Existence of nontrivial solution for quasilinear equations involving the 1-biharmonic operator

Abstract

In this paper, we study the existence results of a quasilinear elliptic problem involving the 1-biharmonic operator in $R^{N}$ , whose nonlinearity satisfies appropriate conditions. The existence theorem is proved through a new version of the Mountain Pass Theorem to locally Lipschitz functionals, where it is considered the Cerami compactness condition rather than the Palais–Smale one.

Keywords

1-biharmonic operator Variational methods

1. Introduction

In this paper, consider the existence of nontrivial solutions to the quasilinear elliptic problems $\begin{array}{rcl} (1.1) & \{\begin{matrix} Δ_{1}^{2} - Δ_{1} u + \frac{u}{| u |} = f (u) & in R^{N}, \\ u \in BL (R^{N}), \end{matrix} \end{array}$ where $N ⩾ 3$ , $BL (R^{N})$ is defined in Section 2, f is a continuous function verifying some conditions, the 1-Laplacian operator is formally defined as $\begin{matrix} Δ_{1} u = div (\frac{D u}{| D u |}), \end{matrix}$ and the 1-biharmonic operator is given by $\begin{matrix} Δ_{1}^{2} u = Δ (\frac{Δ u}{| Δ u |}) . \end{matrix}$

In the last years, elliptic problems involving the 1-Laplacian operator has increased a lot. In fact, there are a lot of papers on this highly singular operator. For example, in [2], the authors made a pioneering study of problems involving this operator, and produced the monograph [3]; in [1,8,10,11], the authors analyzed related questions based on the energy functional of the $BV$ space, and in [14–17], the authors used a method based on approximation through p-Laplacian problems. The problems involving the 1-Laplacian operator can be seen as the limit of the several p-Laplacian ones, as the parameter $p \to 1^{+}$ . As pointed out in [9], this has great mathematical significance because diffusion processes involving this operator do not have diffusion on different levels.

Considering this approximation process, one can consider its higher-order equivalents, including problems involving the 1-biharmonic operator $Δ_{1}^{2} u = Δ (\frac{Δ u}{| Δ u |})$ , which can be seen as the limit of the several p-biharmonic ones, as the parameter $p \to 1^{+}$ . The difference is that few articles discuss the problem of designing this operator. Indeed, in [18], E. Parini, B. Ruf and C.Tarsi first studied the problem of such operator and dealt with the related eigenvalue problem, and the authors indicated that $\begin{matrix} Λ_{1, 1} (Ω) = inf_{u \in B L_{0} (Ω) ∖ {0}} \frac{\int_{Ω} | Δ u |}{‖ u ‖_{1}} \end{matrix}$ is attained by a non-negative and superharmonic function v that belongs to the space $B L_{0} (Ω) = {u \in W_{0}^{1, 1} (Ω); Δ u \in M (Ω)}$ where $M (Ω)$ is the space of the Radon measures defined on Ω. In fact, their result is more complete, as it also provided information about the shape of the domain Ω that maximizes $Λ_{1, 1} (Ω)$ . In [20], the same authors also deal with the 1-biharmonic operator, because they studied the following minimization problem $\begin{matrix} Λ_{1, 1}^{c} (Ω) = inf_{u \in C_{c}^{\infty} (Ω) ∖ {0}} \frac{\int_{Ω} | Δ u |}{‖ u ‖_{1}} . \end{matrix}$ Similarly, in [20] the authors also study the shape of the subset that maximizes the quantity $Λ_{1, 1}^{c} (Ω)$ . In [19], some optimal constants of Sobolev embeddings in some spaces of functions related to 1-biharmonic operator. In [5], Barile and Pimenta study some existence results of bounded variation solutions to the following quasilinear fourth-order problem $\begin{array}{rcl} \{\begin{matrix} Δ_{1}^{2} u = f (x, u) & in Ω, \\ u = \frac{Δ u}{| Δ u |} = 0 & on \partial Ω, \end{matrix} \end{array}$ where f is superlinear and subcritical at infinity which satisfies the Ambrosetti–Rabinowitz condition and a monotonicity one or f is sublinear. In [10], Figueiredo and Pimenta prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, over a set which has its definition inspired in the Nehari manifold and use the 1-laplacian and the 1-biharmonic operator as applications. In particular, Hurtado, Pimenta and Miyagaki also prove some compactness results of the space of radially symmetric functions of $BL (R^{N})$ and the existence of the ground state solution for the quasilinear elliptic problem $\begin{array}{rcl} \{\begin{matrix} Δ_{1}^{2} - Δ_{1} u + \frac{u}{| u |} = f (u) & in R^{N}, \\ u \in BL (R^{N}), \end{matrix} \end{array}$ where the nonlinearity f satisfies the Ambrosetti–Rabinowitz condition and a monotonicity one in [12].

However, we find that some nonlinear term functions do not satisfy the above conditions, through Berestycki, Gallouët and Kavian still got a ground state solution of the relevant problem under the lack of some superquadricity condition and a monotonicity one in [6]. This make one consider if the relevant result can be generalized to problems involving the 1-biharmonic operator.

Motivated by the aforementioned works [6,8], in this paper, we will use some new tricks to deal with the above results to problem (1.1) under the following assumptions:

$f \in C ((0, + \infty), R)$ ;

${lim}_{s \to 0} f (s) = 0$ ;

${lim}_{s \to + \infty} \frac{f (s)}{s^{p - 1}} = 0$ for some $p \in (1, 1^{*})$ , where $1^{*} = \frac{N}{N - 1}$ ;

There exists $δ > 0$ such that $\begin{array}{rcl} F (δ) = \int_{0}^{δ} f (s) d s > δ; \end{array}$

If $F (s) = \int_{0}^{s} f (t) d t$ and $Q (s) = f (s) s - F (s)$ , then there exists $D ⩾ 1$ such that $\begin{array}{rcl} 0 < Q (s) ⩽ D Q (t), 0 < s ⩽ t, \end{array}$ and $\begin{array}{rcl} lim_{s \to + \infty} Q (s) = + \infty . \end{array}$

The main result in this case is the following.

Theorem 1.1.
Suppose that f satisfies $(f_{1})$ – $(f_{5})$ , then in Problem ( 1.1 ) there exists a nontrivial radial solution $u \in {BL}_{rad} (R^{N})$ (the function space see ( 2.11 )).
Remark 1.2.
Note that these conditions allow f to be an asymptotically constant nonlinearity. Obviously, the function $f (u) = ln (1 + u)$ can satisfy the above assumptions $(f_{1})$ – $(f_{5})$ , but does not satisfy the Ambrosetti–Rabinowitz condition.

The proof of Theorem 1.1 is based on an abstract version of the Mountain Pass Theorem which can be applied to the space of functions. The difficulties arise mainly from the following facts:
The energy functional associated with the Problem (1.1) lacks smoothness;

The space $BL (R^{N})$ lacks reflexivity;

The boundedness of the Palais–Smale sequence is not easy to get.
Therefore, we need to use another version of the Mountain Pass Theorem using analytical tools suitable for non-smooth functionals and the Cerami sequence, which will be defined in the following section.

Finally, we discuss the sublinear case. And, of course, we are thinking about the nonlinear terms being replaced by $λ f (u)$ . In addition to $(f_{1})$ , we consider the following additional assumptions on f:

There exist $0 < q < 1$ , a constant $c_{1} > 0$ and a function $c_{2} \in L^{\infty} (R^{N})$ such that $\begin{array}{rcl} | f (s) | ⩽ c_{1} | s |^{q - 1} + c_{2} (x) for a.e. x \in R^{N} and for every s \in R; \end{array}$

$F (t) ⩾ 0$ and there exists $t_{0} > 0$ such that $\int_{R^{N}} F (t_{0}) d x > 0$ .

The main result in the second case is as following.
Theorem 1.3.
Suppose that f satisfies $(f_{1})$ , $(f_{6})$ and $(f_{7})$ , then there exists $λ^{} > 0$ such that Problem (* 1.1 ) with $f (u)$ substituted by $λ f (u)$ has a nontrivial radial solution $u \in {BL}_{rad} (R^{N})$ for all $λ ⩾ λ^{*}$ .

The proof of Theorem 1.3 is based on the minimization techniques. Indeed, we need to use different parameters to show that the energy functional is negative, which ensures that the global minimum is nontrivial.

In this paper, we will denote by $| \cdot |_{q}$ the $L^{q}$ -norm, where $1 ⩽ q < + \infty$ , and employ $o_{n} (1)$ to denote any quantity which tends to 0 as $n \to \infty$ .

This paper is arranged as follows. In Section 2 we give a detailed description of the properties of the working space defined by the energy functional and the variational framework which is needed in the sequel. In Section 3 we treat the case of superlinear nonlinearities and present the proof of Theorem 1.1. In Section 4 we deal with the sublinear case and prove Theorem 1.3.
2. Preliminaries

First of all, the space we are going to deal with is given by $\begin{array}{rcl} B L (R^{N}) : = {u \in W^{1, 1} (R^{N}) : Δ u \in M (R^{N})}, \end{array}$ where we recall $M (R^{N})$ is the set of Radon measures in $R^{N}$ . By [18], it can be proved that $u \in W^{1, 1} (R^{N})$ belongs to $BL (R^{N})$ if and only if $\begin{array}{rcl} \int_{R^{N}} | Δ u | < + \infty, \end{array}$ where $\begin{array}{rcl} \int_{R^{N}} | Δ u | : = sup {\int_{R^{N}} u Δ φ d x : φ \in C_{0}^{\infty} (R^{N}), | φ |_{\infty} ⩽ 1} . \end{array}$

The space $BL (R^{N})$ is a Banach space when endowed with the following norm $\begin{array}{rcl} ‖ u ‖ = \int_{R^{N}} | Δ u | + | \nabla u |_{1} + | u |_{1}, \end{array}$ which is continuously embedded into $L^{r} (R^{N})$ for all $r \in [1, 1^{*}]$ (see [12]).

Moreover, as can be seen, the space of smooth functions is not dense in $BL (R^{N})$ with respect to the topology of the norm. However, it is with respect to the topology induced by the following notion of convergence, which has motivated people to define a weaker sense of convergence in $BL (R^{N})$ . We say that a sequence $(u_{n}) \subset BL (R^{N})$ converges to $u \in BL (R^{N})$ in the sense of the strict convergence if both of the following conditions are satisfied $\begin{array}{rcl} u_{n} \to u in W^{1, 1} (R^{N}), \end{array}$ and $\begin{array}{rcl} \int_{R^{N}} | Δ u_{n} | \to \int_{R^{N}} | Δ u |, \end{array}$ as $n \to + \infty$ . In fact, with respect to the strict convergence, $C^{\infty} (R^{N}) \cap BL (R^{N})$ is dense in $BL (R^{N})$ and $C_{0}^{\infty} (R^{N})$ is dense in $BL (R^{N})$ .

For a vectorial Radon measure $μ \in M (R^{N}, R^{N})$ , we denote by $μ = μ^{a} + μ^{s}$ the usual decomposition stated in the Radon–Nikodym Theorem, where $μ^{a}$ and $μ^{s}$ are, respectively, the absolute continuous and the singular parts with respect to the N-dimensional Lebesgue measure $L^{N}$ . With $| μ |$ as the scalar Radon measure, the usual Lebesgue–Radon–Nikodym derivative of μ with respect to $| μ |$ is given by $\begin{array}{rcl} \frac{μ}{| μ |} (x) = lim_{r \to 0} \frac{μ (B_{r} (x))}{| μ | (B_{r} (x))} . \end{array}$ It is easy to see that $J : BL (R^{N}) \to R$ , given by $\begin{array}{rcl} (2.1) & J (u) = \int_{R^{N}} | Δ u | + \int_{R^{N}} | \nabla u | d x + \int_{R^{N}} | u | d x \end{array}$ is a convex functional which is Lipschitz continuous in its domain and lower semicontinuous with respect to the $W^{1, r} (R^{N})$ topology, for $r \in [1, 1^{*}]$ . Meanwhile, $J$ is lower semicontinuous with respect to the $L^{r} (R^{N})$ topology, for $r \in [1, 1^{*})$ (see [12]). Although nonsmooth, the functional $J$ admits some directional derivatives. More precisely, as is shown in [4], given $u \in BL (R^{N})$ , for all $v \in BL (R^{N})$ such that ${(Δ v)}^{s}$ is absolutely continuous with respect to ${(Δ u)}^{s}$ , ${(Δ v)}^{a}$ vanishes $L^{N}$ -a.e. in ${x \in R^{N}; {(Δ u)}^{a} (x) = 0}$ , $\nabla v$ vanishes a.e. in the set where $\nabla u$ vanishes and $v \equiv 0$ , a.e. in the set where u vanishes, it follows that $\begin{array}{rcl} (2.2) & \begin{matrix} J^{'} (u) v = & \int_{R^{N}} \frac{{(Δ u)}^{a} {(Δ v)}^{a}}{| {(Δ u)}^{a} |} d x + \int_{R^{N}} \frac{Δ u}{| Δ u |} (x) \frac{Δ v}{| Δ v |} (x) | {(Δ v)}^{s} | + \int_{R^{N}} \frac{\nabla u \cdot \nabla v}{| \nabla u |} d x \\ + \int_{R^{N}} sgn (u) v d x, \end{matrix} \end{array}$ where $sgn (u (x)) = 0$ if $u (x) = 0$ and $sgn (u (x)) = u (x) / | u (x) |$ if $u (x) \neq 0$ . In particular, taking (2.2) into account, for all $u \in BL (R^{N})$ , we have $\begin{array}{rcl} (2.3) & J^{'} (u) u = J (u) . \end{array}$

Next the energy functional associated with (1.1) will be presented. Let $Φ : BL (R^{N}) \to R$ be given by $\begin{array}{rcl} Φ (u) = J (u) - F (u), \end{array}$ where $J$ is defined as in (2.1) and $F : BL (R^{N}) \to R$ is defined by $\begin{array}{rcl} F (u) = \int_{R^{N}} F (u) d x . \end{array}$ It can be a plain matter to prove that $F \in C^{1} (BL (R^{N}), R)$ . Moreover, taking $v = u$ in (2.2), it follows that $\begin{array}{rcl} (2.4) & Φ^{'} (u) u = J^{'} (u) u - \int_{R^{N}} f (u) u d x = ‖ u ‖ - \int_{R^{N}} f (u) u d x . \end{array}$

Now a precise definition of the solution we are considering will be given. Since Φ can be written as the difference between the Lipschitz functional $J$ and a smooth functional $F$ , we say that $u_{0} \in BL (R^{N})$ is a solution of (1.1) if $0 \in \partial Φ (u_{0})$ , where $\partial Φ (u_{0})$ denotes the subdifferential of Φ in $u_{0}$ , as defined in [7]. This, in turn, is equivalent to $F^{'} (u_{0}) \in \partial J (u_{0})$ . However, since the convexity of $J$ , it implies that $F^{'} (u_{0}) \in \partial J (u_{0})$ if and only if $\begin{array}{rcl} J (v) - J (u_{0}) ⩾ F^{'} (u_{0}) (v - u_{0}), \forall v \in BL (R^{N}), \end{array}$ or equivalently $\begin{array}{rcl} (2.5) & ‖ v ‖ - ‖ u_{0} ‖ ⩾ \int_{R} f (u_{0}) (v - u_{0}) d x, \forall v \in BL (R^{N}) . \end{array}$ Hence, every $u_{0} \in BL (R^{N})$ such that (2.5) holds is going to be called a solution of (1.1).

In fact, it is possible to show that if $u_{0} \in BL (R^{N})$ satisfies (2.5), then there exist a function $γ \in L_{\infty, N} (R^{N})$ and a vector field $z \in W^{1, 1} (R^{N}) \cap L^{\infty} (R^{N}, R^{N})$ such that $| z |_{\infty} ⩽ 1$ and $\begin{array}{rcl} \{\begin{matrix} div z \in L_{\infty, N} (R^{N}), Δ z \in L_{\infty, N} (R^{N}), \\ \int_{R^{N}} u_{0} Δ z - \int_{R^{N}} u_{0} div z d x = \int_{R^{N}} | Δ u_{0} | + \int_{R^{N}} | \nabla u_{0} | d x, \\ γ | u_{0} | = u_{0} a.e. in R^{N}, \\ Δ z - div z + γ = f (u_{0}), a.e. in R^{N}, \end{matrix} \end{array}$ where $\begin{array}{rcl} L_{\infty, N} (R^{N}) = {g : R^{N} \to R measurable; | g |_{\infty, N} < \infty}, \end{array}$ and $\begin{array}{rcl} | g |_{\infty, N} = sup_{| ϕ |_{1} + | ϕ |_{1^{*}} ⩽ 1} | \int_{R^{N}} g ϕ d x | . \end{array}$

Because of non-smooth, we will give the following details (see [8]). Since the nonlinearity has changed, it is far from becoming trivial to prove the boundedness of the Palais–Smale sequence which comes from the Mountain Pass Theorem. Therefore, it is necessary to use Cerami sequences to prove this theorem.

Definition 2.1 (Cerami sequence).

For a functional $Φ = I_{0} - I$ where $I \in C^{1} (E, R)$ and $I_{0}$ is a locally Lipschitz convex functional defined in a Banach space E. For $d \in R$ , we say that ${(x_{n})}_{n \in N} \subset E$ is a Cerami sequence on the level d, if there exists ${(z_{n})}_{n \in N} \subset E^{*}$ , such that $\begin{array}{l} Φ (x_{n}) = d + o_{n} (1), \\ ‖ z_{n} ‖_{E^{*}} (‖ x_{n} ‖ + 1) = o_{n} (1), \end{array}$ and $\begin{array}{l} I_{0} (y) - I_{0} (x_{n}) ⩾ I^{'} (x_{n}) (y - x_{n}) + {⟨ z_{n}, y - x_{n} ⟩}_{E^{*}, E}, \forall y \in E . \end{array}$

Next, we make use of a version of the Mountain Pass Theorem for Lipschitz functionals.

Lemma 2.2 (Mountain Pass Theorem).

Let E be a Banach space, $Φ = I_{0} - I$ where $I \in C^{1} (E, R)$ and $I_{0}$ a locally Lipschitz convex functional defined in E. Suppose that the functional Φ satisfies the following geometric conditions:

There exist $ρ > 0$ , $α > Φ (0)$ such that $Φ |_{\partial B_{ρ} (0)} ⩾ α$ ,

$Φ (e) < Φ (0)$ for some $e \in E ∖ \overline{B_{ρ} (0)}$ .

Then for all $ϵ > 0$ , there exist $x_{ϵ} \in E$ and $z_{ϵ} \in E^{*}$ , such that $\begin{array}{l} (2.6) & c - 2 ϵ ⩽ Φ (x_{ϵ}) ⩽ c + 2 ϵ, \\ (2.7) & ‖ z_{ϵ} ‖_{E^{*}} (‖ x_{ϵ} ‖ + 1) ⩽ 8 ϵ, \end{array}$ and $\begin{array}{rcl} (2.8) & I_{0} (y) - I_{0} (x_{ϵ}) ⩾ I^{'} (x_{ϵ}) (y - x_{ϵ}) + {⟨ z_{ϵ}, y - x_{ϵ} ⟩}_{E^{*}, E}, \forall y \in E, \end{array}$ where $\begin{array}{rcl} (2.9) & c : = inf_{γ \in Γ} sup_{t \in [0, 1]} Φ (γ (t)), \end{array}$ and $Γ = {γ \in C^{0} ([0, 1], E); γ (0) = 0, γ (1) = e}$ .

The Lions’ Lemma in $BL (R^{N})$ is very important to get the boundness of the Cerami sequence, we will give its proof by drawing on the literature [13]. We consider it is interesting in its own way because it is a classical and largely used tool in the analysis of quasilinear elliptic problems.

Theorem 2.3 (Lions’ Lemma in $BL (R^{N})$ ).

Suppose there exist $R > 0$ , $1 ⩽ q < 1^{*}$ , and a bounded sequence $(u_{n})$ in $BL (R^{N})$ such that $\begin{array}{rcl} sup_{y \in R^{N}} \int_{B_{R} (y)} | u_{n} |^{q} d x \to 0, as n \to \infty . \end{array}$ Then $u_{n} \to 0$ in $L^{s} (R^{N})$ for all $s \in (1, 1^{*})$ .

Proof.
Let $q < s < 1^{}$ and $u \in BL (R^{N})$ . Since $BL (R^{N}) ↪ L^{r} (R^{N})$ for all $r \in [1, 1^{}]$ , then $u \in L^{q} (R^{N})$ and $u \in L^{1^{}} (R^{N})$ .

For $R > 0$ , by interpolation inequality with $θ = \frac{s - q}{1^{} - q} \frac{1^{}}{s}$ and embedding inequality, it implies that $0 < θ < 1$ and $\begin{array}{rcl} | u |_{L^{s} (B_{R} (y))} & ⩽ & | u |_{L^{q} (B_{R} (y))}^{1 - θ} | u |_{L^{1^{} (B_{R} (y))}}^{θ} \\ ⩽ & c | u |_{L^{q} (B_{R} (y))}^{1 - θ} ‖ u ‖_{B L (B_{R} (y))}^{θ} . \end{array}$ Let us cover $R^{N}$ by balls of radius R and center in $(y_{n})$ in such a way that each point in $R^{N}$ belongs to the maximum $N + 1$ balls, we obtain that $\begin{array}{l} \int_{R^{N}} | u |^{s} d x \\ ⩽ \sum_{n = 1}^{+ \infty} \int_{B_{R} (y_{n})} | u |^{s} d x \\ ⩽ c \sum_{n = 1}^{+ \infty} | u |_{L^{q} (B_{R} (y_{n}))}^{(1 - θ)} ‖ u ‖_{B L (B_{R} (y_{n}))}^{θ s} \\ ⩽ c {(sup_{y \in R^{N}} \int_{B_{R} (y)} | u |^{q} d x)}^{\frac{(1 - θ) s}{q}} \\ \times lim_{k \to + \infty} \sum_{n = 1}^{k} {(\int_{B_{R} (y_{n})} | Δ u | + \int_{B_{R} (y_{n})} | \nabla u | d x + \int_{B_{R} (y_{n})} | u | d x)}^{θ s} \\ = c {(sup_{y \in R^{N}} \int_{B_{R} (y)} | u |^{q} d x)}^{\frac{(1 - θ) s}{q}} lim_{k \to + \infty} \sum_{n = 1}^{k} {(\int_{R^{N} χ_{R} (y_{n})} | Δ u | + \int_{R^{N} χ_{R} (y_{n})} (| \nabla u | + | u |) d x)}^{θ s} \\ = c {(sup_{y \in R^{N}} \int_{B_{R} (y)} | u |^{q} d x)}^{\frac{(1 - θ) s}{q}} \\ \times lim_{k \to + \infty} {(\int_{R^{N}} \sum_{n = 1}^{k} χ_{B_{R} (y_{n})} | Δ u | + \int_{R^{N}} \sum_{n = 1}^{k} χ_{B_{R} (y_{n})} (| \nabla u | + | u |) d x)}^{θ s} \\ ⩽ c {(sup_{y \in R^{N}} \int_{B_{R} (y)} | u |^{q} d x)}^{\frac{(1 - θ) s}{q}} (N + 1) ‖ u ‖^{θ s} . \end{array}$ Suppose that $(u_{n})$ is bounded in $BL (R^{N})$ , by the last inequality and the hypothesis, we have $\begin{array}{l} (2.10) & u_{n} \to 0 in L^{s} (R^{N}), \end{array}$ for all $q < s < 1^{}$ .

Then, if $q = 1$ we are done. Otherwise, if $1 < q < 1^{}$ , let us consider $1 < s ⩽ q$ and take $s_{0} \in (q, 1^{})$ in such a way that (2.10) holds. Note that $u \in L^{1} (R^{N}) \cap L^{s_{0}} (R^{N})$ and, since $s \in (1, s_{0})$ , by doing $\begin{matrix} θ = \frac{s_{0} - s}{s (s_{0} - 1)}, \end{matrix}$ we have that $\begin{array}{l} \frac{1}{s} = \frac{θ}{1} + \frac{1 - θ}{s_{0}} and 0 < θ < 1 . \end{array}$ Then, again by interpolation inequality, the embedding of $BL (R^{N})$ and (2.10), it implies that $\begin{matrix} | u_{n} |_{s} ⩽ | u_{n} |_{1} | u_{n} |_{s_{0}} ⩽ ‖ u_{n} ‖ | u_{n} |_{s_{0}} \to 0, \end{matrix}$ as $n \to \infty$ , since $(u_{n})$ is bounded in $BL (R^{N})$ . □

In order to use a variational approach to prove our results, it is necessary for us to study an energy space which is compactly embedded into some Lebesgue space. Hence, let us work with $B L_{rad} (R^{N})$ , defined by $\begin{array}{rcl} (2.11) & B L_{rad} (R^{N}) = {u \in B L (R^{N}); u (x) = u (| x |)} . \end{array}$ It can be known that $B L_{rad} (R^{N})$ is compactly embedded into $L^{r} (R^{N})$ , for all $r \in (1, 1^{})$ (see [12]). From this point on, we are going to consider the functional Φ restricted to this space, $Φ |_{B L_{rad} (R^{N})}$ , and denote it still by Φ.
3. Superlinear nonlinearities

Now we begin to investigate the existence of solutions in the superlinear case, which will give proof of Theorem 1.1.

First of all, we verify that Φ satisfies the conditions of Lemma 2.2.

Proposition 3.1.
The functional Φ satisfies $(g_{1})$ and $(g_{2})$ .
Proof.
We start to verify the first condition. Note that, from $(f_{2})$ – $(f_{3})$ , for all $η > 0$ , there exists $A_{η} > 0$ such that $\begin{array}{rcl} (3.1) & F (s) ⩽ η | s | + A_{η} | s |^{p}, \forall s \in R, \end{array}$ where p is as in $(f_{3})$ . Then, by (3.1) and the continuous embeddings of $B L_{rad} (R^{N})$ we have that $\begin{array}{rcl} Φ (u) & = & \int_{R^{N}} | Δ u | + \int_{R^{N}} | \nabla u | d x + \int_{R^{N}} | u | d x - \int_{R^{N}} F (u) d x \\ = & ‖ u ‖ - \int_{R^{N}} F (u) d x \\ ⩾ & ‖ u ‖ - η \int_{R^{N}} | u | d x - A_{η} \int_{R^{N}} | u |^{p} d x \\ ⩾ & ‖ u ‖ - η C ‖ u ‖ - A_{η} C ‖ u ‖^{p} \\ (3.2) & = & (1 - η C) ‖ u ‖ - A_{η} C ‖ u ‖^{p} . \end{array}$ Let us consider $η > 0$ such that $(1 - η C) > 0$ and ρ such that $\begin{array}{l} 0 < ρ < {(\frac{1 - η C}{A_{η} C})}^{\frac{1}{p - 1}} . \end{array}$ Hence, from (3.2), it implies that $\begin{array}{l} (3.3) & Φ (u) ⩾ α > 0, \end{array}$ for all $u \in B L_{rad} (R^{N})$ such that $‖ u ‖ = ρ$ , where $α = (1 - η C) ρ + A_{η} C ρ^{p} > 0$ . Hence we have verified the condition $(g_{1})$ in Lemma 2.2.

For $(g_{2})$ , the condition $(f_{4})$ will be used to verify that there exists $u \in B L_{rad} (R^{N})$ , such that $\begin{array}{l} \int_{R^{N}} (F (u) - | u |) d x > 0 . \end{array}$ For $R > 0$ to be chosen later, let us define $u_{R} : R^{N} \to R$ by $\begin{array}{l} (3.4) & u_{R} (x) = \{\begin{matrix} δ, & if | x | ⩽ R, \\ δ (R + 1 - r), & if r = | x | \in [R, R + 1], \\ 0, & if | x | > R + 1 . \end{matrix} \end{array}$ Note that, $u_{R} (x) = u_{R} (| x |)$ , $u_{R} \in W_{rad}^{1, 1} (R^{N})$ and, as a distribution, $\begin{array}{l} Δ u_{R} = - δ \frac{(N - 1)}{r} χ_{B_{R + 1} ∖ B_{R}} + δ H_{N - 1} |_{\partial B_{R + 1}} - δ H_{N - 1} |_{\partial B_{R}} \in M (R^{N}) . \end{array}$ Thus $u_{R} \in {BL}_{rad} (R^{N})$ and $\begin{array}{rcl} \int_{B_{R + 1} ∖ B_{R}} (F (u_{R}) - | u_{R} |) d x & ⩾ & - max_{| s | ⩽ δ} (F (s) + | s |) | B_{R + 1} ∖ B_{R} | \\ ⩾ & - max_{| s | ⩽ δ} | F (s) - | s | | | B_{R + 1} ∖ B_{R} |, \end{array}$ and then we obtain $\begin{matrix} \int_{R^{N}} (F (u_{R}) - | u_{R} |) d x & = \int_{B_{R}} (F (u_{R}) - | u_{R} |) d x + \int_{B_{R + 1} ∖ B_{R}} (F (u_{R}) - | u_{R} |) d x \\ ⩾ (F (δ) - δ) | B_{R} | - max_{| s | ⩽ δ} | F (s) - | s | | | B_{R + 1} ∖ B_{R} | . \end{matrix}$ Hence, note that $(f_{4})$ shows that there exists $C_{1}$ , $C_{2} > 0$ such that $\begin{array}{rcl} \int_{R^{N}} (F (u_{R}) - | u_{R} |) d x ⩾ C_{1} R^{N} - C_{2} R^{N - 1} . \end{array}$ Hence choosing $R > 0$ large enough, we have $\begin{array}{rcl} (3.5) & \int_{R^{N}} (F (u_{R}) - | u_{R} |) d x > 0 . \end{array}$ For $t > 0$ , let us define $\begin{array}{l} e_{t} (x) = u_{R} (\frac{x}{t}) . \end{array}$ It is easy to see that $e_{t} \in B L_{rad} (R^{N})$ . In addition, note that $\begin{array}{l} Φ (e_{t}) ⩽ t^{N - 2} \int_{R^{N}} | Δ u_{R} | + t^{N - 1} \int_{R^{N}} | \nabla u_{R} | d x - t^{N} \int_{R^{N}} (F (u_{R}) - | u_{R} |) d x . \end{array}$ Then, from (3.5), it implies that $\begin{array}{l} Φ (e_{t}) \to - \infty, as t \to + \infty . \end{array}$ Hence there exists $t > 0$ such that $\begin{array}{l} Φ (e_{t}) < 0 . \end{array}$ □

Therefore, the functional Φ satisfies the conditions $(g_{1})$ and $(g_{2})$ of Lemma 2.2 and then there exists a sequence $(u_{n}) \subset {BL}_{rad} (R^{N})$ and $(z_{n}) \subset {({BL}_{rad} (R^{N}))}^{}$ such that $\begin{matrix} (3.6) & \begin{matrix} Φ (u_{n}) \to c, as n \to + \infty, \\ ‖ z_{n} ‖_{} (‖ u_{n} ‖ + 1) \to 0, as n \to + \infty, \end{matrix} \end{matrix}$ and $\begin{array}{rcl} (3.7) & ‖ v ‖ - ‖ u_{n} ‖ ⩾ \int_{R^{N}} f (u_{n}) (v - u_{n}) d x + ⟨ z_{n}, v - u_{n} ⟩, \end{array}$ where $‖ \cdot ‖_{}$ denotes the norm in ${({BL}_{rad} (R^{N}))}^{}$ and $\begin{array}{rcl} (3.8) & c = inf_{γ \in Γ} max_{t \in [0, 1]} Φ (γ (t)), \end{array}$ with $\begin{array}{rcl} (3.9) & Γ = {γ \in C ([0, 1], {BL}_{rad} (R^{N}); γ (0) = 0, γ (1) = e} . \end{array}$ Hence, in (3.7), let us take as test function $v = u_{n} + t u_{n}$ , divided by t and make $t \to 0^{+}$ and $t \to 0^{-}$ , it follows from (2.4) that $\begin{array}{rcl} (3.10) & Φ^{'} (u_{n}) u_{n} = ‖ u_{n} ‖ - \int_{R^{N}} f (u_{n}) u_{n} d x = ⟨ z_{n}, u_{n} ⟩ = o_{n} (1), \end{array}$ where the last equality follows by (3.6).

Now let us study related information about the sequence $(u_{n})$ mentioned above. Proposition 3.2.
The sequence $(u_{n})$ is bounded in ${BL}_{rad} (R^{N})$ . Moreover, there exists $u \in B L_{rad} (R^{N})$ such that $u_{n} \to u$ in $L^{q} (R^{N})$ for all $1 < q < 1^{}$ .
Proof.
We prove that $(u_{n})$ is bounded by contradiction that $‖ u_{n} ‖ \to \infty$ as $n \to + \infty$ . Let us define ${\hat{u}}_{n} : = u_{n} / ‖ u_{n} ‖$ and note that ${\hat{u}}_{n}$ is a bounded sequence with $‖ {\hat{u}}_{n} ‖ = 1$ . One of the following two cases occurs:
${lim sup}_{n \to \infty} {sup}_{y \in R^{N}} \int_{B_{1} (y)} | {\hat{u}}_{n} | d x > 0$ ,

${lim sup}_{n \to \infty} {sup}_{y \in R^{N}} \int_{B_{1} (y)} | {\hat{u}}_{n} | d x = 0$ .

Suppose that (2) occurs. For $L > 1$ , note that $\begin{array}{rcl} (3.11) & Φ (\frac{L}{‖ u_{n} ‖} u_{n}) & = & ‖ \frac{L}{‖ u_{n} ‖} u_{n} ‖ - \int_{R^{N}} F (\frac{L}{‖ u_{n} ‖} u_{n}) d x = L - \int_{R^{N}} F (\frac{L}{‖ u_{n} ‖} u_{n}) d x . \end{array}$ Given $ϵ > 0$ , by (3.1), there exists $C_{ϵ} > 0$ such that $\begin{array}{rcl} (3.12) & \int_{R^{N}} F (\frac{L}{‖ u_{n} ‖} u_{n}) d x ⩽ ϵ L \int_{R^{N}} | {\hat{u}}_{n} | d x + C_{ϵ} L^{p} \int_{R^{N}} | {\hat{u}}_{n} |^{p} d x, \end{array}$ where p is as in $(f_{3})$ . By Theorem 2.3, it follows that $\begin{array}{rcl} (3.13) & lim_{n \to + \infty} \int_{R^{N}} | {\hat{u}}_{n} |^{p} d x = 0 . \end{array}$ Thus, by (3.12) and (3.13), $\begin{array}{rcl} (3.14) & \int_{R^{N}} F (\frac{L}{‖ u_{n} ‖} u_{n}) d x ⩽ ϵ L + o_{n} (1) . \end{array}$ For $ϵ = 1 / 2$ , by (3.11) and (3.14), it follows that $\begin{array}{rcl} (3.15) & Φ (\frac{L}{‖ u_{n} ‖} u_{n}) ⩾ \frac{L}{2} - o_{n} (1) . \end{array}$ Since ${lim}_{n \to + \infty} ‖ u_{n} ‖ = + \infty$ , we can assume that $\begin{array}{rcl} \frac{L}{‖ u_{n} ‖} \in (0, 1) for all n large enough. \end{array}$ Hence, we have $\begin{array}{rcl} max_{t \in [0, 1]} Φ (t u_{n}) ⩾ Φ (\frac{L}{‖ u_{n} ‖} u_{n}) ⩾ \frac{L}{2} - o_{n} (1) . \end{array}$ Let $t_{n} \in (0, 1]$ be such that $Φ (t_{n} u_{n}) = {max}_{t \in [0, 1]} Φ (t u_{n})$ . Note that $t_{n} \neq 1$ since, if $t_{n} = 1$ then by (3.15) we obtain that $\begin{array}{rcl} (3.16) & Φ (u_{n}) ⩾ \frac{L}{2} - o_{n} (1) . \end{array}$ Meanwhile, we have $\begin{array}{rcl} (3.17) & Φ (u_{n}) = ‖ u_{n} ‖ - \int_{R^{N}} F (u_{n}) d x \to c, as n \to \infty . \end{array}$ Thus from (3.16) and (3.17) for $L > 1$ large enough we have a contradiction. Hence, $t_{n} \in (0, 1)$ . On the other hand, note that $h (t) = Φ (t u_{n})$ is a $C^{1} (R)$ function, since by $(f_{1})$ and (2.2), $\begin{array}{rcl} h^{'} (t) = Φ^{'} (t u_{n}) u_{n} = ‖ u_{n} ‖ - \int_{R^{N}} f (t u_{n}) u_{n} d x \end{array}$ is continuous. Hence, since $t_{n} \in (0, 1)$ is an extremal point of the smooth function h in $[0, 1]$ , it implies that $Φ^{'} (t_{n} u_{n}) u_{n} = 0$ . Then $\begin{array}{rcl} Φ (t_{n} u_{n}) & = & Φ (t_{n} u_{n}) - Φ^{'} (t_{n} u_{n}) t_{n} u_{n} \\ = & \int_{R^{N}} (f (t_{n} u_{n}) t_{n} u_{n} - F (t_{n} u_{n})) d x \\ ⩽ & D \int_{R^{N}} (f (u_{n}) u_{n} - F (u_{n})) d x \\ = & D (Φ (u_{n}) - Φ^{'} (u_{n}) u_{n}) \\ (3.18) & = & D c + o_{n} (1), \end{array}$ where D is given in $(f_{5})$ . From (3.16) and (3.18), it follows that $\begin{array}{rcl} \frac{L}{2} - o_{n} (1) ⩽ Φ (t_{n} u_{n}) ⩽ D c + o_{n} (1), \end{array}$ and making $L > 1$ large enough, we obtain a contradiction. And the case (2) does not occur.

Now let us consider the case (1). If $(y_{n})$ is such that $| y_{n} | \to \infty$ and $\int_{B_{1} (y_{n})} | {\hat{u}}_{n} | d x > \frac{δ}{2}$ , then $\begin{array}{rcl} (3.19) & \int_{B_{1} (0)} | {\hat{u}}_{n} (x + y_{n}) | d x > \frac{δ}{2} . \end{array}$ Note that ${\hat{u}}_{n} (\cdot + y_{n})$ is bounded in $BL (R^{N})$ and then also in $B L (B_{1} (0))$ . Then, by the compact embedding of $B L (B_{1} (0))$ into Lebesgue spaces, we have $\begin{array}{rcl} {\hat{u}}_{n} (\cdot + y_{n}) \to \hat{u} in L^{p} (B_{1} (0)), \end{array}$ for all $1 ⩽ p < 1^{}$ . Consequently, doing $n \to \infty$ in (3.19), we have $\begin{array}{rcl} (3.20) & \int_{B_{1} (0)} | \hat{u} | d x > \frac{δ}{2} . \end{array}$ Thus, we can infer that $\hat{u} \neq 0$ . Hence, there exists $Ω \subset B_{1} (0)$ such that $| Ω | > 0$ and $\begin{array}{rcl} 0 < | \hat{u} (x) | = lim_{n \to \infty} | \hat{u} (x + y_{n}) | = lim_{n \to \infty} \frac{| u_{n} (x + y_{n}) |}{‖ u_{n} ‖}, a.e. in Ω . \end{array}$ Then, by the last inequality and from the fact that $‖ u_{n} ‖ \to \infty$ , we obtain $\begin{array}{rcl} u_{n} (x + y_{n}) \to \infty, a.e. in Ω . \end{array}$ Hence from $(f_{5})$ and Fatou’s Lemma, we have $\begin{array}{l} \underset{n \to \infty}{lim inf} \int_{R^{N}} (f (u_{n}) u_{n} - F (u_{n})) d x \\ ⩾ \underset{n \to \infty}{lim inf} \int_{Ω} (f (u_{n} (x + y_{n})) u_{n} (x + y_{n}) - F (u_{n} (x + y_{n}))) d x \\ ⩾ \int_{Ω} \underset{n \to \infty}{lim inf} (f (u_{n} (x + y_{n})) u_{n} (x + y_{n}) - F (u_{n} (x + y_{n}))) d x \\ (3.21) & = \infty . \end{array}$ By (3.10) we have $\begin{array}{rcl} (3.22) & \int_{R^{N}} (f (u_{n}) u_{n} - F (u_{n})) d x = Φ (u_{n}) - Φ^{'} (u_{n}) u_{n} = c + o_{n} (1) . \end{array}$ Thus, from (3.21) and (3.22), we have a contradiction in case (1), under the assumption that $| y_{n} | \to \infty$ .

What should be done is to consider the case in which $(y_{n})$ is a bounded sequence in $R^{N}$ . Then, let $R > 1$ such that $| y_{n} | ⩽ R$ , for all $n \in N$ . Then $\begin{array}{rcl} \frac{δ}{2} < \int_{B_{1} (y_{n})} | {\hat{u}}_{n} | d x ⩽ \int_{B_{2 R} (0)} | {\hat{u}}_{n} (x + y_{n}) | d x . \end{array}$ Since $({\hat{u}}_{n})$ is bounded in $B L (B_{2 R} (0))$ (and also in $BL (R^{N})$ ), there exists $\hat{u} \in BL (R^{N})$ such that $\begin{array}{rcl} {\hat{u}}_{n} (x + y_{n}) \to \hat{u} in L^{1} (B_{2 R} (0)), \end{array}$ where $\begin{array}{rcl} \frac{δ}{2} ⩽ \int_{R^{N}} | \hat{u} | d x . \end{array}$ Hence, as in the previous case, there exists $Ω \subset B_{1} (0)$ such that $| Ω | > 0$ and $\begin{array}{rcl} (3.23) & 0 < | \hat{u} (x) | = lim_{n \to \infty} | \hat{u} (x + y_{n}) | = lim_{n \to \infty} \frac{| u_{n} (x + y_{n}) |}{‖ u_{n} ‖}, a.e. in Ω . \end{array}$ Then this will lead us to a contradiction as in (3.21) and (3.22). Hence the sequence $(u_{n})$ is bounded in $B L_{rad} (R^{N})$ .

Moreover, from the compact embeddings of $B L_{rad} (R^{N})$ , it follows that there exists $u \in B L_{rad} (R^{N})$ such that $u_{n} \to u$ in $L^{q} (R^{N})$ for all $1 < q < 1^{}$ . □
Proof of Theorem 1.1.
By Proposition 3.1 and 3.2, we can obtain the following claim: $\begin{array}{rcl} (3.24) & \int_{R^{N}} f (u_{n}) u_{n} d x = \int_{R^{N}} f (u) u d x + o_{n} (1) . \end{array}$ Indeed, since $u_{n} \to u$ in $L^{p} (R^{N})$ , then there exists $R > 0$ such that $\begin{array}{rcl} (3.25) & \int_{B_{R}^{c} (0)} | u_{n} |^{p} d x < η . \end{array}$ Since $(u_{n})$ is bounded in $L^{1} (R^{N})$ , there exists $C > 0$ such that $| u_{n} |_{1} ⩽ C$ , for all $n \in N$ . Thus, from (2.10) and (3.25), we get $\begin{array}{rcl} (3.26) & \int_{B_{R}^{c} (0)} f (u_{n}) u_{n} d x ⩽ (C + A_{η}) η . \end{array}$ Since $u_{n} \to u$ in $L^{r} (B_{R} (0))$ for all $r \in [1, 1^{})$ , by using the Lebesgue Dominated Convergence Theorem we obtain $\begin{array}{rcl} (3.27) & \int_{B_{R} (0)} f (u_{n}) u_{n} d x = \int_{B_{R} (0)} f (u) u d x + o_{n} (1) . \end{array}$ Hence, (3.26), (3.27) and the integrability of $f (u (\cdot)) u (\cdot)$ imply the claim.

Calculating the limsup as $n \to \infty$ in both sides of (3.7) and taking into account the last claim and the lower semicontinuity of the norm in $BL (R^{N})$ with respect to the $W^{1, 1} (R^{N})$ convergence, it follows that $\begin{array}{rcl} (3.28) & ‖ v ‖ - ‖ u ‖ ⩾ \int_{R^{N}} f (u) (v - u) d x, for all v \in {BL}_{rad} (R^{N}) . \end{array}$

Moreover, by the Symmetric Criticality Principle of Palais given in [21], it follows that (3.28) holds for every $v \in BL (R^{N})$ , in such a way that u is in fact a radially symmetric solution of (1.1).

Finally, let us show that $u \in B L_{rad} (R^{N})$ is nontrivial. Taking $v = u + t u$ in (3.28) and doing $t \to 0^{\pm}$ , we have that $\begin{array}{rcl} (3.29) & ‖ u ‖ = \int_{R^{N}} f (u) u d x . \end{array}$ Thus, from (3.10), (3.24) and (3.29), it implies that $\begin{array}{rcl} (3.30) & ‖ u ‖ = \int_{R^{N}} f (u) u d x = \int_{R^{N}} f (u_{n}) u_{n} d x + o_{n} (1) = ‖ u_{n} ‖ + o_{n} (1) . \end{array}$ Hence, from the compact embeddings of ${BL}_{rad} (R^{N})$ , it can be easing seen that $\begin{array}{rcl} (3.31) & lim_{n \to \infty} \int_{R^{N}} F (u_{n}) d x = \int_{R^{N}} F (u) d x . \end{array}$ Then, from (3.6), (3.30) and (3.31) we get $\begin{matrix} Φ (u) & = ‖ u ‖ - \int_{R^{N}} F (u) d x \\ = ‖ u_{n} ‖ - \int_{R^{N}} F (u_{n}) d x + o_{n} (1) \\ = c + o_{n} (1) . \end{matrix}$ Since $c > 0$ , then $u \neq 0$ . Then we have completed the proof of Theorem 1.1. □

4. Sublinear nonlinearities

In this section, let us prove that there exists a nontrivial solution to the following problem $\begin{array}{rcl} (4.1) & \{\begin{matrix} Δ_{1}^{2} - Δ_{1} u + \frac{u}{| u |} = λ f (u) & in R^{N}, \\ u \in {BL}_{rad} (R^{N}), \end{matrix} \end{array}$ where f satisfies $(f_{1})$ , $(f_{6})$ and $(f_{7})$ .

Proof of Theorem 1.3.

Firstly, let us verify that the energy functional Φ is coercive in $B L_{rad} (R^{N})$ . In fact, it follows from $(f_{6})$ , Hölder inequality and the continuous embeddings of ${BL}_{rad} (R^{N})$ that $\begin{array}{rcl} Φ (u) & = & ‖ u ‖ - λ \int_{R^{N}} F (u) d x \\ ⩾ & ‖ u ‖ - c_{1} λ \int_{R^{N}} | u |^{q} d x - λ \int_{R^{N}} c_{2} (x) d x \\ ⩾ & ‖ u ‖ - C | u |_{1}^{q} - C \\ (4.2) & ⩾ & ‖ u ‖ - C ‖ u ‖^{q} - C, \end{array}$ which implies that Φ is coercive and also bounded from below.

Suppose that $(u_{n}) \subset B L_{rad} (R^{N})$ is a minimizing sequence, i.e., $Φ (u_{n}) \to ς$ , where $ς = {inf}_{B L_{rad} (R^{N})} Φ$ . The coerciveness of Φ show that $(u_{n})$ is a bounded sequence in $B L_{rad} (R^{N})$ . Then, by the boundedness of $(u_{n})$ and the compactness of the embeddings of $B L_{rad} (R^{N})$ in $L^{r} (R^{N})$ for all $1 < r < 1^{*}$ , it follows that there exists $u \in B L_{rad} (R^{N})$ such that $\begin{array}{rcl} (4.3) & u_{n} \to u in L^{r} (R^{N}) for all 1 < r < 1^{*} . \end{array}$ Thus, by the lower semicontinuity of Φ with respect to the $W^{1, r} (R^{N})$ $\begin{array}{rcl} Φ (u) & = & ‖ u ‖ - λ \int_{R^{N}} F (u) d x \\ ⩽ & \underset{n \to + \infty}{lim inf} ‖ u_{n} ‖ - lim_{n \to + \infty} λ \int_{R^{N}} F (u_{n}) d x \\ = & ς, \end{array}$ which follows that u is a global minimizer.

Next, let us show that u is a solution of (4.1). For all $v \in B L_{rad} (R^{N})$ and $t > 0$ , note that it follows from the convexity of $J (u)$ $\begin{array}{rcl} J (u) - F (u) & ⩽ & J ((1 - t) u + t v) - F ((1 - t) u + t v) \\ ⩽ & (1 - t) J (u) + t J (v) - F ((1 - t) u + t v), \end{array}$ which is equivalent to $\begin{array}{rcl} J (v) - J (u) & ⩾ & \frac{1}{t} (F (t (v - u) + u) - F (u)) \\ ⩾ & \int_{0}^{1} F^{'} (u + s t (v - u)) d s (v - u) . \end{array}$ and taking $t \to 0^{+}$ , we have that u satisfies (2.5) and u is a solution of (4.1).

Then, let us show that in fact u is nontrivial. We consider $R, δ > 0$ such that $B_{R} (0) \subset \overline{B_{R + δ} (0)} \subset R^{N}$ and $\int_{B_{R} (0)} F (t_{0}) d x > 0$ , where $t_{0}$ is defined in $(f_{7})$ . And let us define the function $w : R^{N} \to R$ by $\begin{array}{rcl} w (x) = \{\begin{matrix} t_{0} & if | x | ⩽ R, \\ \frac{t_{0}}{δ} (R + δ - r) & if r = | x | \in [R, R + δ], \\ 0 & if | x | > R + δ . \end{matrix} \end{array}$ Note that $w \in W_{rad}^{1, 1} (R^{N})$ and, as a distribution, $\begin{array}{rcl} Δ w = - \frac{t_{0}}{δ} \frac{(N - 1)}{r} χ_{B_{R + δ} (0) ∖ B_{R} (0)} + \frac{t_{0}}{δ} H_{N - 1} |_{\partial B_{R + δ} (0)} - \frac{t_{0}}{δ} H_{N - 1} |_{\partial B_{R} (0)} \in M (R^{N}), \end{array}$ which follow that $w \in B L_{rad} (R^{N})$ . Then, $\begin{array}{rcl} Φ (w) & = & \int_{R^{N}} | Δ w | + \int_{R^{N}} | \nabla w | d x + \int_{R^{N}} | w | d x - λ \int_{R^{N}} F (w) d x \\ (4.4) & ⩽ & \int_{R^{N}} | Δ w | + \int_{R^{N}} | \nabla w | d x + \int_{R^{N}} | w | d x - λ \int_{B_{R} (0)} F (t_{0}) d x, \end{array}$ which follows that $Φ (w)$ is negative for $λ ⩾ λ^{*}$ where $λ^{*}$ is sufficiently large, by $(f_{7})$ . Thus, $ς = {inf}_{{BL}_{rad} (R^{N})} Φ < 0$ and u is a nontrivial solution of Problem (4.1). □

Footnotes

Acknowledgement

This work is supported by Research Fund of National Natural Science Foundation of China (No. 11861046), Chongqing Municipal Education Commission (No. KJQN20190081), Chongqing Technology and Business University (No. CTBUZDPTTD201909).

Data availability

No data, models, or code were generated or used during the study.

Conflicts of interest

The authors declare that they have no conflicts of interest.

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Existence of nontrivial solution for quasilinear equations involving the 1-biharmonic operator

Abstract

Keywords

1. Introduction

Definition 2.1 (Cerami sequence).

Lemma 2.2 (Mountain Pass Theorem).

Theorem 2.3 (Lions’ Lemma in BL ( R N ) ).

Footnotes

Acknowledgement

Data availability

Conflicts of interest

References

Theorem 2.3 (Lions’ Lemma in $BL (R^{N})$ ).