . \end{matrix}$

$a (y) = | y |^{p - 2} y + | y |^{q - 2} y$ with $1 < q < p < \infty$ . This map corresponds to the $(p, q)$ -Laplace differential operator defined by $\begin{array}{rcl} Δ_{p} u + Δ_{q} u for all u \in W^{1, p} (Ω) . \end{array}$ Such operators arises in many mathematical models of physical processes and have been studied recently. We refer to the works of Bahrouni–Rǎdulescu–Repoveš [1], Ragusa–Tachikawa [19], Mingione–Rădulescu [8], Papageorgiou–Zhang [17] and the references therein.

$a (y) = {[1 + | y |^{2}]}^{\frac{p - 2}{2}} y$ with $1 < p < \infty$ . This map corresponds to the generalized p-mean curvature differential operator defined by $\begin{matrix} div [{(1 + | D u |^{2})}^{\frac{p - 2}{2}} D u] for all u \in W^{1, p} (Ω) . \end{matrix}$

$a (y) = [1 + \frac{1}{1 + | y |^{p}}] | y |^{p - 2} y$ with $1 < p < \infty$ . This map corresponds to the following differential operator $\begin{matrix} u \mapsto Δ_{p} u + div (\frac{| D u |^{p - 2} D u}{1 + | D u |^{p}}) for all u \in W^{1, p} (Ω) . \end{matrix}$ This operator arises in problems of plasticity theory (see Raubiček [20]).

Let $A : W^{1, p} (Ω) \to W^{1, p} {(Ω)}^{*}$ be the nonlinear operator defined by $\begin{matrix} ⟨ A (u), h ⟩ = \int_{Ω} {(a (D u), D h)}_{R^{N}} d z for all u, h \in W^{1, p} (Ω) . \end{matrix}$ This operator is continuous, monotone (hence maximal monotone too) and of type ${(S)}_{+}$ , that is, if $u_{n} \overset{w}{⟶} u$ in $W^{1, p} (Ω)$ and ${lim sup}_{n \to \infty} ⟨ A (u_{n}), v_{n} - u ⟩ ⩽ 0$ , then $u_{n} \to u$ in $W^{1, p} (Ω)$ (see Gasiński–Papageorgiou [3], p. 279).

Now we introduce the hypotheses on the potential function $ξ (\cdot)$ and the boundary coefficient $β (\cdot)$ .

$ξ \in L^{\infty} (Ω)$ , $ξ (z) ⩾ 0$ for a.a. $z \in Ω$ , and $β \in C^{0, α} (\partial Ω)$ , $0 < α < 1$ , $β (z) ⩾ 0$ for all $z \in \partial Ω$ and $ξ \neq 0$ or $β \neq 0$ .
Remark 5.
Evidently, the case $β \equiv 0$ is also included and corresponds to the Neumann problem.

Let $γ : W^{1, p} (Ω) \to R$ be the $C^{1}$ -function defined by $\begin{matrix} γ (u) = \int_{Ω} p G (D u) d z + \int_{Ω} ξ (z) | u |^{p} d z + \int_{\partial Ω} β (z) | u |^{p} d σ for all u \in W^{1, p} (Ω) . \end{matrix}$ Using Corollary 3, we have that $\begin{matrix} γ (u) ⩾ \frac{c_{1}}{p - 1} ‖ D u ‖_{p}^{p} + \int_{Ω} ξ (z) | u |^{p} d z + \int_{\partial Ω} β (z) | u |^{p} d σ for all u \in W^{1, p} (Ω) . \end{matrix}$ Then using Lemma 4.11 of Mugnai–Papageorgiou [9] and Proposition 2.4 of Gasiński–Papageorgiou [3], we infer that $\begin{matrix} (3) & c_{6} ‖ u ‖^{p} ⩽ γ (u) for all u \in W^{1, p} (Ω) \end{matrix}$ for some $c_{6} > 0$ .

The hypotheses on the reaction $f (z, x)$ are the following:

$f : Ω \times [- θ, θ] \to R$ with $θ > 0$ is a Carathéodory function such that for a.a. $z \in Ω$ , $f (z, \cdot)$ is odd and

there exists $α_{θ} \in L^{\infty} (Ω)$ such that $\begin{matrix} | f (z, x) | ⩽ α_{θ} (z) for a.a. z \in Ω, for all x \in [- θ, θ]; \end{matrix}$

with $τ \in (1, p)$ as in hypothesis $H_{0}$ (iv) we have $\begin{matrix} ℓ (z) ⩽ \underset{x \to 0}{lim inf} \frac{f (z, x)}{| x |^{τ - 2} x} uniformly for a.a. z \in Ω, \end{matrix}$ where $ℓ \in L^{\infty} (Ω)$ is such that $0 ⩽ ℓ (z)$ for a.a. $z \in Ω$ and $ℓ \neq 0$ .
Remark 6.
We see that the conditions on function f are minimal and local in the variable x. Hypothesis $H_{2}$ (ii) implies the existence of a “concave” term near zero. The function $f (x) = | x |^{μ - 2} x$ with $μ < τ$ (the standard concave reaction for problem (1)), satisfies hypotheses $H_{2}$ but not those of [10]. The symmetry condition $f (z, \cdot)$ odd, will lead to a whole sequence of distinct nodal solutions.

We introduce an even cut-off function $χ \in C_{c} (R)$ which has the following properties $\begin{matrix} (4) & 0 ⩽ χ (x) ⩽ 1 for all x \in R, χ |_{[- \frac{θ}{2}, \frac{θ}{2}]} \equiv 1, and supp χ \subseteq [- θ, θ] . \end{matrix}$ We also consider the globally defined Carathéodory function $\begin{matrix} (5) & \hat{f} (z, x) = χ (x) f (z, x) + [1 - χ (x)] | x |^{τ - 2} x for all (z, x) \in Ω \times R . \end{matrix}$ Hence, we have $\begin{matrix} (6) & \{\begin{matrix} | \hat{f} (z, x) | ⩽ c_{7} [1 + | x |^{τ - 1}] & for a.a. z \in Ω, for all x \in R, \\ \hat{f} (z, \cdot) is odd & for a.a. z \in Ω, \end{matrix} \end{matrix}$ for some $c_{7} > 0$ .

Let us introduce the following nonlinear Robin problem $\begin{matrix} (7) & \{\begin{matrix} - div a (D u) + ξ (z) | u |^{p - 2} u = \hat{f} (z, u) & in Ω, \\ \frac{\partial u}{\partial n_{a}} + β (z) | u |^{p - 2} u = 0 & on \partial Ω . \end{matrix} \end{matrix}$ In the sequel by $S_{+}$ (resp. $S_{-}$ ) we denote the set of positive (resp. negative) solutions of problem (7).
3. Constant sign solutions

In this section we are going to prove the nonemptiness of the solution sets $S_{+}$ , $S_{-}$ , and determine the regularity properties of their elements. We also show that $S_{+}$ has a smallest element and $S_{-}$ has a biggest element, respectively. These extremal constant sign solutions will be used in Section 4 to produce nodal solutions for problems (7) and (1).

Proposition 7.
If hypotheses $H_{0}$ , $H_{1}$ and $H_{2}$ hold, then we have $\emptyset \neq S_{+} \subseteq int C_{+}$ and $\emptyset \neq S_{-} \subseteq - int C_{+}$ .
Proof.
Let $\hat{F} (z, x) = \int_{0}^{x} \hat{f} (z, s) d s$ and consider the $C^{1}$ -functional ${\hat{φ}}_{+} : W^{1, p} (Ω) \to R$ defined by $\begin{matrix} {\hat{φ}}_{+} (u) = \frac{1}{p} γ (u) - \int_{Ω} \hat{F} (z, u^{+}) d z for all u \in W^{1, p} (Ω) . \end{matrix}$

On account of (3), (6) and recalling that $τ < p$ , we see that ${\hat{φ}}_{+} (\cdot)$ is coercive. Also, using the Sobolev embedding theorem and the compactness of the trace map, we show that ${\hat{φ}}_{+} (\cdot)$ is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find $u_{0} \in W^{1, p} (Ω)$ such that $\begin{matrix} (8) & {\hat{φ}}_{+} (u_{0}) = inf [{\hat{φ}}_{+} (u) ∣ u \in W^{1, p} (Ω)] . \end{matrix}$ By virtue of hypothesis $H_{2}$ (ii), given $ε > 0$ , we can find $δ = δ (ε) \in (0, \frac{θ}{2})$ such that $\begin{array}{l} F (z, x) ⩾ \frac{1}{τ} [ℓ (z) - ε] | x |^{τ} for a.a. z \in Ω, for all | x | ⩽ δ \\ (9) & \Rightarrow \hat{F} (z, x) ⩾ \frac{1}{τ} [ℓ (z) - ε] | x |^{τ} for a.a. z \in Ω, for all | x | ⩽ δ \end{array}$ (see (4), (5) and recall that $δ < \frac{θ}{2}$ ).

Let $t \in (0, δ)$ . Then hypotheses $H_{1}$ and (9) imply that $\begin{matrix} {\hat{φ}}_{+} (t) ⩽ c_{8} t^{p} - \frac{t^{τ}}{τ} [\int_{Ω} ℓ (z) d z - ε | Ω |_{N}] for some c_{8} > 0 . \end{matrix}$ Choosing $ε \in (0, \frac{1}{| Ω |_{N}} \int_{Ω} ℓ (z) d z)$ (recall $ℓ ⩾ 0$ and $ℓ \neq 0$ ), we have $\begin{matrix} {\hat{φ}}_{+} (t) ⩽ c_{8} t^{p} - c_{9} t^{τ} for some c_{9} > 0 . \end{matrix}$ Since $τ < p$ , choosing $t \in (0, δ)$ small, we have $\begin{matrix} {\hat{φ}}_{+} (t) < 0 \Rightarrow {\hat{φ}}_{+} (u_{0}) < 0 = {\hat{φ}}_{+} (0) (see (8)) \Rightarrow u_{0} \neq 0 . \end{matrix}$ From (8), we have ${\hat{φ}}_{+}^{'} (u_{0}) = 0$ , and $\begin{matrix} (10) & ⟨ γ^{'} (u_{0}), h ⟩ = \int_{Ω} \hat{f} (z, u_{0}^{+}) h d z for all h \in W^{1, p} (Ω) . \end{matrix}$ In (10), we choose $h = - u_{0}^{-} \in W^{1, p}$ to obtain $\begin{matrix} \frac{c_{1}}{p (p - 1)} {‖ u_{0}^{-} ‖}^{p} ⩽ \frac{1}{p} γ (u_{0}^{-}) ⩽ 0 (see (2) and (3)) \Rightarrow u_{0} ⩾ 0 and u_{0} \neq 0 . \end{matrix}$ So, we have $\begin{matrix} (11) & \{\begin{matrix} - div a (D u_{0}) + ξ (ε) u_{0}^{p - 1} = \hat{f} (z, u_{0}) & in Ω, \\ \frac{\partial u_{0}}{\partial n_{a}} + β (z) u_{0}^{p - 1} = 0 & on \partial Ω . \end{matrix} \end{matrix}$ From Proposition 2.12 of Papageorgiou–Rǎdulescu [12], we have $u_{0} \in L^{\infty} (Ω)$ . Then the nonlinear regularity theory of Lieberman [6], implies that $u_{0} \in C_{+} ∖ {0}$ . On account of hypotheses $H_{1}$ and (6), we see that if $ρ = ‖ u_{0} ‖_{\infty}$ , then we can find ${\hat{ξ}}_{ρ} > 0$ such that $\begin{matrix} (12) & \hat{f} (z, x) + {\hat{ξ}}_{ρ} x^{p - 1} ⩾ 0 for a.a. z \in Ω, for all 0 ⩽ x ⩽ ρ . \end{matrix}$ Then from (11) and (12), we have $\begin{array}{l} div a (D u_{0}) ⩽ [‖ ξ ‖_{\infty} + {\hat{ξ}}_{p}] u_{0}^{p - 1} in Ω \\ \Rightarrow u_{0} \in int C_{+} (see Pucci–Serrin [18], pp. 111 and 120) . \end{array}$ So, we conclude that $\begin{matrix} \emptyset \neq S_{+} \subseteq int C_{+} . \end{matrix}$

Similarly using the $C^{1}$ -functional ${\hat{φ}}_{-} : W^{1, p} (Ω) \to R$ defined by $\begin{matrix} {\hat{φ}}_{-} (u) = \frac{1}{p} γ (u) - \int_{Ω} \hat{F} (z, - u^{-}) d z for all u \in W^{1, p} (Ω), \end{matrix}$ we can show that $\emptyset \neq S_{-} \subseteq - int C_{+}$ . □

Hypothesis $H_{1}$ (ii) and (6) imply that given $ε > 0$ , we can find $c_{10} = c_{10} (ε) > 0$ such that $\begin{matrix} (13) & \hat{f} (z, x) x ⩾ [ℓ (z) - ε] | x |^{τ} - c_{10} | x |^{p} for a.a. z \in Ω, for all x \in R . \end{matrix}$ This unilateral growth condition on $\hat{f} (z, x)$ leads to the following auxiliary Robin problem $\begin{matrix} (14) & \{\begin{matrix} - div a (D u) + ξ (z) | u |^{p - 2} u = [ℓ (z) - ε] | u |^{τ - 2} u - c_{10} | u |^{p - 2} u & in Ω, \\ \frac{\partial u}{\partial n_{a}} + β (z) | u |^{p - 2} u = 0 & on \partial Ω . \end{matrix} \end{matrix}$
Proposition 8.
If hypotheses $H_{0}$ and $H_{1}$ hold, then problem ( 14 ) has a unique positive solution $\overline{u} \in int C_{+}$ and since the equation is odd, $\overline{v} = - \overline{u} \in - int C_{+}$ is the unique negative solution of ( 14 ).
Proof.
First, we show the existence of a positive solution for problem (14). To this end, we introduce the $C^{1}$ -functional $ψ_{+} : W^{1, p} (Ω) \to R$ defined by $\begin{matrix} ψ_{+} (u) = \frac{1}{p} γ (u) + \frac{c_{10}}{p} {‖ u^{+} ‖}_{p}^{p} - \frac{1}{τ} \int_{Ω} [ℓ (z) - ε] {(u^{+})}^{τ} d z for all u \in W^{1, p} (Ω) . \end{matrix}$

Since $τ < p$ , using (3), we see that $ψ_{+} (\cdot)$ is coercive. Also it is sequentially weakly lower semicontinuous. So, we can find $\overline{u} \in W^{1, p} (Ω)$ such that $\begin{matrix} (15) & ψ_{+} (\overline{u}) = inf [ψ_{+} (u) ∣ u \in W^{1, p} (Ω)] . \end{matrix}$ Choosing $ε > 0$ small and since $τ < p$ , we see that for $t > 0$ small, it holds $\begin{matrix} ψ_{+} (t) < 0 \Rightarrow ψ_{+} (\overline{u}) < 0 = ψ_{+} (0) (see (15)) \Rightarrow \overline{u} \neq 0 . \end{matrix}$ From (15), we have $\begin{array}{l} ψ_{+}^{'} (\overline{u}) = 0 \\ (16) & \Rightarrow ⟨ γ^{'} (\overline{u}), h ⟩ = \int_{Ω} ([ℓ (z) - ε] {({\overline{u}}^{+})}^{τ - 1} - c_{10} {({\overline{u}}^{+})}^{p - 1}) h d z for all h \in W^{1, p} (Ω) . \end{array}$ In (16), we use the test function $h = - {\overline{u}}^{-} \in W^{1, p} (Ω)$ to get $\begin{matrix} γ ({\overline{u}}^{-}) ⩽ 0 (see (2)) \Rightarrow \overline{u} ⩾ 0 and \overline{u} \neq 0 (see (3)) . \end{matrix}$ So, $\overline{u} \in W^{1, p} (Ω)$ is a positive solution of problem (14). As before the nonlinear regularity theory of Lieberman [6], implies that $\overline{u} \in C_{+} ∖ {0}$ . Moreover, we have $\begin{array}{l} div a (D \overline{u}) ⩽ [‖ ξ ‖_{\infty} + c_{10}] {\overline{u}}^{p - 1} in Ω \\ \Rightarrow \overline{u} \in int C_{+} (see Pucci–Serrin [18], pp. 111 and 120) . \end{array}$

Next we show the uniqueness of this positive solution. For this purpose, we introduce the integral functional $i : L^{1} (Ω) \to \overline{R} = R \cup {+ \infty}$ defined by $\begin{matrix} i (u) = \{\begin{matrix} \frac{1}{p} γ (u^{\frac{1}{τ}}) & if u ⩾ 0 and u^{\frac{1}{τ}} \in W^{1, p} (Ω), \\ + \infty & otherwise . \end{matrix} \end{matrix}$ Let $dom i = {u \in L^{1} (Ω) ∣ i (u) < + \infty}$ (the effective domain of i). Suppose that $u_{1}, u_{2} \in dom i$ and set $y = {[(1 - t) u_{1} + t u_{2}]}^{\frac{1}{τ}}$ with $t \in [0, 1]$ . From Díaz–Saa [2] (see Lemma 2 and its proof) and the monotonicity of $G_{0} (\cdot)$ , we have $\begin{array}{l} | D y | ⩽ {[(1 - t) {| D u_{1}^{\frac{1}{τ}} |}^{τ} + t {| D u_{2}^{\frac{1}{τ}} |}^{τ}]}^{\frac{1}{τ}} a.a. z \in Ω \\ \Rightarrow G_{0} (| D y |) ⩽ G_{0} ({[(1 - t) {| D u_{1}^{\frac{1}{τ}} |}^{τ} + t {| D u_{2}^{\frac{1}{τ}} |}^{τ}]}^{\frac{1}{τ}}) a.a. z \in Ω . \end{array}$ This combined with hypothesis $H_{0}$ (iv) deduces $\begin{array}{l} G_{0} (| D y |) ⩽ (1 - t) G_{0} (| D u_{1}^{\frac{1}{τ}} |) + t G_{0} (| D u_{2}^{\frac{1}{τ}} |) a.a. z \in Ω \\ \Rightarrow G (D y) ⩽ (1 - t) G (D u_{1}^{\frac{1}{τ}}) + t G (D u_{2}^{\frac{1}{τ}}) a.a. z \in Ω \\ \Rightarrow i (\cdot) is convex (since τ < p and ξ ⩾ 0, β ⩾ 0) . \end{array}$ Now let $\hat{u} \in W^{1, p} (Ω)$ be another positive solution of (14). Again we have $\hat{u} \in int C_{+}$ . Using Proposition 4.1.22, p. 274, cf. [14], we have $\begin{matrix} \frac{\overline{u}}{\hat{u}} \in L^{\infty} (Ω) and \frac{\hat{u}}{\overline{u}} \in L^{\infty} (Ω) . \end{matrix}$ Let $h = {\overline{u}}^{τ} - {\hat{u}}^{τ}$ . For $| t | < 1$ small, we have $\begin{matrix} {\overline{u}}^{τ} + t h \in dom i and {\hat{u}}^{τ} + t h \in dom i . \end{matrix}$ Then the convexity of $i (\cdot)$ implies that $i (\cdot)$ is Gâteaux differentiable at ${\overline{u}}^{τ}$ and ${\hat{u}}^{τ}$ in the direction h. Moreover, using the chain rule and the nonlinear Green’s identity (see [14], p. 34), we have $\begin{array}{rcl} i^{'} (\overline{u}) (h) & = & \frac{1}{τ} \int_{Ω} \frac{- div a (D \overline{u}) + ξ (z) {\overline{u}}^{p - 1}}{{\overline{u}}^{τ - 1}} h d z \\ = & \frac{1}{τ} \int_{Ω} ([ℓ (z) - ε] - c_{10} {\overline{u}}^{p - τ}) h d z \end{array}$ and $\begin{array}{rcl} i^{'} (\hat{u}) (h) & = & \frac{1}{τ} \int_{Ω} \frac{- div a (D \hat{u}) + ξ (z) {\hat{u}}^{p - 1}}{{\hat{u}}^{τ - 1}} h d z \\ = & \frac{1}{τ} \int_{Ω} ([ℓ (z) - ε] - c_{10} {\hat{u}}^{p - τ}) h d z . \end{array}$ The convexity of $i (\cdot)$ implies the monotonicity of $i^{'} (\cdot)$ . Hence, from $p > τ$ , we have $\begin{matrix} 0 ⩽ \int_{Ω} c_{10} [{\hat{u}}^{p - τ} - {\overline{u}}^{p - τ}] [{\overline{u}}^{τ} - {\hat{u}}^{τ}] d z ⩽ 0 \Rightarrow \overline{u} = \hat{u} . \end{matrix}$ This proves the uniqueness of the positive solution $\overline{u} \in int C_{+}$ of (14). Notice that problem (14) is odd, therefore $\overline{v} = - \overline{u} \in int C_{+}$ is the unique negative solution of (14). □

The solutions of (14) provide bounds for the sets $S_{+}$ and $S_{-}$ , respectively.
Proposition 9.
If hypotheses $H_{0}$ , $H_{1}$ , $H_{2}$ hold, then $\overline{u} ⩽ u$ for all $u \in S_{+}$ and $v ⩽ \overline{v}$ for all $v \in S_{-}$ .
Proof.
Let $u \in S_{+} \subseteq int C_{+}$ (see Proposition 8). We introduce the Carathéodory function $k_{+} : Ω \times R \to R$ defined by $\begin{matrix} (17) & k_{+} (z, x) = \{\begin{matrix} [ℓ (z) - ε] {(x^{+})}^{τ - 1} - c_{10} {(x^{+})}^{p - 1} & if x ⩽ u (z), \\ [ℓ (z) - ε] u {(z)}^{τ - 1} - c_{10} u {(z)}^{p - 1} & if u (z) ⩽ x . \end{matrix} \end{matrix}$ We set $K_{+} (z, x) = \int_{0}^{x} k_{+} (z, s) d s$ and consider the $C^{1}$ -functional ${\hat{ψ}}_{+} : W^{1, p} (Ω) \to R$ defined by $\begin{matrix} {\hat{ψ}}_{+} (u) = \frac{1}{p} γ (u) - \int_{Ω} K_{+} (z, u) d z for all u \in W^{1, p} (Ω) . \end{matrix}$

From (3) and (17), it is clear that ${\hat{ψ}}_{+} (\cdot)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\hat{u} \in W^{1, p} (Ω)$ such that $\begin{matrix} (18) & {\hat{ψ}}_{+} (\hat{u}) = inf [ψ_{+} (u) ∣ u \in W^{1, p} (Ω)] . \end{matrix}$ Arguing as in the proof of Proposition 8, we have $\hat{u} \neq 0$ . But, from (18), we have $\begin{array}{l} {\hat{ψ}}_{+}^{'} (\hat{u}) = 0 \\ (19) & \Rightarrow ⟨ γ^{'} (\hat{u}), h ⟩ = \int_{Ω} K_{+} (z, \hat{u}) h d z for all h \in W^{1, p} (Ω) . \end{array}$ In (19), first, we choose the test function $h = - {\hat{u}}^{-} \in W^{1, p} (Ω)$ . We obtain $\begin{matrix} \frac{c_{1}}{p (p - 1)} {‖ {\hat{u}}^{-} ‖}^{p} ⩽ 0 (see (3)) \Rightarrow \hat{u} ⩾ 0 and \hat{u} \neq 0 . \end{matrix}$ Next in (19), we use the test function $h = {[\hat{u} - u]}^{+} \in W^{1, p} (Ω)$ and use (13) and (17) to get $\begin{array}{rcl} ⟨ γ^{'} (\hat{u}), {(\hat{u} - u)}^{+} ⟩ & = & \int_{Ω} ([ℓ (z) - ε] u^{τ - 1} - c_{10} u^{p - 1}) {(\hat{u} - u)}^{+} d z \\ ⩽ & \int_{Ω} \hat{f} (z, u) {(\hat{u} - u)}^{+} d z = ⟨ γ^{'} (u), {(\hat{u} - u)}^{+} ⟩, \end{array}$ where we have used the fact that $u \in S_{+}$ . Hence, $u ⩽ \hat{u}$ . So, we have proved that $\begin{matrix} (20) & \hat{u} \in [0, u] and \hat{u} \neq 0 . \end{matrix}$ From (17), (19) and (20), it follows that $\hat{u}$ is a positive solution of (14). By Proposition 8, we have $\hat{u} = \overline{u}$ . Therefore $\begin{matrix} \overline{u} ⩽ u for all u \in S_{+} (see (20)) . \end{matrix}$

Let $v \in S_{-}$ . Starting with the Carathéodory function $k_{-} : Ω \times R \to R$ defined by $\begin{matrix} k_{-} (z, x) = \{\begin{matrix} [ℓ (z) - ε] | \overline{v} (z) |^{τ - 2} \overline{v} (z) - c_{10} | \overline{v} (z) |^{p - 2} \overline{v} (z) & if \overline{v} (z) < x, \\ [ℓ (z) - ε] | x |^{τ - 2} x - c_{10} | x |^{τ - 2} x & if x ⩽ \overline{v} (z), \end{matrix} \end{matrix}$ and reasoning as above, we show that $v ⩽ \overline{v}$ for all $v \in S_{-}$ . □

These bounds lead to extremal constant sign solutions. Proposition 10.
If hypotheses $H_{0}$ , $H_{1}$ and $H_{2}$ hold, then there exists a smallest positive solution $u_{} \in S_{+}$ of (* 7 ) (that is, $u_{} ⩽ u$ for all* $u \in S_{+}$ ) and a biggest negative solution $v_{} \in S_{-}$ of (* 7 ) (that is, $v ⩽ v_{}$ for all* $v \in S_{-}$ ).
Proof.
It follows from the proof of Proposition 7 of Papageorgiou–Rǎdulescu–Repovš [13] that $S_{+}$ is downward directed, that is, if $u_{1}, u_{2} \in S_{+}$ , we can find $u \in S_{+}$ such that $u ⩽ u_{1}$ and $u \in u_{2}$ . Using Lemma 3.10, p. 178 of Hu–Papageorgiou [4], we are able to find a decreasing sequence ${u_{n}}_{n \in N} \subseteq S_{+}$ such that ${inf}_{n \in N} u_{n} = inf S_{+}$ . Hence, we have $\begin{array}{l} (21) & ⟨ γ^{'} (u_{n}), h ⟩ = \int_{Ω} \hat{f} (z, u_{n}) h d z for all h \in W^{1, p} (Ω), for all n \in N \\ (22) & \Rightarrow \overline{u} ⩽ u_{n} ⩽ u_{1} for all n \in N (see Proposition 9) . \end{array}$ Let $h = u_{n} \in W^{1, p} (Ω)$ in (21). Using (6) and (22), we infer that ${u_{n}}_{n \in N} \subseteq W^{1, p} (Ω)$ is bounded. So, we may assume that $\begin{matrix} (23) & u_{n} \overset{w}{⟶} u_{} in W^{1, p} (Ω), and u_{n} \to u_{} in L^{p} (Ω) and in L^{p} (\partial Ω) . \end{matrix}$ We test (21) with $h = u_{n} - u_{} \in W^{1, p} (Ω)$ , pass to the limit as $n \to \infty$ and use (23) to yield ${lim}_{n \to \infty} ⟨ A (u_{n}), u_{n} - u_{} ⟩ = 0$ . By virtue of the fact that A is of type ${(S)}_{+}$ , it holds $\begin{matrix} (24) & u_{n} \to u_{} in W^{1, p} (Ω) . \end{matrix}$

So, if in (21), we pass to the limit as $n \to \infty$ and use (24) we get $\begin{matrix} (25) & ⟨ γ^{'} (u_{}), h ⟩ = \int_{Ω} \hat{f} (z, u_{}) h d z for all h \in W^{1, p} (Ω) . \end{matrix}$ Also from (22) and (24) we have $\begin{matrix} (26) & \overline{u} ⩽ u_{} . \end{matrix}$ From (25) and (26) we infer that $u_{} \in S_{+} \subseteq int C_{+}$ and $u_{} ⩽ u$ for all $u \in S_{+}$ .

We mention that $S_{-}$ is upward directed, that is, if $v_{1}, v_{2} \in S_{-}$ , then there exists $v \in S_{-}$ such that $v_{1} ⩽ v$ and $v_{2} ⩽ v$ . Similarly, working with $S_{-}$ , we produce $v_{} \in S_{-}$ such that $v ⩽ v_{}$ for all $v \in S_{-}$ . □

4. Nodal solutions

In this section we produce a whole sequence ${y_{n}}_{n \in N}$ of distinct nodal solutions such that $y_{n} \to 0$ in $C^{1} (\overline{Ω})$ by using the Theorem 1 of Kajikiya [5]. Observe that any nontrivial solution of (7) in $[v_{*}, u_{*}]$ which is distinct from $v_{*}$ and $u_{*}$ , will be nodal (on account of the extremallity of $u_{*}$ and $v_{*}$ ). Moreover, if the solution can be localized in $[v_{*}, u_{*}] \cap [- \frac{θ}{2}, \frac{θ}{2}]$ , then it will be a nodal solution of problem (1) (see (4) and (5)).

In what follows, let $\hat{φ} : W^{1, p} (Ω) \to R$ be the energy (Euler) functional for problem (7), defined by $\begin{matrix} \hat{φ} (u) = \frac{1}{p} γ (u) - \int_{Ω} \hat{F} (z, u) d z for all u \in W^{1, p} (Ω) . \end{matrix}$ We know that $\hat{φ} \in C^{1} (W^{1, p} (Ω))$ . Moreover $\hat{φ} (\cdot)$ is even and coercive (see (6) and recall that $τ < p$ ). Hence $\hat{φ} (\cdot)$ is bounded from below and satisfies the Palais–Smale condition (see [14], Proposition 5.1.15, p. 369). So $\hat{φ} (\cdot)$ satisfies hypothesis $(A_{1})$ of Theorem 1 of Kajikiya [5]. We will verify that $\hat{φ} (\cdot)$ satisfies hypothesis $(A_{2})$ of Theorem 1 of [5]. To this end, we need to strengthen a little hypotheses $H_{2}$ .

Same as hypotheses $H_{2}$ , but now $ess {inf}_{Ω} ℓ > 0$ .

Proposition 11.

If hypotheses $H_{0}$ , $H_{1}$ and $H_{2}^{'}$ hold, and $V \subseteq W^{1, p} (Ω)$ is a finite dimensional subspace, then there exists $ρ_{V} > 0$ such that $\begin{matrix} sup [\hat{φ} (u) ∣ u \in V with ‖ u ‖ = ρ_{V}] < 0 . \end{matrix}$

Proof.

On account of hypothesis $H_{2}^{'} (ii) = H_{2} (ii)$ , given $ε > 0$ , we can find $δ = δ (ε) \in (0, \frac{θ}{2})$ such that $\begin{array}{l} F (z, x) ⩾ \frac{1}{τ} [ℓ (z) - ε] | x |^{τ} for a.a. z \in Ω, for all | x | ⩽ δ \\ (27) & \Rightarrow \hat{F} (z, x) ⩾ \frac{1}{τ} [ℓ (z) - ε] | x |^{τ} for a.a. z \in Ω, for all | x | ⩽ δ, \end{array}$ where we have used (4) and (5). Hypothesis $H_{0}$ (iv) and Corollary 3, imply that we can find $c_{11} = c_{11} (ε) > 0$ such that $\begin{matrix} (28) & G (y) ⩽ \frac{ε}{τ} | y |^{τ} + c_{11} | y |^{p} for all y \in R^{N} . \end{matrix}$

Since V is finite dimensional, all norms are equivalent. Hence, we can find $ρ_{V} \in (0, 1)$ small such that $\begin{matrix} (29) & u \in V and ‖ u ‖ ⩽ ρ_{V} \Rightarrow | u (z) | ⩽ δ for a.a. z \in Ω . \end{matrix}$ Therefore, if $u \in V$ with $‖ u ‖ = ρ_{V}$ , then from (27)–(29) and hypotheses $H_{1}$ , we can find a constant $c_{12} > 0$ satisfying $\begin{matrix} \hat{φ} (u) ⩽ \frac{ε}{τ} ‖ u ‖^{τ} + c_{12} ‖ u ‖^{p} - \frac{1}{τ} \int_{Ω} ℓ (z) | u |^{τ} d z = c_{12} ‖ u ‖^{p} - c_{13} ‖ u ‖^{τ}, \end{matrix}$ where $c_{13} > 0$ is obtained by using $H_{2}^{'}$ . Recall that $τ < p$ , choosing $ρ_{V} = ‖ u ‖$ even smaller if necessary, we have $\begin{matrix} sup [\hat{φ} (u) ∣ u \in V and ‖ u ‖ = ρ_{V}] < 0 . \end{matrix}$ This completes the proof of the proposition. □

We already established that hypothesis $(A_{1})$ in Theorem 1 of [5] is satisfied. With Proposition 11, we have also satisfied hypothesis $(A_{2})$ in that theorem and so we can use it and have the following theorem concerning nodal solutions of problem (1).

Theorem 12.

If hypotheses $H_{0}$ , $H_{1}$ and $H_{2}^{'}$ hold, then problem ( 1 ) has a sequence ${u_{n}}_{n \in N}$ of distinct nodal solutions with nonpositive energy (that is, $\hat{φ} (u_{n}) ⩽ 0$ for all $n \in N$ ) and $u_{n} \to 0$ in $C^{1} (\overline{Ω})$ .

Proof.

We apply Theorem 1 of Kajikiya [5] and ${u_{n}}_{n \in N} \subseteq K_{\hat{φ}} = {u \in W^{1, p} (Ω) ∣ {\hat{φ}}^{'} (u) = 0}$ such that $\begin{matrix} (30) & \hat{φ} (u_{n}) ⩽ 0 for all n \in N, and ‖ u_{n} ‖ \to 0 as n \to \infty . \end{matrix}$ The nonlinear regularity theory of Lieberman [6] implies that there exist $α \in (0, 1)$ and $c_{14} > 0$ such that $\begin{matrix} u_{n} \in C^{1, α} (\overline{Ω}) and ‖ u_{n} ‖_{C^{1, α} (\overline{Ω})} ⩽ c_{14} for all n \in N . \end{matrix}$ The compact embedding of $C^{1, α} (\overline{Ω})$ into $C^{1} (\overline{Ω})$ and (30) imply that $\begin{matrix} u_{n} \to 0 in C^{1} (\overline{Ω}) \Rightarrow u_{n} \in [v_{*}, u_{*}] \cap [- \frac{θ}{2}, \frac{θ}{2}] for all n ⩾ n_{0} . \end{matrix}$ Therefore, we have that ${u_{n}}_{n ⩾ n_{0}}$ are distinct nodal solutions of (1). □

Remark 13.

If $f (z, x) x < p F (z, x)$ for all $(z, x) \in (Ω ∖ D) \times (R ∖ {0})$ with $| D |_{N} = 0$ , then the nodal solutions $u_{n}$ for $n \in N$ , have negative energy (that is, $\hat{φ} (u_{n}) < 0$ ) for all $n \in N$ (see [5], Remark 1.2). The function $f (x) = | x |^{τ - 2} x$ , $x \in [- θ, θ]$ , $1 < τ < p$ satisfies $H_{2}^{'}$ and this extra condition.

Footnotes

Acknowledgements

This project has received funding from the NNSF of China Grant Nos. 12001478, 12026255 and 12026256, and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH, National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611, and the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2020ZK07. It is also supported by Natural Science Foundation of Guangxi Grant No. 2020GXNSFBA297137, and the Ministry of Science and Higher Education of Republic of Poland under Grants Nos. 4004/GGPJII/H2020/2018/0 and 440328/PnH2/2019.

References

Bahrouni,

V.D.

Rǎdulescu and

D.D.

Repovš, Double phase transonic flow problems with variable growth: Nonlinear patterns and stationary waves, Nonlinearity 32 (2019), 2481–2495. doi:10.1088/1361-6544/ab0b03.

J.I.

Díaz and

J.E.

Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I. Math. 305 (1987), 521–524.

Gasiński and

N.S.

Papageorgiou, Positive solutions for the Robin p-Laplacian problem with competing nonlinearities, Adv. Calc. Var. 12 (2019), 31–56. doi:10.1515/acv-2016-0039.

Hu and

N.S.

Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer Academic Publisher, Dordrecht, The Netherlands, 1997.

Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Functional Anal. 225 (2005), 352–370. doi:10.1016/j.jfa.2005.04.005.

G.M.

Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Diff. Equ. 16 (1991), 311–361. doi:10.1080/03605309108820761.

Z.H.

Liu and

N.S.

Papageorgiou, Asymptotically vanishing nodal solutions for critical double phase problems, Asymp. Anal (2020). doi:10.3233/ASY-201645.

Mingione and

Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501 (2021), ID: 125197. doi:10.1016/j.jmaa.2021.125197.

Mugnai and

N.S.

Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (2012), 729–788.

10.

N.S.

Papageorgiou and

V.D.

Rǎdulescu, Infinitely many nodal solutions for nonlinear nonhomogeneous Robin problems, Adv. Nonlinear Stud. 16 (2016), 287–299. doi:10.1515/ans-2015-5040.

11.

N.S.

Papageorgiou and

V.D.

Rǎdulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math. 28 (2016), 545–571.

12.

N.S.

Papageorgiou and

V.D.

Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction, Adv. Nonlinear Stud. 16 (2016), 737–764. doi:10.1515/ans-2016-0023.

13.

N.S.

Papageorgiou,

V.D.

Rǎdulescu and

D.D.

Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst. 37 (2017), 2589–2618. doi:10.3934/dcds.2017111.

14.

N.S.

Papageorgiou,

V.D.

Rǎdulescu and

D.D.

Repovš, Nonlinear Analysis – Theory and Methods, Springer Nature, Swilzerland, 2019.

15.

N.S.

Papageorgiou,

Vetro and

Vetro, Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms, Topol. Meth. Nonlinear Anal. 50 (2017), 269–286.

16.

N.S.

Papageorgiou and

Zhang, Double phase problem with critial and locally defined rection terms, Asymp. Anal. 116 (2020), 73–92.

17.

N.S.

Papageorgiou and

Zhang, Constant sign and nodal solutions for superlinear

(p, q)

-equations with indefinite potential and a concave boundary term, Adv. Nonlinear Anal. 10 (2021), 76–101. doi:10.1515/anona-2020-0101.

18.

Pucci and

Serrin, The Maximum Principle, Birkhauser, Basel, 2007.

19.

M.A.

Ragusa and

Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal. 9 (2020), 710–728. doi:10.1515/anona-2020-0022.

20.

Roubiček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2005.

21.

Vetro, Parametric and nonparametric A-Laplace problems: Existence of solutions and asymptotic analysis, Asymp. Anal. 122 (2020), 105–118.

22.

Z.Q.

Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Diff. Equ. Appl. 8 (2001), 15–33. doi:10.1007/PL00001436.