Existence and Non-existence of Solutions for Semilinear bi$-\Delta _{\gamma }-$Laplace Equation

Abstract

In this paper, we study existence and non-existence of weak solutions for semilinear bi$-\Delta _{\gamma }-$Laplace equation

$$\begin{aligned} \Delta ^2_\gamma u=f(x,u) \ \text { in }\Omega , \quad u= \partial _\nu u =0 \; \text { on }\partial \Omega , \end{aligned}$$

where $\Omega$ is a bounded domain with smooth boundary in $\mathbb {R}^N \ (N \ge 2), f(x,\xi )$ is a Carathéodory function and $\Delta _{\gamma }$ is the subelliptic operator of the type

$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}}, \gamma = (\gamma _1, \gamma _2, ..., \gamma _N),\quad \Delta ^2_\gamma : =\Delta _\gamma (\Delta _\gamma ). \end{aligned}$$

1 Introduction

In the last years, the biharmonic problems

$$\begin{aligned} \Delta ^2 u+ c\Delta u=f(x,u) \; \text { in }\Omega , \; u= \Delta u =0 \;\text { on }\partial \Omega , \text{ or } u= \partial _\nu u =0 \; \text { on }\partial \Omega , \end{aligned}$$

have been studied by many authors, see [1,2,3,4, 20, 33,34,35,36] and the references therein.

In this paper, we study the following bi$-\Delta _{\gamma }-$Laplace problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2_\gamma u=f(x,u) \ \text { in }\Omega , \\ u= \partial _\nu u =0 \;\;\quad \text { on }\partial \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega \subset {\mathbb {R}}^N$ is a smooth bounded domain, $\nu =(\nu _1, \cdots , \nu _N)$ is the unit outward normal on $\partial \Omega$ and

$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}}, \gamma = (\gamma _1, \gamma _2, ..., \gamma _N),\quad \Delta ^2_\gamma : =\Delta _\gamma (\Delta _\gamma ). \end{aligned}$$

The $\Delta _\gamma -$operator was considered by B. Franchi and E. Lanconelli in [7] and recently reconsidered by A. E. Kogoj and E. Lanconelli in [14] under the additional assumption that the operator is homogeneous of degree two with respect to a group dilation in ${\mathbb {R}}^N$. The $\Delta _\gamma -$operator contains many degenerate elliptic operators such as the Grushin-type operator (see [9, 26, 28])

$$\begin{aligned} G_{\alpha }: = \Delta _x + |x|^{2\alpha }\Delta _y, \quad (x,y) \in \mathbb {R}^{N_1} \times \mathbb {R}^{N_2},\quad \alpha \ge 0, \end{aligned}$$

and the strongly degenerated elliptic operator $P_{\alpha ,\beta }$ (see [25, 31]) of the form

$$\begin{aligned} P_{\alpha ,\beta }: = \Delta _{x} + \Delta _{y} + |x|^{2\alpha }|y|^{2\beta }\Delta _{z}, \quad (x,y,z) \in \mathbb {R}^{N_1} \times \mathbb {R}^{N_2}\times \mathbb {R}^{N_3},\alpha , \beta \ge 0. \end{aligned}$$

Note that $G_0 \equiv P_{0,0}\equiv \Delta$ is the Laplace operator. $G_{\alpha }$, when $\alpha > 0$, is not elliptic in domains intersecting the surface $x=0$, and $P_{\alpha ,\beta }$, when $\alpha> 0, \beta >0$, is not elliptic in domains intersecting the surface $x=0, y=0$. Many aspects of the theory of degenerate elliptic differential operators are presented in monographs [31, 32] (see also some recent results in [9,10,11,12,13,14,15,16, 18, 19, 21, 23, 27] and the references therein).

This paper is organized as follows: In Sect. 2, we recall function spaces, embedding theorems and Mountain Pass Theorem. In Sect. 3, if the nonlinear term grows faster than a power function and the domain $\Omega$ is $\delta _t$-starshape, we establish nonexistence theorem via an identity of Pohozaev’s type. It seems to us that this identity is even new in the case $\gamma \equiv 1$ when $\Delta _\gamma \equiv \Delta$ is the Laplace operator. In Sect. 4, we obtain theorems on the existence of nontrivial solutions and infinitely many nontrivial solutions to the problem (1.1) by using the variational method in weighted Sobolev spaces.

2 Preliminary Results

2.1 Function Spaces and Embedding Theorems

We consider the operator of the form

$$\begin{aligned} {{\Delta }_{\gamma }}:=\sum \limits _{j=1}^{N}{{{\partial }_{{{x}_{j}}}} \left( \gamma _{j}^{2}{{\partial }_{{{x}_{j}}}} \right) ,\quad {{\partial }_{{{x}_{j}}}}: =\frac{\partial }{\partial {{x}_{j}}}},\; j = 1, 2,\dots , N. \end{aligned}$$

Here, $\gamma _j: \mathbb {R}^N \longrightarrow \mathbb {R}, \gamma _j \in C^\infty (\mathbb {R}^N,{\mathbb {R}})$ and $\gamma _j\ne 0$ in $\mathbb {R}^N\backslash \Pi$ for all $j = 1, 2,\dots , N,$ where

$$\begin{aligned} \Pi := \left\{ x = (x_1, x_2, \dots , x_N) \in \mathbb {R}^N: \prod \limits _{j=1}^{N}{{{x}_{j}}}=0\right\} . \end{aligned}$$

As in [14], we assume that $\gamma _j$ satisfy the following properties:

• (i) ${{\gamma }_{1}}=1,{{\gamma }_{j}}\left( x \right) ={{\gamma }_{j}}\left( {{x}_{1}},{{x}_{2}},\dots ,{{x}_{j-1}} \right) , \ j=2,\dots ,N.$

• (ii) There exists a constant $\rho \ge 0$ such that

  $$\begin{aligned} 0\le {{x}_{k}}{{\partial }_{{{x}_{k}}}}{{\gamma }_{j}}\left( x \right) \le \rho {{\gamma }_{j}}\left( x \right) ,\forall k\in \left\{ 1,2,\dots ,j-1 \right\} ,\forall j=2,\dots ,N, \end{aligned}$$

  and for every $x\in \overline{ \mathbb {R}}_{+}^{N}:=\left\{ (x_1, \dots , x_N) \in \mathbb {R}^N: x_j \ge 0, \forall j = 1,2, \dots , N \right\}$.

• (iii) Equalities ${{\gamma }_{j}}\left( x \right) ={{\gamma }_{j}}\left( x^* \right) \ (j= 1,2, \dots , N)$ are satisfied for every $x\in {{\mathbb {R}}^{N}}$, where

  $$\begin{aligned} x^*=\left( \left| {{x}_{1}} \right| ,\dots ,\left| {{x}_{N}} \right| \right) \text{ if } x= (x_1, x_2, \dots , x_N). \end{aligned}$$

• (iv) There exists a group of dilations $\{\delta _t\}_{t>0}$ such that

  $$\begin{aligned} {\delta _t}:{ \mathbb {R}^N}&\longrightarrow \mathbb {R}^N \\ \left( {{x_1},\dots ,{x_N}} \right)&\longmapsto {\delta _t}\left( {{x_1},\dots ,{x_N}} \right) = \left( {{t^{{\varepsilon _1}}}{x_1},\dots ,{t^{{\varepsilon _N}}}{x_N}} \right) , \end{aligned}$$

  where $1 = {\varepsilon _1} \le {\varepsilon _2} \le \dots \le {\varepsilon _N}$ such that $\gamma _j$ is $\delta _t -$homogeneous of degree $\varepsilon _j - 1$, i. e.,

  $$\begin{aligned} {{\gamma }_{j}}\left( {{\delta }_{t}}\left( x \right) \right) ={{t}^{{{\varepsilon }_{j}}-1}}{{\gamma }_{j}}\left( x \right) , \forall x\in {{ \mathbb {R}}^{N}},\forall t>0,\ j=1,\dots ,N. \end{aligned}$$

  The number

  $$\begin{aligned} {\widetilde{N}}:=\sum \limits _{j=1}^{N}{{{\varepsilon }_{j}}} \end{aligned}$$

  (2.1)

is called the homogeneous dimension of $\mathbb {R}^N$. The homogeneous dimension ${\widetilde{N}}$ plays a crucial role, both in the geometry and the functional associated to the operator $\Delta _\gamma$ (see [37]).

Definition 2.1

Let $\Omega$ be a bounded domain in ${\mathbb {R}}^N$. Denote by $S_{\gamma ,0}^{1,p}(\Omega ) \ (1 \le p < +\infty )$ the closure of $C_0^\infty (\Omega )$ in the norm

$$\begin{aligned} {{\left\| u \right\| }_{S_{\gamma ,0 }^{1,p}(\Omega )}} =\left\{ \int \limits _{\Omega }\left( \sum \limits _{j=1}^{N}{{{\left| {{\gamma }_{j}}{{\partial }_{{{x}_{j}}}} u \right| }^{p}}}\right) \mathrm {d}x \right\} ^{\frac{1}{p}}. \end{aligned}$$

By $S_\gamma ^{2,p}(\Omega ) \ (1 \le p < +\infty )$ we will denote the set of all functions $u \in L^p(\Omega )$ such that ${{\gamma }_{j}}{{\partial }_{{{x}_{j}}}}u \in L^p(\Omega )$ and $\gamma _j \partial _{x_j}(\gamma _i\partial _{x_i}u) \in L^p(\Omega )$ for all $i,j=1,\dots ,N$. We define the norm in this space as follows

$$\begin{aligned} {{\left\| u \right\| }_{S_{\gamma }^{2,p}(\Omega )}} =\left\{ \int \limits _{\Omega }\left( |u|^p + \sum \limits _{j=1}^{N}{{{\left| {{\gamma }_{j}}{{\partial }_{{{x}_{j}}}} u \right| }^{p}}}+ \sum \limits _{i,j=1}^{N} \left| \gamma _j \partial _{x_j}(\gamma _i\partial _{x_i}u) \right| ^p \right) \mathrm {d}x \right\} ^{\frac{1}{p}}. \end{aligned}$$

If $p = 2$ we can also define the scalar product in ${S_\gamma ^{2,2}(\Omega )}$ as follows

$$\begin{aligned} (u,v)_{{S_{\gamma }^{{2,2}} (\Omega )}}= & {} (u,v)_{{L^{2} (\Omega )}} + \sum \limits _{{j = 1}}^{N} {(\gamma _{j} \partial _{{x_{j} }} u,\gamma _{j} \partial _{{x_{j} }} v)_{{L^{2} (\Omega )}} } \\&+ \sum \limits _{{i,j = 1}}^{N} {(\gamma _{j} \partial _{{x_{j} }} (} \gamma _{i} \partial _{{x_{i} }} u),\gamma _{j} \partial _{{x_{j} }} (\gamma _{i} \partial _{{x_{i} }} v))_{{L^{2} (\Omega )}} \end{aligned}$$

The space $S_{\gamma ,0 }^{2,2}(\Omega )$ is defined as the closure of $C_0^\infty (\Omega )$ in the space $S_{\gamma }^{2,2}(\Omega )$.

Set

$$\begin{aligned} {{\nabla }_{\gamma }}u: =\left( {{\gamma }_{1}}{{\partial }_{{{x}_{1}}}}u, {{\gamma }_{2}}{{\partial }_{{{x}_{2}}}}u, \dots ,{{\gamma }_{N}}{{\partial }_{{{x}_{N}}}}u \right) , {{\left| {{\nabla }_{\gamma }}u \right| }} :=\Bigg (\sum \limits _{j=1}^{N}{{{\left| {{\gamma }_{j}}{{\partial }_{{{x}_{j}}}} u \right| }^{2}}}\Bigg )^{\frac{1}{2}}. \end{aligned}$$

From Proposition 3.2 in [14], we obtain

Proposition 2.2

Let $\Omega$ be a bounded domain in ${\mathbb {R}}^N, {\widetilde{N}} > 4$. Then the embedding

$$\begin{aligned} S_{\gamma ,0}^{2,2}( \Omega ) \hookrightarrow L^q( \Omega ), \text{ where } 1 \le q \le 2_{*}^{\gamma }:=\frac{2\widetilde{N}}{{{\widetilde{N}}} - 4} \end{aligned}$$

is continuous, i.e., there exist $C_q>0$ such that

$$\begin{aligned} \left\| u\right\| _{L^q(\Omega )}\le C_q \left\| u\right\| _{S_{\gamma ,0}^{2,2}( \Omega )}, \quad \forall u \in S_{\gamma ,0}^{2,2}( \Omega ), \end{aligned}$$

Moreover, the embedding

$$\begin{aligned} S^{2,2}_{\gamma ,0}(\Omega ) \hookrightarrow \hookrightarrow L^q(\Omega ) \end{aligned}$$

is compact for every $1 \le q < 2_{*}^{\gamma }.$

Remark 1

Following [24], note that the two norms $\left\| u\right\| _{S_{\gamma ,0}^{2,2}( \Omega )}$ and

$$\begin{aligned} ||| u|||_{S_{\gamma ,0}^{2,2}( \Omega )} = \left( \int \limits _{\Omega }|\Delta _\gamma u|^2 \mathrm {d}x \right) ^{\frac{1}{2}} \end{aligned}$$

are equivalent.

2.2 Mountain Pass Theorem

For the reader’s convenience, we recall the revised form of the Mountain Pass Theorem see [5, 6].

Definition 2.3

(see [5, 6]) Let $\mathbb {X}$ be a real Banach space with its dual space $\mathbb {X}^*$ and $\Phi \in C^1(\mathbb {X}, \mathbb {R})$. For $c \in \mathbb {R}$ we say that $\Phi$ satisfies the $(C)_c$ condition if for any sequence $\{x_n\}_{n=1}^{\infty }\subset \mathbb {X}$ with

$$\begin{aligned} \Phi (x_n) \rightarrow c \text{ and } (1+ \left\| x_n\right\| _{\mathbb {X}})\left\| \Phi '(x_n)\right\| _{\mathbb {X}^*}\rightarrow 0, \end{aligned}$$

then there exists a subsequence $\{x_{n_k}\}_{k=1}^\infty$ that converges strongly in $\mathbb {X}$. If $\Phi$ satisfies the $(C)_c$ condition for all $c>0$, then we say that $\Phi$ satisfies the Cerami condition.

Lemma 2.4

(see [5, 6]) Let $\mathbb {X}$ be a real Banach space and let $\Phi \in C^1(\mathbb {X}, \mathbb {R})$ satisfy the $(C)_c$ condition for any $c\in {\mathbb {R}}, \Phi (0) =0$, and

(i) There are constants $\rho , \alpha > 0$ such that $\Phi (u) \ge \alpha$ for all $\left\| u\right\| _{\mathbb {X}} =\rho$;

(ii) There is an $u_1 \in {\mathbb {X}}, \left\| u_1\right\| _{\mathbb {X}} > \rho$ such that $\Phi (u_1)\le 0$.

Then $\Phi$ has a critical value $\beta$ which can be characterized as

$$\begin{aligned} \beta : =\inf \limits _{\lambda \in \Lambda }\max \limits _{0\le t\le 1} \Phi (\lambda (t)) \ge \alpha , \end{aligned}$$

where $\Lambda :=\{\lambda \in C([0,1],{\mathbb {X}}): \lambda (0)=0, \lambda (1)=u_1\}.$

Lemma 2.5

(see [22]) Let $\mathbb {X}$ be an infinite-dimensional Banach space, $\mathbb {X}= \mathbb {Y}\bigoplus \mathbb {Z},$ where $\mathbb {Y}$ is finite-dimensional subspace and let $\Phi \in C^1(\mathbb {X}, \mathbb {R})$ satisfy the $(C)_c$ condition for all $c>0, \Phi (0) =0, \Phi (-u) = \Phi (u)$ for all $u \in \mathbb {X}$, and

(i) There are constants $\rho , \alpha > 0$ such that $\Phi (u) \ge \alpha$ for all $u \in \mathbb {Z}$ and $\left\| u\right\| _{\mathbb {X}} =\rho$;

(ii) For any finite-dimensional subspace $\widehat{\mathbb {X}}\subset \mathbb {X}$, there is $R= R(\widehat{\mathbb {X}}) > 0$ such that $\Phi (u)\le 0$ on $\widehat{\mathbb {X}} \backslash B_R$.

Then $\Phi$ possesses an unbounded sequence of critical values.

3 Nonexistence Results

In this section, we prove the non-existence result for our problem when the domain $\Omega$ is $\delta _t-$starshaped in the following sense. We consider the vector field

$$\begin{aligned} T:=\sum \limits _{i=1}^N\varepsilon _i x_i\partial _{x_i}. \end{aligned}$$

(3.1)

Our first integral identity is the following one.

Lemma 3.1

For every $u \in C^4(\Omega )\cap C^3({\overline{\Omega }}),$ we have

$$\begin{aligned} \int \limits _\Omega \!\Big [T(u)\Delta ^2_\gamma u + T(\Delta _\gamma u)\Delta _\gamma u \Big ]\mathrm {d}x =&\!\int \limits _{\partial \Omega }\!\Big [T(u) \langle \nabla _\gamma \Delta _\gamma u,\nu _\gamma \rangle +T(\Delta _\gamma u) \langle \nabla _\gamma u,\nu _\gamma \rangle \Big ]\mathrm {d}S \nonumber \\&- \int \limits _{\partial \Omega } \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \langle T,\nu \rangle \mathrm {d}S\nonumber \\&+({{\widetilde{N}}} -2)\int \limits _\Omega \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \mathrm {d}x, \end{aligned}$$

(3.2)

where T is the vector field (3.1), $\langle \cdot ,\cdot \rangle$ stands for the Euclidean inner product, $\nu =(\nu _1, \cdots , \nu _N)$ is the unit outward normal on $\partial \Omega$ and $\nu _\gamma =(\gamma _1\nu _1, \cdots , \gamma _N\nu _N)$.

Proof

Integrating by parts, we get

$$\begin{aligned} \int \limits _\Omega T(u)\Delta ^2_\gamma u\mathrm {d}x&=\sum \limits _{i,j=1}^N \int \limits _{\partial \Omega } \varepsilon _i x_i \partial _{x_i}u \gamma _j^2 \partial _{x_j}\Delta _\gamma u \nu _j \mathrm {d}S - \int \limits _\Omega \gamma _j^2 \partial _{x_j}\Delta _\gamma u \partial _{x_j}(\varepsilon _i x_i \partial _{x_i}u)\mathrm {d}x \nonumber \\&=:I_1+I_2. \end{aligned}$$

(3.3)

From (3.1), we have

$$\begin{aligned} I_1= \sum \limits _{i=1}^N \int \limits _{\partial \Omega } \varepsilon _i x_i \partial _{x_i}u \langle \nabla _\gamma \Delta _\gamma u, \nu _\gamma \rangle \mathrm {d}S =\int \limits _{\partial \Omega }T(u) \langle \nabla _\gamma \Delta _\gamma u, \nu _\gamma \rangle \mathrm {d}S. \end{aligned}$$

(3.4)

It is easily seen that

$$\begin{aligned} I_2= -\sum \limits _{j=1}^N \int \limits _\Omega \varepsilon _j \gamma _j^2\partial _{x_j}\Delta _\gamma u \partial _{x_j}u \mathrm {d}x -\sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}\Delta _\gamma u \partial ^2_{x_ix_j}u\mathrm {d}x= :I_{2,1}+I_{2,2}. \end{aligned}$$

Moreover, an integration by parts in $I_{2,2}$ gives

$$\begin{aligned} I_{2,2}&= -\sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}\Delta _\gamma u \partial ^2_{x_ix_j}u\mathrm {d}x\\&=-\int \limits _{\partial \Omega } \langle \nabla _\gamma u, \nabla _\gamma \Delta _\gamma u \rangle \langle T, \nu \rangle \mathrm {d}S +{{\widetilde{N}}} \int \limits _\Omega \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \mathrm {d}x \\&\quad +\sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}u \partial ^2_{x_ix_j}\Delta _\gamma u \mathrm {d}x + \sum \limits _{j=1}^N \int \limits _\Omega \partial _{x_j}u\partial _{x_j}\Delta _\gamma u T\gamma _j^2\mathrm {d}x. \end{aligned}$$

Since $\gamma _j$ is $\delta _t-$homogeneous of degree $\varepsilon _j-1$,

$$\begin{aligned} T\gamma ^2_j =2\gamma _jT\gamma _j =2(\varepsilon _j-1)\gamma _j^2. \end{aligned}$$

Therefore, we get

$$\begin{aligned} I_{2,2}&=-\int \limits _{\partial \Omega } \langle \nabla _\gamma u, \nabla _\gamma \Delta _\gamma u \rangle \langle T, \nu \rangle \mathrm {d}S +{{\widetilde{N}}} \int \limits _\Omega \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \mathrm {d}x \nonumber \\&\quad +\sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}u \partial ^2_{x_ix_j}\Delta _\gamma u \mathrm {d}x + \sum \limits _{j=1}^N \int \limits _\Omega 2(\varepsilon _j-1) \partial _{x_j}u\partial _{x_j}\Delta _\gamma u \gamma _j^2\mathrm {d}x \nonumber \\&=-\int \limits _{\partial \Omega } \langle \nabla _\gamma u, \nabla _\gamma \Delta _\gamma u \rangle \langle T, \nu \rangle \mathrm {d}S +\Big ({{\widetilde{N}}} -2\Big ) \int \limits _\Omega \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \mathrm {d}x \nonumber \\&\quad +\sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}u \partial ^2_{x_ix_j}\Delta _\gamma u \mathrm {d}x - 2 I_{2,1}. \end{aligned}$$

(3.5)

It follows from (3.3)-(3.5) that

$$\begin{aligned} \int \limits _\Omega T(u)\Delta ^2_\gamma u\mathrm {d}x&= \int \limits _{\partial \Omega }T(u) \langle \nabla _\gamma \Delta _\gamma u, \nu _\gamma \rangle \mathrm {d}S -\int \limits _{\partial \Omega } \langle \nabla _\gamma u, \nabla _\gamma \Delta _\gamma u \rangle \langle T, \nu \rangle \mathrm {d}S\nonumber \\&\quad +({{\widetilde{N}}} -2) \int \limits _\Omega \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \mathrm {d}x -I_{2,1}\nonumber \\&\quad + \sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}u \partial ^2_{x_ix_j}\Delta _\gamma u \mathrm {d}x \end{aligned}$$

(3.6)

By similar computations, we also obtain

$$\begin{aligned} \int \limits _\Omega T(\Delta _\gamma u)\Delta _\gamma u\mathrm {d}x&= \int \limits _{\partial \Omega }T(\Delta _\gamma u) \langle \nabla _\gamma u, \nu _\gamma \rangle \mathrm {d}S - \sum \limits _{j=1}^N \int \limits _\Omega \varepsilon _j\gamma _j^2 \partial _{x_j}\Delta _\gamma u \partial _{x_j} u\mathrm {d}x \nonumber \\&\quad - \sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}u \partial ^2_{x_ix_j}\Delta _\gamma u \mathrm {d}x \nonumber \\&=\int \limits _{\partial \Omega }T(\Delta _\gamma u) \langle \nabla _\gamma u, \nu _\gamma \rangle \mathrm {d}S +I_{2,1}\nonumber \\&\quad - \sum \limits _{i,j=1}^N \int \limits _\Omega \varepsilon _i x_i \gamma _j^2 \partial _{x_j}u \partial ^2_{x_ix_j}\Delta _\gamma u \mathrm {d}x. \end{aligned}$$

(3.7)

Combining (3.6) and (3.7) implies (3.2). The proof of Lemma 3.1 is complete. $\square$

Definition 3.2

(see [14, 28]) A domain $\Omega$ is called $\delta _t -$starshaped with respect to the origin if $0\in \Omega$ and $\langle T,\nu \rangle \ge 0$ at every point of $\partial \Omega$.

Definition 3.3

A function $u\in C^4(\Omega )\cap C^3({\overline{\Omega }})$ is called a solution of the problem (1.1) if $\Delta ^2_\gamma u=f(x,u) \text { in }\Omega$ and $u= \partial _\nu u =0 \text { on }\partial \Omega .$

If $u\equiv 0$ then u is called a trivial solution of the problem (1.1).

Lemma 3.4

Suppose that $f(x,\xi ) \equiv f(\xi )$ and $f(0)=0$. Let $u\in C^4(\Omega )\cap C^3({\overline{\Omega }})$ be a solution of the problem (1.1). Then the solution u satisfies the identity

$$\begin{aligned} \int \limits _\Omega \Big ({{\widetilde{N}}} F(u) - \frac{\widetilde{N}-4}{2} uf(u)\Big )\mathrm {d}x =\frac{1}{2}\int \limits _{\partial \Omega } \left| \Delta _\gamma u\right| ^2 \langle T, \nu \rangle \mathrm {d}S. \end{aligned}$$

(3.8)

Proof

By $u=0$ on $\partial \Omega$, then

$$\begin{aligned}&\int \limits _\Omega \Big [T(u)\Delta ^2_\gamma u + T(\Delta _\gamma u)\Delta _\gamma u \Big ]\mathrm {d}x =\int \limits _\Omega \Big [T(u)f(u) + T(\Delta _\gamma u)\Delta _\gamma u \Big ]\mathrm {d}x \nonumber \\&\quad =-\frac{{{\widetilde{N}}}}{2}\int \limits _\Omega \left| \Delta _\gamma u \right| ^2 \mathrm {d}x +\frac{1}{2}\int \limits _{\partial \Omega } \left| \Delta _\gamma u\right| ^2 \langle T, \nu \rangle \mathrm {d}S -{{\widetilde{N}}} \int \limits _\Omega F(u)\mathrm {d}x. \end{aligned}$$

(3.9)

On the other hand, by $u=\partial _\nu u =0$ on $\partial \Omega$, we get

$$\begin{aligned}&\int \limits _\Omega u f(u) \mathrm {d}x =-\int \limits _\Omega \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \mathrm {d}x, \quad \int \limits _\Omega \left| \Delta _\gamma u \right| ^2 \mathrm {d}x =- \int \limits _\Omega \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \mathrm {d}x. \nonumber \\&\int \limits _{\partial \Omega }\Big [T(u) \langle \nabla _\gamma \Delta _\gamma u,\nu _\gamma \rangle +T(\Delta _\gamma u) \langle \nabla _\gamma u,\nu _\gamma \rangle - \langle \nabla _\gamma u,\nabla _\gamma \Delta _\gamma u \rangle \langle T,\nu \rangle \Big ]\mathrm {d}S =0. \end{aligned}$$

(3.10)

Therefore, from Lemma 3.1 and (3.9)-(3.10), we obtain

$$\begin{aligned} \int \limits _\Omega \Big ({{\widetilde{N}}} F(u) - \frac{\widetilde{N}-4}{2} uf(u)\Big )\mathrm {d}x =\frac{1}{2}\int \limits _{\partial \Omega } \left| \Delta _\gamma u\right| ^2 \langle T, \nu \rangle \mathrm {d}S. \end{aligned}$$

$\square$

From Lemma 3.4 we can easily deduce the following two theorems:

Theorem 3.5

Suppose that $f(x,\xi ) \equiv f(\xi ), f(0)=0$. Let $\Omega$ is $\delta _t -$starshaped with respect to the origin and

$$\begin{aligned} {{\widetilde{N}}} F(\xi ) - \frac{{{\widetilde{N}}}-4}{2} \xi f(\xi ) < 0,\quad \forall \xi \ne 0. \end{aligned}$$

Then the problem (1.1) has no nontrivial solution $u\in C^4(\Omega )\cap C^3({\overline{\Omega }})$.

Theorem 3.6

Suppose that $f(x,\xi ) \equiv \left| \xi \right| ^{p-1}\xi$ and ${{\widetilde{N}}} > 4$. Let $\Omega$ is $\delta _t -$starshaped with respect to the origin and

$$\begin{aligned} p\ge \frac{{{\widetilde{N}}} +4}{{{\widetilde{N}}} -4}. \end{aligned}$$

Then the problem (1.1) has no nontrivial solution $u\in C^4(\Omega )\cap C^3({\overline{\Omega }})$.

4 Existence Results

From now on we suppose that $f(x,\xi )$ has only polynomial growth in $\xi$.

Definition 4.1

A function $u\in S^{2,2}_{\gamma ,0}(\Omega )$ is called a weak solution of the problem (1.1) if the identity

$$\begin{aligned} \int \limits _{\Omega }{ \Delta _\gamma u \Delta _\gamma \varphi \mathrm {d}x} -\int \limits _{\Omega }{f\left( x,u(x) \right) \varphi \mathrm {d}x }= 0, \end{aligned}$$

is satisfied for every $\varphi \in S^{2,2}_{\gamma ,0}(\Omega ).$

We try to find weak solutions of the problem (1.1) as critical points of a nonlinear functional. To this end we define the functional $\Phi$ on the space $S^{2,2}_{\gamma ,0}(\Omega )$ as follows

$$\begin{aligned} \Phi (u) = \frac{1}{2} \int \limits _\Omega \left| \Delta _\gamma u\right| ^2\mathrm {d}x -\int \limits _\Omega F(x,u)\mathrm {d}x. \end{aligned}$$

(4.1)

Using Hölder’s inequality and Proposition 2.2, we can easily obtain (see [17])

Lemma 4.2

Assume that $f: \Omega \times \mathbb {R} \rightarrow \mathbb {R}$ is a Carathéodory function such that there exist $p\in (2,2_{*}^{\gamma })$, $f_1(x)\in L^{p_1}(\Omega ), f_2(x)\in L^{p_2}(\Omega ),$ where $p_1/(p_1-1) < 2_{*}^{\gamma }, pp_2/(p_2-1)\le 2_{*}^{\gamma }, p_1> \max \{1,\frac{2_{*}^{\gamma } p_2}{p_2(p-1) +2_{*}^{\gamma }}\}, p_2> 1$ such that

$$\begin{aligned} \left| f(x,\xi )\right| \le f_1(x) + f_2(x) \left| \xi \right| ^{p-1} \text{ almost } \text{ everywhere } \text{ in } \Omega \times \mathbb {R}. \end{aligned}$$

Then $\Phi _1(u) \in C^1(S^{2,2}_{\gamma ,0}(\Omega ), \mathbb {R} )$ and

$$\begin{aligned} \Phi _1'(u)(v) = \int \limits _{\Omega }f(x,u)v\mathrm {d}x \end{aligned}$$

for all $v \in S^{2,2}_{\gamma ,0}(\Omega )$, where

$$\begin{aligned} \Phi _1\left( u\right) = \int \limits _\Omega F\left( x,u\right) \mathrm {d}x, \end{aligned}$$

and $F(x,\xi ) = \displaystyle \int \limits _{0}^\xi f(x,\tau ) \mathrm {d}\tau$.

We assume that $f: \Omega \times \mathbb {R} \rightarrow \mathbb {R}$ is a Carathéodory function satisfying

• (A1) There exist $p\in (2, 2_{*}^{\gamma })$, $f_1(x)\in L^{p_1}(\Omega ), f_2(x)\in L^{p_2}(\Omega ),$ where $p_1/(p_1-1)< 2_{*}^{\gamma }, pp_2/(p_2-1)< 2_{*}^{\gamma }, p_1> \max \{1,\frac{ 2_{*}^{\gamma } p_2}{p_2(p-1) + 2_{*}^{\gamma }}\}, p_2> 1,$ such that

  $$\begin{aligned} \left| f(x,\xi )\right| \le f_1(x) + f_2(x) \left| \xi \right| ^{p-1} \text{ almost } \text{ everywhere } \text{ in } \Omega \times \mathbb {R}. \end{aligned}$$

• (A2) $\lim \limits _{\xi \rightarrow 0} \frac{f(x,\xi ) }{\xi } =0,$ uniformly for $x \in \Omega$.

• (A3) $\lim \limits _{\left| \xi \right| \rightarrow \infty } \frac{\left| F(x,\xi )\right| }{\xi ^2} =\infty ,$ for almost every $x \in \Omega$, and

  $$\begin{aligned} F(x,\xi ) \ge 0 \quad \text{ for } \text{ all } (x,\xi ) \in \Omega \times \mathbb {R}. \end{aligned}$$

• (A4) There are constants $\mu > 2$ and $r_1 > 0$ such that

  $$\begin{aligned} \mu F(x,\xi ) \le \xi f(x,\xi ) \quad \text{ for } \text{ all } (x,\xi ) \in \Omega \times \mathbb {R}, \left| \xi \right| \ge r_1. \end{aligned}$$

• (A’4) There are constants $C_0, r_2>0$ and $\kappa >{\max \{1,\frac{{\widetilde{N}}}{2}}\}$ such that

  $$\begin{aligned} |F(x,\xi )|^\kappa \le C_0|\xi |^{2\kappa }\mathcal {F}(x,\xi ),\quad \forall (x,\xi )\in \Omega \times \mathbb {R},\; |\xi |\ge r_2, \end{aligned}$$

  where $\mathcal {F}(x,\xi )=\frac{1}{2}f(x,\xi )\xi -F(x,\xi ).$

• (A5) $f\left( x, \xi \right)$ is an odd function in $\xi .$

From Lemma 4.2 by f satisfying (A1), we have $\Phi$ as well defined on $S^{2,2}_{\gamma ,0}(\Omega )$ and $\Phi \in C^1( S^{2,2}_{\gamma ,0}(\Omega ), \mathbb {R})$ with

$$\begin{aligned} \Phi '(u)(v) = \int \limits _{\Omega }\Delta _\gamma u \Delta _\gamma v \mathrm {d}x - \int \limits _\Omega f\left( x,u\right) v \mathrm {d}x \end{aligned}$$

for all $v \in S^{2,2}_{\gamma ,0}(\Omega ).$ It is clear that the weak solutions of the problem (1.1) are the critical points of $\Phi$. We are now in a position to state our main results.

Theorem 4.3

Assume that f satisfies (A1)–(A4). Then the problem (1.1) has a nontrivial weak solution.

Further, if the condition (A5) is added, then the problem (1.1) possesses infinitely many nontrivial solutions.

Theorem 4.4

Assume that f satisfies (A1)–(A3) and (A’4). Then the problem (1.1) has a nontrivial weak solution.

Further, if the condition (A5) is added, then the problem (1.1) possesses infinitely many nontrivial solutions.

We prove Theorems 4.3 and 4.4 by verifying that all conditions of Lemmas 2.4 and 2.5 are satisfied. First, we check the Cerami condition in those lemmas.

Lemma 4.5

Assume that f satisfies (A1), (A3) and (A4). Then $\Phi$ satisfies the $(C)_c$ condition for all $c \in {\mathbb {R}}$ on $S^{2,2}_{\gamma ,0}(\Omega )$.

Proof

Let $\{u_m\}_{m=1}^\infty$ be a sequence in $S^{2,2}_{\gamma ,0}(\Omega )$ such that

$$\begin{aligned} \left( 1+\left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}\right) \left\| \Phi '(u_m)\right\| _{(S^{2,2}_{\gamma ,0}(\Omega ))^*} \rightarrow 0 \text{ and } \Phi (u_m) \rightarrow c \text{ as } m \rightarrow \infty ; \end{aligned}$$

hence,

$$\begin{aligned} \Phi '(u_m)(u_m) \rightarrow 0\ \text{ and } \ \frac{1}{2}\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} - \int \limits _{\Omega }F(x,u_m)\mathrm {d}x \rightarrow c \text{ as } m \rightarrow \infty . \end{aligned}$$

(4.2)

When m is large enough, we have

$$\begin{aligned} c +1&\ge \Phi (u_m) - \frac{1}{\mu }\Phi '(u_m)(u_m)\nonumber \\&=\left( \frac{1}{2} - \frac{1}{\mu }\right) \left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}+\int \limits _{\Omega }\left( \frac{1}{\mu }f(x,u_m)u_m - F(x,u_m)\right) \mathrm {d}x \nonumber \\&=\left( \frac{1}{2} - \frac{1}{\mu }\right) \left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}+\int \limits _{\Omega _m(0,r_1)}\left( \frac{1}{\mu }f(x,u_m)u_m - F(x,u_m)\right) \mathrm {d}x \nonumber \\&\quad +\int \limits _{\Omega _m(r_1, \infty )}\left( \frac{1}{\mu }f(x,u_m)u_m - F(x,u_m)\right) \mathrm {d}x \nonumber \\&\ge \left( \frac{1}{2} - \frac{1}{\mu }\right) \left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}+\int \limits _{\Omega _m(0,r_1)}\left( \frac{1}{\mu }f(x,u_m)u_m - F(x,u_m)\right) \mathrm {d}x,\nonumber \\ \end{aligned}$$

(4.3)

where $\Omega _m(a,b) =\{x\in \Omega : a\le \left| u_m(x)\right| < b\}$ for $0\le a < b$.

We first show that $\{u_m\}_{m=1}^\infty$ is bounded in $S^{2,2}_{\gamma ,0}(\Omega )$ by a contradiction argument. Indeed, we can (by passing to a subsequence if necessary) suppose for any m that $\left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}>1$ and

$$\begin{aligned} \left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} \rightarrow \infty \text{ as } m \rightarrow \infty . \end{aligned}$$

(4.4)

Setting

$$\begin{aligned} v_m =\frac{u_m}{\left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}}, \end{aligned}$$

then $\left\| v_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}=1$. Passing to a subsequence, we may assume that $v_m \rightharpoonup v$ weakly in ${S^{2,2}_{\gamma ,0}(\Omega )}$ as $m\rightarrow \infty$ then by Proposition 2.2 implies

$$\begin{aligned} {\left\{ \begin{array}{ll} v_m \rightarrow v \text{ strongly } \text{ in } L^q(\Omega )\ \text{ as } m \rightarrow \infty , 2 \le q < 2_{*}^{\gamma },\\ v_m \rightarrow v \text{ a.e. } \text{ on } \Omega \text{ as } m \rightarrow \infty . \end{array}\right. } \end{aligned}$$

(4.5)

From (4.3) and (4.4), we obtain

$$\begin{aligned} \limsup \limits _{m\rightarrow \infty } \frac{1}{\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}}\int \limits _{\Omega _m(0,r_1)}\left( \frac{1}{\mu }f(x,u_m)u_m - F(x,u_m)\right) \mathrm {d}x \le \frac{1}{\mu }- \frac{1}{2} < 0. \end{aligned}$$

(4.6)

Now, we consider two possible cases: $v\equiv 0$ or $v\ne 0$.

Case 1: $v \equiv 0.$ From (A1) and (4.5), we get

$$\begin{aligned}&\left| \int \limits _{\Omega _m(0,r_1)} \frac{f(x,u_m)u_m - \mu F(x,u_m)}{\mu \left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}}\mathrm {d}x \right| \\&\quad \le C \int \limits _{\Omega _m(0,r_1)}\left( \left| f_1(x,y)\right| \frac{\left| u_m \right| }{\left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}}+ \left| f_2(x,y)\right| \left| v_m\right| ^2 \right) \mathrm {d}x \\&\quad \le C \left( \left\| f_1\right\| _{L^{p_1}(\Omega )} \left\| v_m\right\| _{L^{\frac{p_1}{p_1-1}}(\Omega ) } + \left\| f_2\right\| _{L^{p_2}(\Omega )} \left\| v_m\right\| ^2_{L^{\frac{2p_2}{p_2-1}}(\Omega ) }\right) \\&\qquad \rightarrow 0 \text{ as } m \rightarrow \infty , \end{aligned}$$

which contradicts (4.6).

Case 2: $v\ne 0$. Set ${\Omega ^*}= \{ x\in \Omega : v(x) \ne 0\}$ then $\text{ meas }(\Omega ^*) > 0$. For almost every $x \in \Omega ^*$, we have

$$\begin{aligned} \lim \limits _{m \rightarrow \infty }\left| u_m(x)\right| =\lim \limits _{m\rightarrow \infty } \left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} v_m(x,y)=\infty . \end{aligned}$$

It follows from (A1), (A3), (4.2) and Fatou’s Lemma that

$$\begin{aligned} \frac{1}{2}= & {} \lim \limits _{m \rightarrow \infty }\int \limits _\Omega \frac{F(x,u_m)}{\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}} \mathrm {d}x \nonumber \\\ge & {} \liminf \limits _{m\rightarrow \infty } \int \limits _{\Omega ^*} \frac{F(x,u_m)}{\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}}\mathrm {d}x \nonumber \\\ge & {} \int \limits _{\Omega ^*} \liminf \limits _{m\rightarrow \infty }\frac{F(x,u_m)}{\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}}\mathrm {d}x \nonumber \\= & {} \int \limits _{\Omega ^*} \liminf \limits _{m\rightarrow \infty }\frac{F(x,u_m)}{\left| u_m\right| ^2} v^2_m\mathrm {d}x = +\infty , \end{aligned}$$

(4.7)

which is a contradiction. Thus, $\{u_m\}_{m=1}^\infty$ is bounded in $S^{2,2}_{\gamma ,0}(\Omega )$. Because of the above result, without loss of generality, we can suppose that

$$\begin{aligned} u_m&\rightharpoonup&u \text{ in } S^{2,2}_{\gamma ,0}(\Omega )\ \text{ as } m \rightarrow \infty \nonumber \\ u_m\rightarrow & {} u \text{ strongly } \text{ in } L^q(\Omega )\ \text{ as } m \rightarrow \infty , 2\le q < 2_{*}^{\gamma }. \end{aligned}$$

(4.8)

By (A1), we obtain

$$\begin{aligned} \left| \int \limits _{\Omega } f(x,u_m)(u_m -u) \mathrm {d}x \right| \le \int \limits _{\Omega }\left| f_1(x)\right| \left| u_m -u\right| \mathrm {d}x + \int \limits _{\Omega }\left| u_m -u\right| \left| u_m\right| ^{p-1} \left| f_2(x)\right| \mathrm {d}x \nonumber \\ \le \left\| u_m -u\right\| _{L^{\frac{p_1}{p_1-1}}(\Omega )} \left\| f_1\right\| _{L^{p_1}(\Omega )} + \left\| u_m -u\right\| _{L^{\frac{pp_2}{p_2-1}}(\Omega )} \left\| u_m\right\| ^{p-1}_{L^{\frac{pp_2}{p_2-1}}(\Omega )} \left\| f_2\right\| _{L^{p_2}(\Omega )}. \end{aligned}$$

(4.9)

From (4.8) and (4.9), hence

$$\begin{aligned} \int \limits _{\Omega } f(x,u_m)(u_m -u) \mathrm {d}x \rightarrow 0 \text{ as } m \rightarrow \infty . \end{aligned}$$

Therefore,

$$\begin{aligned} \int \limits _{\Omega } \left[ f(x,u_m) - f(x, u)\right] (u_m -u) \mathrm {d}x \rightarrow 0 \text{ as } m \rightarrow \infty . \end{aligned}$$

(4.10)

From $\Phi '(u_m)(u_m) \rightarrow 0$ as $m \rightarrow \infty$ and (4.8), we get

$$\begin{aligned} \left\langle \Phi '(u_m) -\Phi '(u) , u_m -u \right\rangle \rightarrow 0 \text{ as } m \rightarrow \infty . \end{aligned}$$

(4.11)

Moreover,

$$\begin{aligned}&\left\| u_m - u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} \nonumber \\&\quad = \left\langle \Phi '(u_m) -\Phi '(u) , u_m -u \right\rangle + \int \limits _{\Omega }\left[ f(x,u_m) - f(x,u)\right] (u_m -u) \mathrm {d}x.\nonumber \\ \end{aligned}$$

(4.12)

From (4.10), (4.11) and (4.12), we have

$$\begin{aligned} \left\| u_m - u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} \rightarrow 0 \text{ as } m \rightarrow \infty . \end{aligned}$$

Therefore, we conclude that $u_m \rightarrow u$ strongly in $S^{2,2}_{\gamma ,0}(\Omega )$ as $m\rightarrow \infty$. The proof of Lemma 4.5 is complete. $\square$

Lemma 4.6

Assume that f satisfies (A1), (A3) and (A’4). Then $\Phi$ satisfies the $(C)_c$ condition for all $c \in \mathbb R$ on $S^{2,2}_{\gamma ,0}(\Omega )$.

Proof

Let $\{u_m\}_{m=1}^\infty$ be a sequence in $S^{2,2}_{\gamma ,0}(\Omega )$ such that

$$\begin{aligned} \left( 1+\left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}\right) \left\| \Phi '(u_m)\right\| _{(S^{2,2}_{\gamma ,0}(\Omega ))^*} \rightarrow 0 \text{ and } \Phi (u_m) \rightarrow c \text{ as } m \rightarrow \infty ; \end{aligned}$$

hence,

$$\begin{aligned} \Phi '(u_m)(u_m) \rightarrow 0\ \text{ and } \ \frac{1}{2}\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} - \int \limits _{\Omega }F(x,u_m)\mathrm {d}x \rightarrow c \text{ as } m \rightarrow \infty . \end{aligned}$$

(4.13)

We first show that $\{u_m\}_{m=1}^\infty$ is bounded in $S^{2,2}_{\gamma ,0}(\Omega )$ by a contradiction argument. Indeed, we can (by passing to a subsequence if necessary) suppose for any m such that $\left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}>1$ and

$$\begin{aligned} \left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} \rightarrow \infty \text{ as } m \rightarrow \infty . \end{aligned}$$

Setting

$$\begin{aligned} v_m =\frac{u_m}{\left\| u_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}}, \end{aligned}$$

then $\left\| v_m\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}=1$. Passing to a subsequence, we may assume that

$$\begin{aligned} \begin{aligned}&v_m \rightharpoonup v \text{ weakly } \text{ in } S^{2,2}_{\gamma ,0}(\Omega )\ \text{ as } m \rightarrow \infty , \\&v_m \rightarrow v \text{ a.e. } \text{ in } \Omega \ \text{ as } m \rightarrow \infty .\\&v_m \rightarrow v \text{ strongly } \text{ in } L^q(\Omega ) \ \text{ as } m \rightarrow \infty , 2 \le q < 2_{*}^{\gamma }. \end{aligned} \end{aligned}$$

(4.14)

From (4.13), when m is large enough, we have

$$\begin{aligned} c+1\ge \Phi (u_m)-\frac{1}{2} \Phi '(u_m)(u_m) =\int \limits _{\Omega }\mathcal {F}(x,u_m) \mathrm {d}x. \end{aligned}$$

(4.15)

Now, we consider two possible cases: $v\equiv 0$ or $v\ne 0$.

Case 1: If $v\equiv 0$ then $v_m\rightarrow 0$ in $L^p(\Omega )$, $2\le p<2_{*}^{\gamma }$ and $v_m\rightarrow 0$ a.e. on $\Omega$. From (4.13), it implies that

$$\begin{aligned} \lim \limits _{m\rightarrow \infty }\int \limits _{\Omega }\frac{F(x,u_m)}{\Vert u_m\Vert ^2_{S^{2,2}_{\gamma ,0}(\Omega )}}\mathrm {d}x =\frac{1}{2}. \end{aligned}$$

(4.16)

On the other hand, we obtain

$$\begin{aligned}&\int \limits _{\Omega _m(0,r_2)}\frac{F(x,u_m)}{\Vert u_m\Vert ^2_{S^{2,2}_{\gamma ,0}(\Omega )}} \mathrm {d}x \nonumber \\&\quad \le C \Big ( \left\| f_1\right\| _{L^{p_1}(\Omega )} \left\| v_m\right\| _{L^{\frac{p_1}{p_1-1}}(\Omega ) } + \left\| f_2\right\| _{L^{p_2}(\Omega )} \left\| v_m\right\| ^2_{L^{\frac{2p_2}{p_2-1}}(\Omega ) }\Big ) \rightarrow 0 \text{ as } m \rightarrow \infty . \end{aligned}$$

(4.17)

Set $\kappa '=\kappa /(\kappa -1)$, $\kappa >\max \{1,{\widetilde{N}}/2\}$, then ${2\kappa '}$ $\in [2,2_{*}^{\gamma })$. Hence, from (A’4), (4.15) and (4.14), we have

$$\begin{aligned}&\int \limits _{\Omega _m(r_2,+\infty )}\frac{F(x,u_m)}{\Vert u_m\Vert ^2_{S^{2,2}_{\gamma ,0}(\Omega )}} \mathrm {d}x =\int \limits _{\Omega _m(r_2,+\infty )}\frac{F(x,u_m)}{|u_m|^2}|v_m|^2 \mathrm {d}x\nonumber \\&\quad \le \left[ \int \limits _{\Omega _m(r_2,+\infty )} \left( \frac{F(x,u_m)}{|u_m|^2}\right) ^{\kappa } \mathrm {d}x\right] ^{1/\kappa } \left[ \int \limits _{\Omega _m(r_2,+\infty )}|v_m|^{2\kappa '} \mathrm {d}x\right] ^{1/\kappa '}\nonumber \\&\quad \le {C_0}^{1/\kappa }\left[ \int \limits _{\Omega _m(r_2,+\infty )}\mathcal {F}(x,u_m) \mathrm {d}x \right] ^{1/\kappa }\left[ \int \limits _{\Omega _m(r_2,+\infty )}|v_m|^{2\kappa '} \mathrm {d}x\right] ^{1/\kappa '}\nonumber \\&\quad \le \left[ C_0(C+1)\right] ^{1/\kappa } \left[ \int \limits _{\Omega _m(r_0,+\infty )}|v_m|^{2\kappa '} \mathrm {d}x\right] ^{1/\kappa '}\nonumber \\&\quad \le \left[ C_0(C+1)\right] ^{1/\kappa }\Vert v_m\Vert ^2_{L^{2\kappa '}(\Omega )}\rightarrow 0, \quad \text {as } m\rightarrow \infty . \end{aligned}$$

(4.18)

Combining (4.17) with (4.18), we have

$$\begin{aligned}&\int \limits _{\Omega }\frac{F(x,u_m)}{\Vert u_m\Vert ^2_{S^{2,2}_{\gamma ,0}(\Omega )}}\mathrm {d}x \\&\quad =\int \limits _{\Omega _m(0,r_2)}\frac{F(x,u_m)}{|u_m|^2}|v_m|^2 \mathrm {d}x +\int \limits _{\Omega _m(r_2,+\infty )}\frac{F(x,u_m)}{|u_m|^2}|v_m|^2 \mathrm {d}x\rightarrow 0, \quad \text {as } m\rightarrow \infty , \end{aligned}$$

which contradicts (4.16).

Case 2: $v\ne 0$. The proof is the same as in Case 2 of Lemma 4.5.

Further arguments are similar to those of Lemma 4.5, so we omit them. The proof of Lemma 4.6 is complete. $\square$

Lemma 4.7

Let (A1) and (A2) be satisfied. Then there exist $\alpha , \rho >0$ such that

$$\begin{aligned} \Phi (u)\ge \alpha , \quad \forall u \in S^{2,2}_{\gamma ,0}(\Omega ), \;\; \left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}=\rho . \end{aligned}$$

Proof

By (A1) and (A2), for any $\varepsilon >0$, there is a constant $C(\varepsilon )>0$ such that

$$\begin{aligned} \left| f(x,\xi )\right| \le \varepsilon \left| \xi \right| +C(\varepsilon ) \left| \xi \right| ^{p-1}, \quad \forall (x,\xi ) \in \Omega \times {\mathbb {R}}. \end{aligned}$$

By Lemma 2.2, we have

$$\begin{aligned} \Phi (u)&\ge \frac{1}{2} \left\| u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} -\frac{\varepsilon }{2} \left\| u\right\| ^2_{L^2(\Omega )} -\frac{C(\varepsilon )}{p}\left\| u\right\| ^p_{L^p(\Omega )}\\&\ge \frac{1}{2} \left\| u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} -\frac{\varepsilon }{2} C^2_2 \left\| u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} -\frac{C(\varepsilon )}{p} C^p_p\left\| u\right\| ^p_{S^{2,2}_{\gamma ,0}(\Omega )}. \end{aligned}$$

Since $\varepsilon$ is enough small and $p>2,$ we can choose $\alpha , \rho >0$ such that $\Phi (u) \ge \alpha$ when $\left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}=\rho .$ $\square$

Lemma 4.8

Let (A1) and (A3) be satisfied. Then for any finite-dimensional subspace $\widehat{\mathbb {X}}\subset {S^{2,2}_{\gamma ,0}(\Omega )}$, there is $R= R(\widehat{\mathbb {X}}) > 0$ such that

$$\begin{aligned} \Phi (u) \le 0, \quad \forall u \in \widehat{\mathbb {X}}, \;\; \left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} \ge R. \end{aligned}$$

Proof

Arguing by contradiction, suppose that for some sequence $\{u_n\}_{n=1}^{\infty }\subset \widehat{\mathbb {X}}$ with $\left\| u_n\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} \rightarrow \infty$, there is $M > 0$ such that $\Phi (u_n) \ge -M$ for all $n \in {\mathbb {N}}.$ Set

$$\begin{aligned} v_n(x) = \frac{u_n}{\left\| u_n\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} }, \end{aligned}$$

then $\left\| v_n\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} = 1.$ Therefore, we can (by passing to a subsequence if necessary) suppose that

$$\begin{aligned} v_n&\rightharpoonup&v \text{ weakly } \text{ in } S^{2,2}_{\gamma ,0}(\Omega )\ \text{ as } n \rightarrow \infty , \\ v_n\rightarrow & {} v \text{ a.e. } \text{ in } \Omega \ \text{ as } n \rightarrow \infty . \\ v_n\rightarrow & {} v \text{ strongly } \text{ in } L^q(\Omega ) \ \text{ as } n \rightarrow \infty , 2 \le q < 2_{*}^{\gamma }. \end{aligned}$$

Since $\widehat{\mathbb {X}}$ is finite-dimensional, then

$$\begin{aligned} v_n \rightarrow v \text{ strongly } \text{ in } \widehat{\mathbb {X}}\ \text{ as } n \rightarrow \infty \end{aligned}$$

and $v \in \widehat{\mathbb {X}}, \left\| v\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}=1$. Therefore, it follows from (4.7) that

$$\begin{aligned} 0= & {} \lim \limits _{m \rightarrow \infty }\frac{-M}{\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}} \le \lim \limits _{m\rightarrow \infty }\frac{\Phi (u_m)}{\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}} \\\le & {} \frac{1}{2}- \int \limits _{\Omega }\liminf \limits _{m\rightarrow \infty }\frac{F(x,u_m)}{\left\| u_m\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )}}\mathrm {d}x= -\infty . \end{aligned}$$

Hence, we arrive at a contradiction. So, there is $R = R(\widehat{\mathbb {X}})>0$ such that $\Phi (u) \le 0$ for $u \in \widehat{\mathbb {X}}$ and $\left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}\ge R.$ $\square$

Let $\{e_j\}_{j=1}^\infty$ be a total orthonormal basis of $S^{2,2}_{\gamma ,0}(\Omega )$ and define $\mathbb {X}_j = {\mathbb {R}} e_j,$

$$\begin{aligned} {\mathbb {Y}}_k = \bigoplus \limits _{j=1}^k {\mathbb {X}}_j , \quad {\mathbb {Z}}_k = \bigoplus \limits _{j=k+1}^\infty {\mathbb {X}}_j, \quad k \in {\mathbb {N}}. \end{aligned}$$

Let

$$\begin{aligned} \beta _k = \sup _{\begin{array}{c} u \in {\mathbb {Z}}_k \\ \left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega ) }=1 \end{array}} \left\| u\right\| _{L^q(\Omega )}, 2\le q <2_{*}^{\gamma } \end{aligned}$$

(4.19)

then $\beta _k \rightarrow 0$ as $k\rightarrow \infty$. Indeed, suppose that this is not the case. Then there is an $\varepsilon _0 > 0$ and $\{u_j\}_{j=1}^\infty \subset {S^{2,2}_{\gamma ,0}(\Omega )}, \left\| u_j\right\| _{S^{2,2}_{\gamma ,0}(\Omega ) }=1$, with $u_j \bot {\mathbb {Y}}_{k_j}, \left\| u_j\right\| _{L^q(\Omega )}\ge \varepsilon _0$ where $k_j \rightarrow \infty$ as $j \rightarrow \infty$. For any $v \in S^{2,2}_{\gamma ,0}(\Omega )$, we may find a $w_j \in \mathbb Y_{k_j}$ such that $w_j \rightarrow v$ as $j \rightarrow \infty$. Therefore,

$$\begin{aligned} \left| (u_j,v)_{S^{2,2}_{\gamma ,0}(\Omega )}\right| =\left| (u_j,w_j -v)_{S^{2,2}_{\gamma ,0}(\Omega )}\right| \le \left\| w_j -v\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} \end{aligned}$$

as $j \rightarrow \infty ,$ i.e., $u_j \rightharpoonup 0$ weakly in $S^{2,2}_{\gamma ,0}(\Omega )$. Hence, $u_j \rightarrow 0$ in $L^q(\Omega )$, a contradiction.

Lemma 4.9

Let (A1) and (A3) be satisfied. Then there exist constants $\rho , \alpha , k > 0$ such that $\Phi (u) \ge \alpha$ for all $u \in \mathbb {Z}_k$ and $\left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} =\rho$.

Proof

For any $u \in {\mathbb {Z}}_k$ and $\left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} >0$, using Hölder’s inequality, we have

$$\begin{aligned} \Phi (u) \ge&\frac{1}{2} \left\| u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} -\left\| f_1\right\| _{L^{p_1}(\Omega )}\left\| u\right\| _{L^{\frac{p_1}{p_1 -1}}(\Omega )} -\left\| f_2\right\| _{L^{p_2}(\Omega )}\left\| u\right\| ^p_{L^{\frac{pp_1}{p_1 -1}}(\Omega )} \\ =&\frac{1}{2} \left\| u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} -\left\| f_1\right\| _{L^{p_1}(\Omega )} \left\| \frac{u}{\left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}} \right\| _{L^{\frac{p_1}{p_1 -1}}(\Omega )} \left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} \\&-\left\| f_2\right\| _{L^{p_2}(\Omega )} \left\| \frac{u}{\left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )}} \right\| ^p_{L^{\frac{pp_1}{p_1 -1}}(\Omega )} \left\| u\right\| ^p_{S^{2,2}_{\gamma ,0}(\Omega )}. \end{aligned}$$

Because $2\le 2p_1/(p_1-1)< 2_{*}^{\gamma }, 2\le p p_2/(p_2-1) < 2_{*}^{\gamma }$, we have

$$\begin{aligned} \Phi (u) \ge \frac{1}{2} \left\| u\right\| ^2_{S^{2,2}_{\gamma ,0}(\Omega )} -\left\| f_1\right\| _{L^{p_1}(\Omega )} \beta _k \left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} -\left\| f_2\right\| _{L^{p_2}(\Omega )} \beta _k^p \left\| u\right\| ^p_{S^{2,2}_{\gamma ,0}(\Omega )}. \end{aligned}$$

By (4.19), we can choose k large enough, and $\left\| u\right\| _{S^{2,2}_{\gamma ,0}(\Omega )} =\frac{1}{2}$ such that

$$\begin{aligned} \frac{1}{8} - \frac{1}{2} \left\| f_1\right\| _{L^{p_1}(\Omega )} \beta _k -\left\| f_2\right\| _{L^{p_2}(\Omega )} \beta _k^p\frac{1}{2^p} =\alpha > 0. \end{aligned}$$

$\square$

Proof of Theorem 4.3

By Lemmas 4.5, 4.7 and 4.8, all conditions of Lemma 2.4 are satisfied. Thus, the problem (1.1) has a nontrivial weak solution

If $f(x, -\xi ) =-f(x, \xi )$ then choose ${\mathbb {X}} \equiv S^{2,2}_{\gamma ,0}(\Omega ), {\mathbb {Y}} \equiv {\mathbb {Y}}_k, {\mathbb {Z}} \equiv {\mathbb {Z}}_k$. By Lemmas 4.5, 4.8 and 4.9 all conditions of Lemma 2.5 are satisfied. Thus, the problem (1.1) possesses infinitely many nontrivial solutions. $\square$

Proof of Theorem 4.4

By Lemmas 4.6, 4.7 and 4.8, all conditions of Lemma 2.4 are satisfied. Thus, the problem (1.1) has a nontrivial weak solution

If $f(x, -\xi ) =-f(x, \xi )$ then choose ${\mathbb {X}} \equiv S^{2,2}_{\gamma ,0}(\Omega ), {\mathbb {Y}} \equiv {\mathbb {Y}}_k, {\mathbb {Z}} \equiv {\mathbb {Z}}_k$. By Lemmas 4.6, 4.8 and 4.9 all conditions of Lemma 2.5 are satisfied. Thus, the problem (1.1) possesses infinitely many nontrivial solutions. $\square$