Existence of Solutions for a Perturbed N-Laplacian Boundary Value Problem with Critical Growth

Abstract

In this paper, we investigate a perturbed elliptic boundary value problem that exhibits critical growth characterized by a Trudinger–Moser-type inequality. Our primary focus is to establish the existence of two nontrivial solutions. This is achieved by employing a combination of the Trudinger–Moser-type inequality and a linking theorem based on the ${\mathbb{Z}}_2$-cohomological index. The main feature and novelty of this paper lies in extending the equation to N-Laplacian boundary value problems utilizing the aforementioned methods. This extension not only broadens the applicability of these techniques but also enriches the research outcomes in the field of nonlinear analysis.

1. Introduction

In this paper, we study the existence of solutions for the following perturbed N-Laplacian boundary value problem:

$$\left\{\begin{array}{ccc}-{\Delta }_{N}u=\lambda {\left|u\right|}^{N-2}u+f(x,u)+h(x),\hfill & \hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=0,\hfill & \hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N$, $u=u(x),x\in \mathsf{\Omega }\subset {\mathbb{R}}^N$, $-{\Delta }_{N}u={div(|\nabla u|}^{N-2}\nabla u)$ is the N-Laplacian operator, $N\ge 2, N\in \mathbb{N}$, and $\lambda$ is not the eigenvalue of $-{\Delta }_N$ with a Dirichlet boundary condition; that is, there exists $m\in \mathbb{N}$ such that ${\lambda }_m<\lambda <{\lambda }_{m+1}$. The perturbation term h belongs to the dual space of $W_0^{1,N}(\mathsf{\Omega })$, which is denoted by $W_0^{-1,N^{\prime }}(\mathsf{\Omega })$ with the norm ${∥·∥}_{*}$, $N^{\prime }=\frac{N}{N-1}$.

Next, we assume that f satisfies the following prerequisite conditions:

($F_1$)

$f(x,t)\in C(\overline{\mathsf{\Omega }}\times \mathbb{R},\mathbb{R})$, $f(x,0)=0$ and f is of critical growth; that is, there exists ${\alpha }_0>0$ such that

$$\underset{t\to \infty }{lim}\frac{|f(x,t)|}{exp(\alpha |t{|}^{N^{\prime }})}=\left\{\begin{array}{ccc}0,\hfill & \hfill & \alpha >{\alpha }_0,\hfill \\ \infty ,\hfill & \hfill & \alpha <{\alpha }_0;\hfill \end{array}\right.$$

($F_2$)

There exists a constant $0<\ell <1$ such that

$$f(x,t)t>\frac{N}{\ell }F(x,t)+\frac{\lambda }{\ell }{\left|t\right|}^N>\frac{N}{\ell }F(x,t)>0,\forall t\in \mathbb{R}\setminus \left\{0\right\},x\in \mathsf{\Omega },$$

where $F(x,t)={\int }_0^{t}f(x,s)ds$;

($F_3$)

There exists ${\beta }_0>\frac{1-\ell }{2L_k^N\mathcal{G}}{(\frac{N}{{\gamma }_0})}^{N-1}$ such that

$$\underset{t\to +\infty }{lim\; inf}\frac{tf(x,t)}{exp(\gamma t^{N^{\prime }})}\ge {\beta }_0,$$

where $\gamma >{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0>{\gamma }_0>{\alpha }_0$, and ${\gamma }_0$ is close to ${\alpha }_0$, $L_k=\frac{1}{2k^{k+1}}$ for some $k\in {\mathbb{Z}}^{+}$, $\mathcal{G}=\underset{r\to +\infty }{lim}logr{\int }_0^{1}exp[Nlogr({\tau }^{N^{\prime }}-\tau )]d\tau$, which is a positive number according to do Ó [1];

($F_4$)

$\underset{t\to 0}{lim}sup\frac{F(x,t)}{{\left|t\right|}^N}\le \frac{{\lambda }_{m+1}}{N\ell }$, where ${\lambda }_{m+1}$ is mentioned in (19).

A typical example is defining $F(x,t)={a\left|t\right|}^{q}exp({\alpha }_0{\left|t\right|}^{N^{\prime }})$ by choosing the constant a to be large enough, with $q>N$ and q close to N.

Recently, the elliptic equations with critical growth have been extensively studied. As these have many practical applications in physics, biology, chemistry and other fields, these applications can be seen in [2,3,4]. do Ó [1] obtained the nontrivial solutions to the following class of elliptic problems:

$$-{\Delta }_{N}u=f(x,u),u\ge 0\mathrm{in}\mathsf{\Omega },$$

under the condition $\lambda <{\lambda }_1$, where ${\lambda }_1$ is the first eigenvalue of $-{\Delta }_N$ with a Dirichlet boundary condition (see Drábek and Robinson [5]) by the mountain pass theorem and concentration compactness method.

Yang and Perera [6] employed new abstract results based on the ${\mathbb{Z}}_2$-cohomological index and a related pseudo-index to establish the existence of a nontrivial solution for the following problem:

$$\left\{\begin{array}{ccc}-{\Delta }_{N}u=\lambda {\left|u\right|}^{N-2}u{e}^{{\left|u\right|}^{N^{\prime }}},\hfill & \hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=0,\hfill & \hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

where $\lambda >{\lambda }_1$. Then, Liu and Zhang et al. [7] used a nonstandard linking theorem to demonstrate the existence of a nontrivial solution for the following problem:

$$-{\Delta }_{N}u=\lambda {\left|u\right|}^{N-2}u+f(x,u),u(x)\ge 0\mathrm{in}\mathsf{\Omega },$$

where $u=u(x), x\in \mathsf{\Omega }\subset {\mathbb{R}}^N$, $\lambda >{\lambda }_1$, and f is of critical growth.

For the study of elliptic equations with critical growth, we noted that Laplacian equations with critical exponential growth were discussed in [8,9,10,11], which greatly aided our analysis of the nonlinear properties in this paper, particularly in Lemmas 2 and 3. Moreover, Do Ó’s pioneering work [1] was essential for energy level estimates. Similarly, Crandall et al. [12] and Zhang et al. [13] explored eigenvalue relationships, contributing to the construction of $A_0$ and the nonlinear estimates in Section 4. Furthermore, Perera’s critical point theorem [14], based on cohomological index theory, forms the foundation of our method. Additionally, Yang [15] and Yang and Perera [6] provided key insights for estimating nonlinear terms and constructing the spaces $A_0$ and $B_0$ under the ${\mathbb{Z}}_2$-cohomological index. Finally, Yang and Perera [16] extended the N-Laplacian operator to the $(N,q)$-Laplacian operator, offering further inspiration for our nonlinear estimates and future research.

With the perturbation term, Tonkes [17] investigated the following elliptic problem, involving a perturbed term and a function with critical growth:

$$-{\Delta }_{N}u=e(x,u)+h(x),u\in W_0^{1,N}(\mathsf{\Omega }),$$

where $0<{∥h∥}_{*}<h^{**}$, and $h^{**}$ is a chosen number. Two different solutions are obtained by the mountain pass theorem and a local minimization technique. De Souza and do Ó [18] extended this to the following singular and nonhomogeneous elliptic equation:

$$\left\{\begin{array}{ccc}-{\Delta }_{N}u=\frac{f(x,u)}{{\left|x\right|}^a}+h(x),\hfill & \hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=0,\hfill & \hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

And they applied minimax methods to obtain the existence and multiplicity of weak solutions. Zhang and Yao [19] obtained at least two nontrivial solutions for the following elliptic equations, with critical growth and singular potentials in ${\mathbb{R}}^N$:

$$\left\{\begin{array}{ccc}-{\Delta }_{N}u+V(x){\left|u\right|}^{N-2}u=\lambda \frac{{\left|u\right|}^{N-2}u}{{\left|x\right|}^{\beta }}+\frac{f(x,u)}{{\left|x\right|}^{\beta }}+\epsilon h(x),\hfill & \hfill & x\in {\mathbb{R}}^N,\hfill \\ u\ne 0,\hfill & \hfill & x\in {\mathbb{R}}^N,\hfill \end{array}\right.$$

(2)

where $N\ge 2$ and $0<\lambda <{\lambda }_1$. In particular, Perera [20] proved a general perturbation theorem, introduced in Theorem 2, which can be used to obtain pairs of nontrivial solutions for the following critical p-Laplacian problem:

$$\left\{\begin{array}{ccc}-{\Delta }_{p}u={\lambda \left|u\right|}^{p-2}u+{\mu \left|u\right|}^{q-2}u+{\left|u\right|}^{p^{*}-2}u+h(x),\hfill & \hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=0,\hfill & \hfill & \mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

From the above statement, it is evident that the mountain pass theorem and local minimization techniques can be employed to address N-Laplacian boundary value problems with perturbation terms. Furthermore, an abstract critical point theorem presented by Perera [20] has been utilized to study p-Laplacian boundary value problems, successfully establishing the existence of two nontrivial solutions. However, this method has not yet been extended to N-Laplacian boundary value problems. Moreover, the question of whether problem (1) has two nontrivial solutions when $\lambda >{\lambda }_1$ has remained open. So, in this paper, inspired by Perera [20], Yang [6], and the papers mentioned above, we prove that this is indeed the case when $\lambda >{\lambda }_1$. For more equations with perturbations, we recommend that readers read studies [20,21,22,23,24] and their references.

We observed that due to the influence of the perturbed term in problem (1), the energy level c in the compactness condition becomes smaller. This complicates the estimation of the functional in the linking theorem. Therefore, the present paper strengthened the conditions of the nonlinear term, thus ensuring the validity of the linking theorem under compactness conditions.

The organization of this paper is as follows. Section 2 delivers a comprehensive overview of the conceptual framework and preliminary ideas that underpin this research. Section 3 is dedicated to elucidating the proof of the compactness outcome for problem (1). Section 4 considers the spatial construction of the abstract critical point theorem by Perera [20], while Section 5 focuses on analyzing the behavior of the functional without the perturbation term in problem (1). Ultimately, Section 6 presents the main results of this paper and provides comprehensive proofs.

2. Variational Framework and Preliminaries

In this section, we first state some useful notations. Let $\mathsf{\Omega }$ be a smooth bounded domain in ${\mathbb{R}}^N$ and $L^p(\mathsf{\Omega })$ be a Lebesgue space with the norm

$${\left|u\right|}_p=({\int }_{\mathsf{\Omega }}{\left|u\right|}^p{dx)}^{\frac{1}{p}}\mathrm{for}1\le p<\infty ,$$

Let $W_0^{1,N}(\mathsf{\Omega })(N\in \mathbb{N}, N\ge 2)$ be a Sobolev space with the norm

$$∥u∥=({\int }_{\mathsf{\Omega }}{|\nabla u|}^N{dx)}^{\frac{1}{N}}.$$

The investigation of elliptic equations featuring nonlinearity characterized by critical exponential growth is connected to the Trudinger–Moser inequality established in the study conducted by Moser [25].

Theorem 1 

([25]). Defined in the aforementioned space, for $N\ge 2$

$$\forall u\in W_0^{1,N}{(\mathsf{\Omega }),exp(|u|}^{N^{\prime }})\in L^p(\mathsf{\Omega }),\forall p\in [1,+\infty )$$

and

$$\underset{∥u∥\le 1}{sup}{\int }_{\mathsf{\Omega }}{exp(\alpha |u|}^{N^{\prime }})dx<\infty ,\mathit{if} \mathit{and} \mathit{only} \mathit{if} \alpha \le {\alpha }_N=N{\omega }_{N-1}^{\frac{1}{N-1}},$$

(3)

where ${\omega }_{N-1}$ is the area of the unit sphere in ${\mathbb{R}}^N$.

Consider the following functional:

$${\mathsf{\Phi }}_h(u)=\frac{1}{N}{\int }_{\mathsf{\Omega }}{|\nabla u|}^{N}dx-\frac{\lambda }{N}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{N}dx-{\int }_{\mathsf{\Omega }}F(x,u)dx-{\int }_{\mathsf{\Omega }}hudx,u\in W_0^{1,N}(\mathsf{\Omega }),$$

We say $u\in W_0^{1,N}(\mathsf{\Omega })$ is a weak solution of problem (1) if for all $v\in W_0^{1,N}(\mathsf{\Omega })$,

$$\langle {\mathsf{\Phi }}_h^{\prime }(u),v\rangle ={\int }_{\mathsf{\Omega }}{|\nabla u|}^{N-2}\nabla u\nabla vdx-\lambda {\int }_{\mathsf{\Omega }}{\left|u\right|}^{N-2}uvdx-{\int }_{\mathsf{\Omega }}f(x,u)vdx-{\int }_{\mathsf{\Omega }}hvdx=0.$$

(4)

Hence, the solutions of problem (1) are associated with the critical points of the functional ${\mathsf{\Phi }}_h$.

Let W be a Banach space and $\mathcal{M}=\{u\in W:∥u∥=1\}$, $\mathcal{M}\subset W\setminus \left\{0\right\}$ is a bounded complete symmetric $C^1$-Fisher manifold radially homeomorphic to the unit sphere S in W. This means that the radial projection $\pi :W\setminus \left\{0\right\}\to S$, defined as $\pi (u)=\frac{u}{∥u∥}$, is a homeomorphism when restricted to $\mathcal{M}$. The radial projection from $W\setminus \left\{0\right\}$ onto $\mathcal{M}$ can then be expressed as ${\pi }_{\mathcal{M}}={(\pi {|}_{\mathcal{M}})}^{-1}\circ \pi$. Meanwhile, denoting by $i(M)$ the ${\mathbb{Z}}_2$-cohomological index (see Fadell and Rabinowitz [26]), the following is the abstract critical point theorem.

Theorem 2 

([20]). Let Φ be a $C^1$-Functional on W and let $A_0$ and $B_0$ be disjoint close symmetric subsets of $\mathcal{M}$ such that

$$i(A_0)=i(\mathcal{M}\setminus B_0)=m<\infty .$$

Assume that there exists ${\omega }_0\in \mathcal{M}\setminus A_0$, $0<\rho <R$, and $a<b$ such that setting

$$A_1=\{{\pi }_{\mathcal{M}}((1-\tau )v+\tau {\omega }_0):v\in A_0,0\le \tau \le 1\},A^{*}=\{tu:u\in A_1,0\le t\le R\},$$

$$A=\{tv:v\in A_0,0\le t\le R\}\cup \{Ru:u\in A_1\},B=\{\rho \omega :\omega \in B_0\},$$

$$B^{*}=\{t\omega :\omega \in B_0,0\le t\le \rho \},$$

we have

$$a<\underset{B^{*}}{inf}\mathsf{\Phi },\underset{A}{sup}\mathsf{\Phi }<\underset{B}{inf}\mathsf{\Phi },\underset{A^{*}}{sup}\mathsf{\Phi }<b.$$

If Φ satisfies the ${(PS)}_c$ condition for all $c\in (a,b)$, then Φ has two critical points $u_1$ and $u_2$ with

$$\underset{B^{*}}{inf}\mathsf{\Phi }\le \mathsf{\Phi }(u_1)\le \underset{A}{sup}\mathsf{\Phi },\underset{B}{inf}\mathsf{\Phi }\le \mathsf{\Phi }(u_2)\le \underset{A^{*}}{sup}\mathsf{\Phi }.$$

3. Compactness Result

We recall that the functional ${\mathsf{\Phi }}_h\in C^1(W_0^{1,N}(\mathsf{\Omega }),\mathbb{R})$ satisfies the ${(PS)}_c$ condition if any sequence $\left\{u_j\right\}\subset W_0^{1,N}(\mathsf{\Omega })$ such that ${\mathsf{\Phi }}_h(u_j)\to c,{\mathsf{\Phi }}_h^{\prime }(u_j)\to 0$ as $j\to \infty$ has a convergent subsequence where $c\in \mathbb{R}$. Now, we aim to determine a threshold level below which the functional ${\mathsf{\Phi }}_h$ satisfies the ${(PS)}_c$ condition.

Proposition 1. 

${\mathsf{\Phi }}_h$ satisfies the ${(PS)}_c$ condition for all

$$0\ne c<\frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}-{\left(\frac{1-\ell }{2}\right)}^{\frac{-1}{N-1}}\frac{N-1}{N}{∥h∥}_{*}^{N^{\prime }}.$$

Proof. 

Let $(u_j)$ be a sequence in $W_0^{1,N}(\mathsf{\Omega })$ such that ${\mathsf{\Phi }}_h(u_j)\to c,{\mathsf{\Phi }}_h^{\prime }(u_j)\to 0$ as $j\to \infty$; that is,

$${\mathsf{\Phi }}_h(u_j)=\frac{1}{N}{\int }_{\mathsf{\Omega }}{|\nabla u_j|}^{N}dx-\frac{\lambda }{N}{\int }_{\mathsf{\Omega }}{\left|u_j\right|}^{N}dx-{\int }_{\mathsf{\Omega }}F(x,u_j)dx-{\int }_{\mathsf{\Omega }}h{u}_{j}dx=c+o(1),$$

(5)

and $\forall v\in W_0^{1,N}(\mathsf{\Omega }),$

$$\begin{array}{cc}\hfill \langle {\mathsf{\Phi }}_h^{\prime }(u_j),v\rangle ={\int }_{\mathsf{\Omega }}{|\nabla u_j|}^{N-2}\nabla u_j\nabla vdx-\lambda {\int }_{\mathsf{\Omega }}{\left|u_j\right|}^{N-2}{u}_{j}vdx-{\int }_{\mathsf{\Omega }}f(x,u_j)vdx-& {\int }_{\mathsf{\Omega }}hvdx\hfill \\ & =o(1)∥v∥.\hfill \end{array}$$

(6)

Taking $v=u_j$ in (6), we have

$${\int }_{\mathsf{\Omega }}{|\nabla u_j|}^{N}dx-\lambda {\int }_{\mathsf{\Omega }}{\left|u_j\right|}^{N}dx-{\int }_{\mathsf{\Omega }}f(x,u_j)u_{j}dx-{\int }_{\mathsf{\Omega }}h{u}_{j}dx=o(1)∥u_j∥.$$

(7)

From (5), we obtain that

$${\int }_{\mathsf{\Omega }}{|\nabla u_j|}^{N}dx-\lambda {\int }_{\mathsf{\Omega }}{\left|u_j\right|}^{N}dx=Nc+N{\int }_{\mathsf{\Omega }}F(x,u_j)dx+N{\int }_{\mathsf{\Omega }}h{u}_{j}dx+o(1).$$

(8)

In applying (8) and $(F_2)$ to (7),

$$\begin{array}{cc}\hfill \frac{1}{N}{\int }_{\mathsf{\Omega }}f(x,u_j)u_{j}dx-{\int }_{\mathsf{\Omega }}F(x,u_j)dx& \le (1-\frac{1}{N})∥u_j∥{∥h∥}_{*}+c+o(1)+o(1)∥u_j∥,\hfill \\ \hfill \frac{1-\ell }{\ell }{\int }_{\mathsf{\Omega }}F(x,u_j)dx& \le ∥u_j∥{∥h∥}_{*}+c+o(1)+o(1)∥u_j∥,\hfill \\ \hfill {\int }_{\mathsf{\Omega }}F(x,u_j)dx& \le \frac{\ell }{1-\ell }\left(∥u_j∥{∥h∥}_{*}+c\right)+(1+∥u_j∥)o(1).\hfill \end{array}$$

(9)

From $(F_2)$, we can deduce that there exist $c_1$ and $c_2$ such that

$${\left|u\right|}_N^N\le c_1{\int }_{\mathsf{\Omega }}F(x,u)dx+c_2,$$

(10)

Combining (5), (9), and (10), we obtain that

$$\begin{array}{cc}\hfill \frac{1}{N}{∥u_j∥}^N& =\frac{\lambda }{N}{\left|u_j\right|}_N^N+{\int }_{\mathsf{\Omega }}F(x,u_j)dx+{\int }_{\mathsf{\Omega }}h{u}_{j}dx+c+o(1)\hfill \\ & \le \frac{\lambda }{N}\left(c_1{\int }_{\mathsf{\Omega }}F(x,u_j)dx+c_2\right)+{\int }_{\mathsf{\Omega }}F(x,u_j)dx+∥u_j∥{∥h∥}_{*}+c+o(1)\hfill \\ & \le \left(\frac{\lambda c_1}{N}+1\right)\left(\frac{\ell }{1-\ell }\right)\left(∥u_j∥{∥h∥}_{*}+c\right)+∥u_j∥{∥h∥}_{*}+\frac{\lambda }{N}{c}_2+c+(2+∥u_j∥)o(1),\hfill \end{array}$$

(11)

From (11), we know that $(u_j)$ is bounded in $W_0^{1,N}(\mathsf{\Omega })$. Moreover, passing to a subsequence still denoted by $(u_j)$, we have

$$\begin{array}{cc}\hfill & u_j\rightharpoonup uin{W}_0^{1,N}(\mathsf{\Omega });\hfill \\ \hfill & u_j\to uin{L}^q(\mathsf{\Omega }),\forall q\ge 1;\hfill \\ \hfill & u_j\to ua.e.in\mathsf{\Omega },\hfill \end{array}$$

In multiplying (7) by $\frac{\ell }{N}$ and subtracting (5),

$$\begin{array}{cc}\hfill \frac{1-\ell }{N}{∥u_j∥}^N+\frac{\lambda }{N}\left(\ell -1\right){\left|u\right|}_N^N+\frac{\ell }{N}{\int }_{\mathsf{\Omega }}f(x,u_j)u_{j}dx-{\int }_{\mathsf{\Omega }}F(x,u_j)dx-\left(1-\frac{\ell }{N}\right)& {\int }_{\mathsf{\Omega }}h{u}_{j}dx=\hfill \\ \hfill & c+o(1)∥u_j∥,\hfill \end{array}$$

By $(F_2)$ and the Hölder inequality, we have

$$\begin{array}{cc}\hfill c+o(1)∥u_j∥& \ge \frac{1-\ell }{N}{∥u_j∥}^N-\frac{\lambda }{N}{\left|u_j\right|}_N^N+\frac{\ell }{N}{\int }_{\mathsf{\Omega }}f(x,u_j)u_{j}dx-{\int }_{\mathsf{\Omega }}F(x,u_j)dx-{∥h∥}_{*}∥u_j∥\hfill \\ & \ge \frac{1-\ell }{N}{∥u_j∥}^N-{∥h∥}_{*}∥u_j∥,\hfill \end{array}$$

Next, we use Young’s inequality for any $\epsilon >0$:

$$\frac{1-\ell }{N}{∥u_j∥}^N-\epsilon {∥u_j∥}^N-C(\epsilon ){∥h∥}_{*}^{N^{\prime }}\le c+o(1)∥u_j∥,$$

where $C(\epsilon )={\left(\epsilon N\right)}^{\frac{-1}{N-1}}\frac{N-1}{N}$. Taking $\epsilon =\frac{1-\ell }{2N}$, we obtain that

$$\underset{j\to \infty }{lim\; sup}{∥u_j∥}^N\le \frac{2N}{1-\ell }\left[c+{\left(\frac{1-\ell }{2}\right)}^{\frac{-1}{N-1}}\frac{N-1}{N}{∥h∥}_{*}^{N^{\prime }}\right],$$

It follows from $c<\frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}-{\left(\frac{1-\ell }{2}\right)}^{\frac{-1}{N-1}}\frac{N-1}{N}{∥h∥}_{*}^{N^{\prime }}$ that

$$\underset{j\to \infty }{lim\; sup}{∥u_j∥}^N<{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}.$$

(12)

In the following, we aim to establish that $u\ne 0$. For the sake of argument, let us suppose that, contrary to our claim, $u=0$. Invoking the Lebesgue dominated convergence theorem, we have

$${\int }_{\mathsf{\Omega }}h{u}_{j}dx\to 0.$$

(13)

From $(F_1)$, we can set $\alpha ={\gamma }_0>{\alpha }_0$; thus, there exists $M>0$ such that

$$\left|f(x,t)\right|\le c_{3}exp({\gamma }_0{\left|t\right|}^{N^{\prime }}),\left|t\right|>M.$$

(14)

From (12), we choose $s>1$ to be sufficiently close to 1 such that

$${\gamma }_{0}s{∥u_j∥}^{N^{\prime }}\le {\alpha }_N,$$

Moreover, from (14), we obtain that

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\left|f(x,u_j)\right|}^{s}dx& \le {\int }_{\left\{\left|u_j\right|>M\right\}}{c}_{3}exp\left({\gamma }_{0}s{\left|u_j\right|}^{N^{\prime }}\right)dx+{\int }_{\left\{\left|u_j\right|\le M\right\}}{\left|f(x,u_j)\right|}^{s}dx\hfill \\ & \le {\int }_{\left\{\left|u_j\right|>M\right\}}{c}_{3}exp\left[{\gamma }_{0}s{∥u_j∥}^{N^{\prime }}{\left(\frac{\left|u_j\right|}{∥u_j∥}\right)}^{N^{\prime }}\right]dx+{\int }_{\left\{\left|u_j\right|\le M\right\}}{\left|f(x,u_j)\right|}^{s}dx,\hfill \end{array}$$

Due to (3), it is obvious that there exists $s>1$ such that

$${\int }_{\mathsf{\Omega }}{\left|f(x,u_j)\right|}^{s}dx<\infty .$$

(15)

From Tonkes [17], $F(x,u_j)\to F(x,u)$ in $L^1(\mathsf{\Omega })$, and by (7) and the Hölder inequality $\left(\frac{1}{s}+\frac{1}{s^{\prime }}=1\right)$,

$$\begin{array}{cc}\hfill o(1)& ={\int }_{\mathsf{\Omega }}{|▿u_j|}^{N}dx-\lambda {\int }_{\mathsf{\Omega }}{\left|u_j\right|}^{N}dx-{\int }_{\mathsf{\Omega }}f(x,u_j)u_{j}dx-{\int }_{\mathsf{\Omega }}h{u}_{j}dx\hfill \\ & \ge {\int }_{\mathsf{\Omega }}{|▿u_j|}^{N}dx-\lambda {\int }_{\mathsf{\Omega }}{\left|u_j\right|}^{N}dx-{\left({\int }_{\mathsf{\Omega }}{\left|f(x,u_j)\right|}^{s}dx\right)}^{\frac{1}{s}}{\left({\int }_{\mathsf{\Omega }}{\left|u_j\right|}^{s^{\prime }}\right)}^{\frac{1}{s^{\prime }}}-{\int }_{\mathsf{\Omega }}h{u}_{j}dx,\hfill \end{array}$$

Furthermore, from (13) and (15), we obtain that

$$\underset{j\to \infty }{lim}{∥u_j∥}^N=0.$$

By (5), we have

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim}{∥u_j∥}^N& ={\lambda \left|u\right|}_N^N+N\left(c+{\int }_{\mathsf{\Omega }}F(x,u)dx+{\int }_{\mathsf{\Omega }}hudx\right)\hfill \\ & =Nc,\hfill \end{array}$$

However, it is a contradiction to $c\ne 0$; thus, $u\ne 0$.

Now, we set $u_j=u+{\omega }_j$; it is obvious that

$$\begin{array}{cc}\hfill & {\omega }_j\rightharpoonup 0in{W}_0^{1,N}(\mathsf{\Omega });\hfill \\ \hfill & {\omega }_j\to 0in{L}^q(\mathsf{\Omega }),\forall q\ge 1,\hfill \end{array}$$

By the Brézis–Lieb lemma (see Brézis and Lieb [27]),

$${∥u_j∥}^N={∥u∥}^N+{∥{\omega }_j∥}^N+o(1).$$

From Tonkes [17], $f(x,u_j)\to f(x,u)$ in $L^1(\mathsf{\Omega })$; thus, we obtain

$${\int }_{\mathsf{\Omega }}f(x,u_j)udx\to {\int }_{\mathsf{\Omega }}f(x,u)udx,$$

Therefore,

$$\begin{array}{cc}\hfill \langle {\mathsf{\Phi }}_h^{\prime }(u_j),u_j\rangle & ={∥u_j∥}^N-\lambda {\left|u_j\right|}_N^N-{\int }_{\mathsf{\Omega }}f(x,u_j)u_{j}dx-{\int }_{\mathsf{\Omega }}h{u}_{j}dx\hfill \\ & ={∥u∥}^N+{∥{\omega }_j∥}^N-\lambda {\left|u_j\right|}_N^N-{\int }_{\mathsf{\Omega }}f(x,u_j)({\omega }_j+u)dx-{\int }_{\mathsf{\Omega }}h{u}_{j}dx+o(1)\hfill \\ & =\langle {\mathsf{\Phi }}_h^{\prime }(u),u\rangle +{∥{\omega }_j∥}^N-{\int }_{\mathsf{\Omega }}f(x,u_j){\omega }_{j}dx+o(1).\hfill \end{array}$$

(16)

From (15) and the Hölder inequality,

$${\int }_{\mathsf{\Omega }}f(x,u_j){\omega }_{j}dx\le {\left|f(x,u_j)\right|}_s{\left|{\omega }_j\right|}_{s^{\prime }}\to 0.$$

(17)

Since $\langle {\mathsf{\Phi }}_h^{\prime }(u),u\rangle =0$, combining (16) and (17), we obtain that

$${\omega }_j\to 0,$$

and the proposition is proved. □

4. The Structure of ${\mathit{A}}_{\mathbf{0}}$

The nonlinear eigenvalue problem

$$\left\{\begin{array}{ccc}-{\Delta }_{N}u=\lambda {\left|u\right|}^{N-2}u,\hfill & \hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ u=0,\hfill & \hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

(18)

plays an important role in our result.

Let $\mathcal{M}=\{u\in W_0^{1,N}(\mathsf{\Omega }):∥u∥=1\}$, where $\mathcal{M}\subset W_0^{1,N}(\mathsf{\Omega })\setminus \left\{0\right\}$ is a bounded complete symmetric $C^1$-Fisher manifold radially homeomorphic to the unit sphere in $W_0^{1,N}(\mathsf{\Omega })$. The eigenvalues of problem (18) coincide with critical values of the $C^1$-Functional

$$\mathsf{\Psi }(u)=\frac{1}{{\left|u\right|}_N^N},u\in \mathcal{M}.$$

We define $\mathcal{F}$ as the class of symmetric subsets of $\mathcal{M}$; let ${\mathcal{F}}_m=\left\{M\in \mathcal{F}:i(M)\ge m\right\}$ and set

$${\lambda }_m:=\underset{M\in {\mathcal{F}}_m}{inf}\underset{u\in M}{sup}\mathsf{\Psi }(u),m\in \mathbb{N},$$

(19)

Furthermore, we use the standard notations

$${\mathsf{\Psi }}^a=\{u\in \mathcal{M}:\mathsf{\Psi }(u)\le a\}\mathrm{and}{\mathsf{\Psi }}_a=\{u\in \mathcal{M}:\mathsf{\Psi }(u)\ge a\},a\in \mathbb{R}$$

for the sublevel sets and superlevel sets.

By Yang and Perera [6], the sublevel set ${\mathsf{\Psi }}^{{\lambda }_m}$ has a compact symmetric subset C of index m that is bounded in $C^1(\mathsf{\Omega })$. In general, we can assume that $0\in \mathsf{\Omega }$.

Let

$${\eta }_k(x)=\left\{\begin{array}{ccc}0,\hfill & \hfill & \left|x\right|\le \frac{1}{2k^{k+1}};\hfill \\ 2k^k\left(\left|x\right|-\frac{1}{2k^{k+1}}\right),\hfill & \hfill & \frac{1}{2k^{k+1}}<\left|x\right|\le \frac{1}{k^{k+1}};\hfill \\ {\left(k\left|x\right|\right)}^{\frac{1}{k}},\hfill & \hfill & \frac{1}{k^{k+1}}<\left|x\right|\le \frac{1}{k};\hfill \\ 1,\hfill & \hfill & \left|x\right|>\frac{1}{k}.\hfill \end{array}\right.$$

Lemma 1 

([6]). As $k\to \infty$,

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\left|{\eta }_{k}u\right|}^{N}dx={\int }_{\mathsf{\Omega }}{\left|u\right|}^{N}dx+O\left(\frac{1}{k^N}\right);\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}{\left|\nabla ({\eta }_{k}u)\right|}^{N}dx=1+O\left(\frac{1}{k^{N-1}}\right);\hfill \\ \hfill & \mathsf{\Psi }\left({\pi }_{\mathcal{M}}\left({\eta }_{k}u\right)\right)=\mathsf{\Psi }(u)+O\left(\frac{1}{k^{N-1}}\right)\hfill \end{array}$$

(20)

uniformly in $u\in C$.

Set $C_k=\left\{{\pi }_{\mathcal{M}}\left({\eta }_{k}u\right):u\in C\right\}$; since $C\in {\mathsf{\Psi }}^{{\lambda }_m}$ by and (20), we have

$$\mathsf{\Psi }\left({\pi }_{\mathcal{M}}\left({\eta }_{k}u\right)\right)\le {\lambda }_m+O\left(\frac{1}{k^{N-1}}\right),\forall u\in C,$$

Using ${\lambda }_m<\lambda$ and fixing k to be large enough, we obtain

$$\mathsf{\Psi }(u)\le \lambda ,\forall u\in C_k.$$

(21)

Since $\lambda <{\lambda }_{m+1}$, we know that $C_k\subset \mathcal{M}\setminus {\mathsf{\Psi }}_{{\lambda }_{m+1}}$; therefore,

$$i(C_k)\le i(\mathcal{M}\setminus {\mathsf{\Psi }}_{{\lambda }_{m+1}})=m,$$

On the other hand, $C\to C_k$ and $u\mapsto \mathsf{\Psi }\left({\pi }_{\mathcal{M}}\left({\eta }_{k}u\right)\right)$, which is an odd continuous map; hence

$$i(C_k)\ge i(C)=m,$$

Thus,

$$i(C_k)=m.$$

Let $A_0=C_k$, $B_0={\mathsf{\Psi }}_{{\lambda }_{m+1}}$ in Theorem 2; then, $i(A_0)=i(\mathcal{M}\setminus B_0)=m$.

5. The Behavior of ${\mathsf{\Phi }}_{\mathbf{0}}$

In this section, we consider the behavior of ${\mathsf{\Phi }}_0$, which is defined by

$${\mathsf{\Phi }}_0(u)=\frac{1}{N}{\int }_{\mathsf{\Omega }}{|▿u|}^{N}dx-\frac{\lambda }{N}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{N}dx-{\int }_{\mathsf{\Omega }}F(x,u)dx,$$

and we can obtain the following lemmas:

Lemma 2. 

For all $u\in \mathcal{M}$ and $t\ge 0$,

$${\mathsf{\Phi }}_0(tu)\le \frac{1}{N}\left(1-\frac{\lambda }{\mathsf{\Psi }(u)}\right)t^N-c_3{\left|\mathsf{\Omega }\right|}^{\frac{\ell -1}{\ell }}{\left(\frac{1}{\mathsf{\Psi }(u)}\right)}^{\frac{1}{\ell }}{t}^{\frac{N}{\ell }}+c_4\left|\mathsf{\Omega }\right|.$$

Proof. 

From $(F_2)$, we can obtain that there exists $c_3,c_4>0$ such that

$$F(x,u)\ge c_3{\left|u\right|}^{\frac{N}{\ell }}-c_4.$$

Since $u\in \mathcal{M}$,

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_0(tu)& =\frac{t^N}{N}{\int }_{\mathsf{\Omega }}{|\nabla u|}^{N}dx-\frac{\lambda t^N}{N}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{N}dx-{\int }_{\mathsf{\Omega }}F(x,tu)dx\hfill \\ \hfill & =\frac{t^N}{N}-\frac{\lambda t^N}{N\mathsf{\Psi }(u)}-{\int }_{\mathsf{\Omega }}F(x,tu)dx\hfill \\ \hfill & \le \frac{1}{N}\left(1-\frac{\lambda }{\mathsf{\Psi }(u)}\right)t^N-c_3{t}^{\frac{N}{\ell }}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{\frac{N}{\ell }}dx+c_4\left|\mathsf{\Omega }\right|,\hfill \end{array}$$

By the Hölder inequality, we know that

$${\left|\mathsf{\Omega }\right|}^{\frac{1-\ell }{\ell }}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{\frac{N}{\ell }}dx\ge {\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^{N}dx\right)}^{\frac{1}{\ell }}={\left(\frac{1}{\mathsf{\Psi }(u)}\right)}^{\frac{1}{\ell }},$$

Thus, we can obtain

$${\mathsf{\Phi }}_0(tu)\le \frac{1}{N}\left(1-\frac{\lambda }{\mathsf{\Psi }(u)}\right)t^N-c_3{\left|\mathsf{\Omega }\right|}^{\frac{\ell -1}{\ell }}{\left(\frac{1}{\mathsf{\Psi }(u)}\right)}^{\frac{1}{\ell }}{t}^{\frac{N}{\ell }}+c_4\left|\mathsf{\Omega }\right|.$$

□

Lemma 3. 

For $u\in B_0$, taking $B=\{\rho \omega :\omega \in B_0\}$, we have

$$inf{\mathsf{\Phi }}_0(B)>0,$$

if ρ is sufficiently small.

Proof. 

By $(F_4)$, in taking $0<\epsilon <(\ell +1){\lambda }_{m+1}-\ell \lambda ,\delta =\delta (\epsilon )$,

$$\left|F(x,t)\right|<\frac{{\lambda }_{m+1}-\epsilon }{N\ell }{\left|t\right|}^N,\forall \left|t\right|\le \delta ,$$

(22)

Since $f(x,t)$ has critical growth at ∞, by $(F_1)$, we obtain that

$$\left|F(x,t)\right|<c_5{\left|t\right|}^q{exp(\alpha |t|}^{N^{\prime }}),q>N,\alpha >{\alpha }_0,\forall \left|t\right|\ge \delta ,$$

(23)

In combining (22) and (23),

$$\left|F(x,t)\right|<\frac{{\lambda }_{m+1}-\epsilon }{N\ell }{\left|t\right|}^N+c_5{\left|t\right|}^q{exp(\alpha |t|}^{N^{\prime }}),$$

By the Hölder inequality and (3), we choose $\rho$ to be small enough such that $2\alpha {\rho }^{N^{\prime }}\le {\alpha }_N$ for $u\in B_0$:

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_0(\rho u)& =\frac{1}{N}{\int }_{\mathsf{\Omega }}{|\nabla (\rho u)|}^{N}dx-\frac{\lambda {\rho }^N}{N}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{N}dx-{\int }_{\mathsf{\Omega }}F(x,\rho u)dx\hfill \\ \hfill & >\frac{{\rho }^N}{N}\left(1-\frac{\lambda -\frac{{\lambda }_{m+1}}{\ell }+\frac{\epsilon }{\ell }}{\mathsf{\Psi }(u)}\right)-c_5{\rho }^q{\int }_{\mathsf{\Omega }}{\left|u\right|}^{q}exp\left(\alpha {\rho }^{N^{\prime }}{\left|u\right|}^{N^{\prime }}\right)dx\hfill \\ \hfill & >\frac{{\rho }^N}{N}\left(1-\frac{\lambda -\frac{{\lambda }_{m+1}}{\ell }+\frac{\epsilon }{\ell }}{{\lambda }_{m+1}}\right)-c_5{\rho }^q{\left|u\right|}_{2q}^q{\left[{\int }_{\mathsf{\Omega }}exp\left(2\alpha {\rho }^{N^{\prime }}{\left|u\right|}^{N^{\prime }}\right)\right]}^{\frac{1}{2}}\hfill \\ \hfill & >\frac{{\rho }^N}{N}\left(1-\frac{\lambda -\frac{{\lambda }_{m+1}}{\ell }+\frac{\epsilon }{\ell }}{{\lambda }_{m+1}}\right)-c_6{\rho }^q.\hfill \end{array}$$

Since we have $\lambda -\frac{{\lambda }_{m+1}}{\ell }+\frac{\epsilon }{\ell }<{\lambda }_{m+1}$ and $q>N$, it follows that $inf{\mathsf{\Phi }}_0(B)>0$ if $\rho$ is sufficiently small. □

Lemma 4. 

For $u\in A_0$, there exists $R>0$ such that by taking $A=\{tv:v\in A_0,0\le t\le R\}\cup \{Ru:u\in A_1\},\mathrm{where}{A}_1=\{{\pi }_{\mathcal{M}}((1-\tau )v+\tau {\omega }_0):v\in A_0,0\le \tau \le 1\}$, we obtain

$$sup{\mathsf{\Phi }}_0(A)<0.$$

Proof. 

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_0(tu)& =\frac{1}{N}{\int }_{\mathsf{\Omega }}{\left|\nabla \left(tu\right)\right|}^{N}dx-\frac{\lambda t^N}{N}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{N}dx-{\int }_{\mathsf{\Omega }}F(x,tu)dx\hfill \\ \hfill & <\frac{t^N}{N}\left(1-\frac{\lambda }{\mathsf{\Psi }(u)}\right),\hfill \end{array}$$

and by (21), we obtain ${\mathsf{\Phi }}_0(tu)<0$ for $t\ge 0,u\in A_0$.

Moreover, C is compact, the map $C\to A_0$ given by $u\mapsto {\pi }_{\mathcal{M}}({\eta }_{k}u)$ is continuous, $A_0$ is compact, and so is the set $\left\{{\pi }_{\mathcal{M}}((1-\tau )v+\tau {\omega }_0):v\in A_0\right\}$. Therefore, $\mathsf{\Psi }$ is bounded on this set, and there exists $R>\rho$ such that ${\mathsf{\Phi }}_0<0$ on $\left\{R{\pi }_{\mathcal{M}}((1-\tau )v+\tau {\omega }_0):v\in A_0\right\}$ by Lemma 2. □

Let

$${\omega }_r(x)=\frac{1}{{\omega }_{N-1}^{\frac{1}{N}}}\left\{\begin{array}{ccc}{\left(logr\right)}^{\frac{1}{N^{\prime }}},\hfill & \hfill & \left|x\right|<\frac{1}{r};\hfill \\ \frac{{log\left|x\right|}^{-1}}{{\left(logr\right)}^{\frac{1}{N}}},\hfill & \hfill & \frac{1}{r}\le \left|x\right|\le 1;\hfill \\ 0,\hfill & \hfill & \left|x\right|>1,\hfill \end{array}\right.$$

It satisfies $∥{\omega }_r∥=1$ and $|{\omega }_r{|}_N^N=\left(\frac{1}{logr}\right)$ as $r\to \infty$. Now, we define ${\omega }_0(x)={\tilde{\omega }}_r(x):={\omega }_r(\frac{x}{L_k})$ with $L_k=\frac{1}{2k^{k+1}}$.

Lemma 5. 

Taking ${\omega }_0={\tilde{\omega }}_r$ and $A^{*}=\{tu:u\in A_1,0\le t\le R\}$, we can obtain

$$sup{\mathsf{\Phi }}_0(A^{*})<c^{*}=\frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}.$$

Proof. 

Obviously, $B_{L_k}(0)\subset \mathsf{\Omega }$, ${\tilde{\omega }}_r\in W_0^{1,N}(\mathsf{\Omega })$ and $∥{\tilde{\omega }}_r∥=1$. For large enough r, we obtain

$$\mathsf{\Psi }({\tilde{\omega }}_r)=\frac{1}{L_k^N{\left|{\omega }_r\right|}_N^N}>\lambda ,$$

so ${\tilde{\omega }}_r\notin A_0$ by (21). For $v\in A_0$ and $t,\tau \ge 0$, set $u={\pi }_{\mathcal{M}}((1-\tau )v+\tau {\tilde{\omega }}_r)\in A_1$; now, we aim to prove that

$$\underset{u\in A_1,t\ge 0}{sup}{\mathsf{\Phi }}_0(tu)<\frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}.$$

(24)

We note that (24) is equivalent to

$$\underset{v\in A_0,s,t\ge 0}{sup}{\mathsf{\Phi }}_0(sv+t{\tilde{\omega }}_r)<\frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1},$$

Since $v=0$ on $B_{L_k}(0)$ and ${\tilde{\omega }}_r=0$ on $\mathsf{\Omega }\setminus B_{L_k}(0)$, we obtain that

$${\mathsf{\Phi }}_0(sv+t{\tilde{\omega }}_r)\le {\mathsf{\Phi }}_0(sv)+{\mathsf{\Phi }}_0(t{\tilde{\omega }}_r),$$

Therefore,

$$\underset{v\in A_0,s,t\ge 0}{sup}{\mathsf{\Phi }}_0(sv+t{\tilde{\omega }}_r)\le \underset{v\in A_0,s\ge 0}{sup}{\mathsf{\Phi }}_0(sv)+\underset{t\ge 0}{sup}{\mathsf{\Phi }}_0(t{\tilde{\omega }}_r),$$

Since we obtain that ${\mathsf{\Phi }}_0(sv)<0$ from the proof above, it suffices to show that $sup{\mathsf{\Phi }}_0(t{\tilde{\omega }}_r)<\frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}$. Due to Lemma 2, ${\mathsf{\Phi }}_0(t{\tilde{\omega }}_r)\to -\infty$ as $t\to +\infty$, there exists $t_r\ge 0$ such that

$${\mathsf{\Phi }}_0(t_r{\tilde{\omega }}_r)=\frac{1}{N}{\int }_{\mathsf{\Omega }}{\left|\nabla \left(t_r{\tilde{\omega }}_r\right)\right|}^{N}dx-\frac{\lambda }{N}{\int }_{\mathsf{\Omega }}{\left|t_r{\tilde{\omega }}_r\right|}^{N}dx-{\int }_{\mathsf{\Omega }}F(x,t_r{\tilde{\omega }}_r)dx=sup{\mathsf{\Phi }}_0(t{\tilde{\omega }}_r)$$

and

$${\mathsf{\Phi }}_0^{\prime }(t_r{\tilde{\omega }}_r){\tilde{\omega }}_r=t_r^{N-1}{\int }_{\mathsf{\Omega }}{\left|\nabla {\tilde{\omega }}_r\right|}^{N}dx-\lambda t_r^{N-1}{\int }_{\mathsf{\Omega }}{\left|{\tilde{\omega }}_r\right|}^{N}dx-{\int }_{\mathsf{\Omega }}f(x,t_r{\tilde{\omega }}_r){\tilde{\omega }}_{r}dx=0,$$

(25)

We suppose for contradiction that $sup{\mathsf{\Phi }}_0(t{\tilde{\omega }}_r)\ge \frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}$; thus,

$$\begin{array}{cc}\hfill \frac{t_r^N}{N}-\frac{\lambda }{N}{t}_r^N{\left|{\tilde{\omega }}_r\right|}_N^N-{\int }_{\mathsf{\Omega }}F(x,t_r{\tilde{\omega }}_r)dx& \ge \frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1},\hfill \\ \hfill \frac{t_r^N}{N}& \ge \frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}.\hfill \end{array}$$

(26)

From (26),

$$t_r^N\ge {\int }_{B_{L_k}(0)}f(x,t_r{\tilde{\omega }}_r)t_r{\tilde{\omega }}_{r}dx=L_k^N{\int }_{B_1(0)}f(x,t_r{\omega }_r)t_r{\omega }_{r}dx\ge L_k^N{\int }_{B_{\frac{1}{r}}(0)}f(x,t_r{\omega }_r)t_r{\omega }_{r}dx,$$

(27)

By ($F_3$), for a given $\kappa >0$, there exists ${\rho }_{\kappa }>0$ such that

$$f(x,t)t\ge ({\beta }_0-\kappa ){exp(\gamma |t|}^{N^{\prime }}),\forall t\ge {\rho }_{\kappa },$$

(28)

We choose r to be large enough such that $t_r{\omega }_r=t_r\frac{{\left(logr\right)}^{\frac{1}{N^{\prime }}}}{{\omega }_{N-1}^{\frac{1}{N}}}\ge {\rho }_{\kappa }$ in $B_{\frac{1}{r}}(0)$ and $r\gg 1$. Using (28) in (27), we have

$$\begin{array}{cc}\hfill t_r^N& \ge L_k^N({\beta }_0-\kappa ){\int }_{B_{\frac{1}{r}}(0)}exp\left(\gamma |t_r{\omega }_r{|}^{N^{\prime }}\right)dx\hfill \\ \hfill & \ge L_k^N({\beta }_0-\kappa ){\int }_{B_{\frac{1}{r}}(0)}exp\left(\gamma t_r^{N^{\prime }}{\omega }_{N-1}^{-\frac{1}{N-1}}logr\right)dx\hfill \\ \hfill & \ge \frac{L_k^N({\beta }_0-\kappa ){\omega }_{N-1}}{N}exp\left[\left(\gamma t_r^{N^{\prime }}{\omega }_{N-1}^{-\frac{1}{N-1}}-N\right)logr\right],\hfill \end{array}$$

(29)

Another condition we know is $\gamma >{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0$ from ($F_3$); in combining this condition with (26),

$$\begin{array}{cc}\hfill \gamma & >{\gamma }_0+\left[1-{\left(\frac{1-\ell }{2}\right)}^{\frac{1}{N-1}}\right]{\left(\frac{2}{1-\ell }\right)}^{\frac{1}{N-1}}{\gamma }_0\hfill \\ \hfill & >{\gamma }_0+\left[1-{\left(\frac{1-\ell }{2}\right)}^{\frac{1}{N-1}}\right]{\left(\frac{2}{1-\ell }\right)}^{\frac{1}{N-1}}\frac{N{\omega }_{N-1}^{\frac{1}{N-1}}}{{\alpha }_N}{\gamma }_0\hfill \\ \hfill & >{\gamma }_0+\left[1-{\left(\frac{1-\ell }{2}\right)}^{\frac{1}{N-1}}\right]N{\omega }_{N-1}^{\frac{1}{N-1}}{t}_r^{-N^{\prime }}\hfill \end{array}$$

which means that

$$\gamma t_r^{N^{\prime }}{\omega }_{N-1}^{-\frac{1}{N-1}}-N>{\gamma }_0{t}_r^{N^{\prime }}{\omega }_{N-1}^{-\frac{1}{N-1}}-{\left(\frac{1-\ell }{2}\right)}^{\frac{1}{N-1}}N\ge 0,$$

(30)

By (29) and (30), we have

$$1>\frac{L_k^N({\beta }_0-\kappa ){\omega }_{N-1}}{N}exp(G_r),$$

(31)

where $G_r=\left[{\gamma }_0{t}_r^{N^{\prime }}{\omega }_{N-1}^{-\frac{1}{N-1}}-{\left(\frac{1-\ell }{2}\right)}^{\frac{1}{N-1}}N\right]logr-Nlog{t}_r$. This implies that $(t_r)$ is bounded. If not, then up to a subsequence we know that $G_r\to +\infty$, which is in contradiction to (31). On the other hand, from (29) and (30), we have

$$t_r^N>\frac{L_k^N({\beta }_0-\kappa ){\omega }_{N-1}}{N}exp\left\{\left[{\gamma }_0{t}_r^{N^{\prime }}{\omega }_{N-1}^{-\frac{1}{N-1}}-{\left(\frac{1-\ell }{2}\right)}^{\frac{1}{N-1}}N\right]logr\right\},$$

(32)

By (26), (32), and the fact that $(t_r)$ is bounded, we have

$$t_r^N\to \frac{1-\ell }{2}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}.$$

(33)

Next, we construct two sets:

$$X_r=\{x\in \overline{B_1(0)}:t_r{\omega }_r\ge {\rho }_{\kappa }\};Y_r=\overline{B_1(0)}-X_r,$$

Since $\gamma >{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0$, we have

$$\begin{array}{cc}\hfill t_r^N>& {\ell }_k^N\left\{({\beta }_0-\kappa ){\int }_{\left|x\right|\le 1}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx+{\int }_{Y_r}{t}_r{\omega }_{r}f(x,t_r{\omega }_r)dx\right.\hfill \\ \hfill & \left.-({\beta }_0-\kappa ){\int }_{Y_r}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx\right\}.\hfill \end{array}$$

(34)

Using the fact that ${\omega }_r\to 0$ and the characteristic function ${\chi }_{Y_r}\to 1a.e.in\overline{B_1(0)}$, we have

$$\begin{array}{cc}\hfill & {\int }_{Y_r}{t}_r{\omega }_{r}f(x,t_r{\omega }_r)dx\to 0,\hfill \\ \hfill & {\int }_{Y_r}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx\to \frac{{\omega }_{N-1}}{N}.\hfill \end{array}$$

Obviously,

$$\begin{array}{cc}\hfill {\int }_{\left|x\right|\le 1}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx& ={\int }_{\left|x\right|\le \frac{1}{r}}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx\hfill \\ \hfill & +{\int }_{\frac{1}{r}\le \left|x\right|\le 1}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx,\hfill \end{array}$$

By the definition of ${\omega }_r$, we obtain

$${\int }_{\left|x\right|\le \frac{1}{r}}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx\to \frac{{\omega }_{N-1}}{N}.$$

Using a straightforward computation from do Ó [1], we have

$${\int }_{\frac{1}{r}\le \left|x\right|\le 1}exp\left[{(\frac{2}{1-\ell })}^{\frac{1}{N-1}}{\gamma }_0{\left|t_r{\omega }_r\right|}^{N^{\prime }}\right]dx\to {\omega }_{N-1}\mathcal{G}.$$

Now, in taking $r\to \infty$ in (34) and using (33),

$$\frac{1-\ell }{2}{\left(\frac{{\alpha }_N}{{\gamma }_0}\right)}^{N-1}\ge L_k^N({\beta }_0-\kappa ){\omega }_{N-1}\mathcal{G},$$

Consequently,

$${\beta }_0\le \frac{1-\ell }{2L_k^N\mathcal{G}}{\left(\frac{N}{{\gamma }_0}\right)}^{N-1},$$

which is a contradiction to ($F_3$); thus, this part is proved. □

6. The Main Result

Now, we can state our result.

Theorem 3. 

Assume that ($F_1$)–($F_4$) hold, λ is not the eigenvalue of $-{\Delta }_N$ with a Dirichlet boundary condition, and ${\lambda }_m<\lambda <{\lambda }_{m+1}$. Then, there exists $h_0>0$ such that $0<{∥h∥}_{*}<h_0$, and problem (1) has two nontrivial solutions.

The proof of Theorem 3 is based on Theorem 2. From Proposition 1, we assume that there is a threshold level $c_h^{*}=\frac{1-\ell }{2N}{(\frac{{\alpha }_N}{{\gamma }_0})}^{N-1}-{\left(\frac{1-\ell }{2}\right)}^{\frac{-1}{N-1}}\frac{N-1}{N}{∥h∥}_{*}^{N^{\prime }}$ such that ${\mathsf{\Phi }}_h$ satisfies the ${(PS)}_c$ condition at all levels $c<c_h^{*}$. We also define

$$c^{*}=\underset{{∥h∥}_{*}\to 0}{lim\; inf}{c}_h^{*},$$

(35)

Due to the definition of ${\mathsf{\Phi }}_h$ and ${\mathsf{\Phi }}_0$, we know

$$\left|{\mathsf{\Phi }}_0(u)-{\mathsf{\Phi }}_h(u)\right|\le {∥h∥}_{*}∥u∥.$$

(36)

Lemma 6. 

There exists $h_1$ such that $inf{\mathsf{\Phi }}_h(B)>0$ for ρ small enough with ${∥h∥}_{*}<h_1$.

Proof. 

For $u\in B_0$,

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_h(\rho u)& \ge {\mathsf{\Phi }}_0(\rho u)-\rho {∥h∥}_{*}∥u∥\hfill \\ \hfill & \ge {\mathsf{\Phi }}_0(\rho u)-\rho {∥h∥}_{*},\hfill \end{array}$$

combined with Lemma 3; thus, this Lemma is proved. □

Lemma 7. 

There exists $h_2$ such that ${sup}_A{\mathsf{\Phi }}_h<0$ with ${∥h∥}_{*}<h_2$.

Proof. 

For $u\in A_0$, $0\le t\le R$,

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_h(tu)& \le {\mathsf{\Phi }}_0(tu)+t{∥h∥}_{*}∥u∥\hfill \\ & \le {\mathsf{\Phi }}_0(tu)+t{∥h∥}_{*},\hfill \end{array}$$

Moreover, from Lemma 4, there exists $h_2$ such that ${sup}_A{\mathsf{\Phi }}_h<0$ with ${∥h∥}_{*}<h_2$. □

From Lemma 5, (35), and (36), there exists an $h_3>0$ small enough such that

$$sup{\mathsf{\Phi }}_h(A^{*})<c_h^{*}$$

with ${∥h∥}_{*}<h_3$. Furthermore, by Lemmas 6 and 7, we can obtain an $h_0=min\left\{h_1,h_2,h_3\right\}$ that satisfies

$$\underset{B}{inf}{\mathsf{\Phi }}_h>0,\underset{A}{sup}{\mathsf{\Phi }}_h<\underset{B}{inf}{\mathsf{\Phi }}_h,\underset{A^{*}}{sup}{\mathsf{\Phi }}_h<c_h^{*},$$

for all $h\in W_0^{-1,N^{\prime }}(\mathsf{\Omega })$ with ${∥h∥}_{*}<h_0$. Since $B^{*}$ is bounded and ${\int }_{\mathsf{\Omega }}F(x,u)dx$ is also bounded on the bounded set,

$$\underset{B^{*}}{inf}{\mathsf{\Phi }}_h>-\infty ,$$

Therefore, we can apply Theorem 2 with $inf\mathsf{\Phi }(B^{*})>a$ to obtain two critical points, and then Theorem 3 is proved.

7. Conclusions

During our study, we focused on a perturbed N-Laplacian boundary value problem. We first constructed an appropriate variational framework of (1) and analyzed the properties of the energy functional to determine the structure of the solutions. Then, we applied the linking theorem in conjunction with the ${\mathbb{Z}}_2$-cohomological index to establish the conditions for the existence of two solutions.

Nonetheless, several challenging issues remain to be explored. One area that requires further investigation is the boundary conditions. This paper focused on a specific bounded boundary condition $\mathsf{\Omega }$ in relation to the existence of solutions for N-Laplacian problems. Additionally, we could consider incorporating logarithmic nonlinearities into the boundary problem. These topics will be the subject of our future research efforts.