On Fractional p-Laplacian Equations at Resonance

Abstract

This article shows the existence of weak solutions of a resonant problem for a fractional p-Laplacian equation in a bounded domain in ${\mathbb {R}}^N$. Our arguments are based on the Minimum principle, saddle point theorem and rely on a generalization of the Landesman–Lazer-type condition.

1 Introduction and Preliminaries

Let $\Omega$ be a bounded domain in ${\mathbb {R}}^N,$$(N\ge 3)$ with smooth boundary $\partial \Omega$. In this article, we study the existence of weak solutions of the following Dirichlet problem at resonance for fractional p-Laplacian equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s_p u=\lambda _1|u|^{p-2} u+f(x,u)-k(x),&{} x\in \Omega \\ u=0 &{}\text {in } {\mathbb {R}}^N\setminus \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $p\ge 2, s\in (0;1)$ [1,2,3,4,5].

$$\begin{aligned} (-\Delta )^s_p u(x)=2\lim _{\epsilon \rightarrow +0} \int _{{\mathbb {R}}^N\setminus B_{\epsilon }(x)}\frac{|u(x)-u(y)|^{p-2}\left( u(x)-u(y)\right) }{|x-y|^{N+sp}}\mathrm{d}y,\quad x\in {\mathbb {R}}^N, \end{aligned}$$

(1.2)

and $f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a Carathéodory function, $\lambda _1$ denotes the first eigenvalue of the eigenvalue problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s_p u=\lambda |u|^{p-2} u&{} \text {in } \Omega \\ u=0&{} \text {on } {\mathbb {R}}^N\setminus \Omega . \end{array}\right. }\end{aligned}$$

(1.3)

The properties of eigenvalue problem will be specialited below.

Remark that the operation $(-\Delta )^s_p$ known as the fractional p-Laplacian leads naturally to the study of the quasilinear problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s_p u(x)=g(x,u)&{} \text {in } \Omega \\ u=0&{} \text {on } {\mathbb {R}}^N\setminus \Omega . \end{array}\right. }\end{aligned}$$

(1.4)

One feature of the aforementioned operator is the nonlocality in the sense that the value of $(-\Delta )^s_p u(x)$ at any point $x\in \Omega$ depends not only on the values of u on the whole $\Omega$, but also on the whole ${\mathbb {R}}^N$, since u(x) represents the expected value of a random variable tied to a process randomly jumping arbitrarily far from the points. The fractional p-Laplacian operator $(-\Delta )^s_p u(x)$, $p\ge 2$, and more generally pseudo differential operators, have been a classical topic in Hamonic analysis and partial differential equations. Nonlocal operator $(-\Delta )^s_p$ such as naturally arise in continuum mechanics, phase transition phenomena, population dynamics,...

In the literature, there are many works on the existence of solutions for fractional p-Laplacian equation, $p\ge 2$. The authors applied some different methods to study the existence, nonexistence or multiplicity results of weak solutions for nonlocal equations involving the fractional p-Laplacian in domain $\Omega \subset {\mathbb {R}}^N$. We refer the reader to some following paper. In [6], the authors investigated the fractional p-Laplacian equation (1.4) and established the existence and multiplicity results of weak solutions by using Morse Theory. In [7], the authors established the existence of multiple weak solutions for (1.4) with nonlinearity in form

$$\begin{aligned} \lambda f(x,u)+\mu g(x,u). \end{aligned}$$

In [7,8,9,10,11,12,13,14,15,16], the authors applied some different methods (as Variational method via the Mountain Pass Theorem, fixed point method, etc.) to study the existence, nonexistence or multiplicity results of weak solutions for nonlocal equations involving the fractional p-Laplacian in domain $\Omega \subset {\mathbb {R}}^N$.

Our aim in this paper is to study the existence of weak solutions for a fractional p-Laplacian problem (1.1) by using the Minimum principle, the saddle point theorem together with a generalization of the Landesman–Lazer-type condition.

Now, let us introduce a variational setting for the problem (1.1).

We first recall some results related to the fractional Sobolev space and the fractional p-Laplacian, for more details see [6, 17].

Let $\Omega \subset {\mathbb {R}}^N$ be a bounded domain with smooth boundary $\partial \Omega$. For $p\in (1;+\infty ), s\in (0;1)$, the fractional critical exponent is defined as

$$\begin{aligned} p_s^ * = \left\{ {\begin{array}{*{20}{c}} {\frac{{Np}}{{N - sp}}}&{}\quad {{\text {if }}sp < N}\\ { + \infty }&{}\quad {{\text {if }}sp \ge N.} \end{array}} \right. \end{aligned}$$

Define the Gagliardo seminorm by

$$\begin{aligned} {\left[ u \right] _{s,p}} = {\left( {\int \limits _{{{\mathbb {R}}^N}} {\int \limits _{{{\mathbb {R}}^N}} {\frac{{{{\left| {u(x) - u(y)} \right| }^p}}}{{{{\left| {x - y} \right| }^{N + sp}}}}\mathrm{d}x\mathrm{d}y} } } \right) ^{\frac{1}{p}}}, \end{aligned}$$

where $u:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a measurable function, and we define the fractional Sobolev space

$$\begin{aligned} {W^{s,p}}\left( {{{\mathbb {R}}^N}} \right) = \left\{ {u \in {L^p}\left( {{{\mathbb {R}}^N}} \right) : u { \text { measurable, }}{{\left[ u \right] }_{s,p}} < + \infty } \right\} , \end{aligned}$$

endowed with the norm

$$\begin{aligned} {\left\| u \right\| _{s,p}} = {\left( {\left\| u \right\| _p^p + \left[ u \right] _{s,p}^p} \right) ^{\frac{1}{p}}}, \end{aligned}$$

where $\left\| . \right\| _p$ denotes the norm of $L^p(\Omega )$.

Denote $X(\Omega )$ as the closed linear subspace

$$\begin{aligned} X\left( \Omega \right) = \left\{ {u \in {W^{s,p}}\left( {{{\mathbb {R}}^N}} \right) :u(x) = 0 \text { a.e. } x \in {{\mathbb {R}}^N}\setminus \Omega } \right\} . \end{aligned}$$

which can be equivalently renormed by setting $\Vert . \Vert = [.]_{s,p}$ (see [6, 17]).

Moreover $\left( X\left( \Omega \right) ,\left\| {.} \right\| \right)$ is a uniformly convex Banach space and that the embedding $X(\Omega )$ into $L^q(\Omega )$ is continuous for all $1\le q\le p^*_s$ and compact for all $1\le q< p^*_s$ (see [6, 17]).

We set the nonlinear operator $A: X(\Omega ) \rightarrow X(\Omega )^*$ defined for all $u,v\in X(\Omega )$ by

$$\begin{aligned} \left\langle {A(u),v} \right\rangle = \int \limits _{{{\mathbb {R}}^N}} {\int \limits _{{{\mathbb {R}}^N}} {\frac{{{{\left| {u(x) - u(y)} \right| }^{p - 2}}\left( {u(x) - u(y)} \right) \left( {v(x) - v(y)} \right) }}{{{{\left| {x - y} \right| }^{N + sp}}}}} } \mathrm{d}x\mathrm{d}y. \end{aligned}$$

Remark that, if u is smooth enough, this definition coincides with that of (1.2).

Clearly for all $u\in X(\Omega )$, we have

$$\begin{aligned} \left\langle {A(u),u} \right\rangle = {\left\| u \right\| ^p},\qquad \left\| {A(u)} \right\| _{*} \le {\left\| u \right\| ^{p - 1}}. \end{aligned}$$

Since $X(\Omega )$ is uniformly convex Banach space, operator A satisfies the following compactness condition (see [6]).

Lemma 1.1

(S-property) If $\{u_m\}$ is a sequence weakly converging to u in $X(\Omega )$ such that

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

Then $\{u_m\}$ strongly converges to u in $X(\Omega )$.

Moreover A is the Gateaux derivative of the functional

$$\begin{aligned} u \rightarrow J(u) = \frac{{{{\left\| u \right\| }^p}}}{p} \text { in } X(\Omega ). \end{aligned}$$

Now, we consider the nonlinear eigenvalue problem in $X(\Omega )$, namely

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s_p u=\lambda |u|^{p-2} u,&{} \text {in } \Omega \\ u=0&{} \text {on } {\mathbb {R}}^N\setminus \Omega . \end{array}\right. }\end{aligned}$$

(1.5)

Many properties of the eigenvalue problem (1.5) have been detected by several authors (can see [6, 18, 19]). Hence we can recall only the properties that using in our arguments below.

Let

$$\begin{aligned} {\lambda _1} = \mathop {\inf }\limits _{u \in X(\Omega ) \setminus \left\{ 0 \right\} } \frac{{{{\left\| u \right\| }^p}}}{{\left\| u \right\| _p^p}} = \mathop {\inf }\limits _{u \in X(\Omega ) \setminus \left\{ 0 \right\} } \frac{{\left\langle {A(u),u} \right\rangle }}{{\left\| u \right\| _p^p}}, \end{aligned}$$

(1.6)

where $\left\| u \right\| = {\left[ u \right] _{s,p}},u \in X\left( \Omega \right)$. Then $\lambda _1\in (0;+\infty )$ is the first eigenvalue of the eigenvalue problem (1.5). The number $\lambda _1$ plays an important role in arguments for our problem.

• $\lambda _1=\min \sigma (s,p)$ is an isolated point of $\sigma (s,p)$, where $\sigma (s,p)$ is the spectrum of the operator $\left( -\Delta \right) ^s_p$ in $X(\Omega )$. Moreover $\lambda _1-$ eigenfunctions are proportionale.

• $\varphi _1(x)$ is a $\lambda _1-$ eigenfunction, then either $\varphi _1(x)>0$ a.e. in $\Omega$ or $\varphi _1(x)<0$ a.e. in $\Omega$. In below we always assume that $\varphi _1(x)>0$ for a.e. $x\in \Omega$.

Definition 1.1

A function $u(x)\in X(\Omega )$ is said a weak solution of the problem (1.1) if only if

$$\begin{aligned} \left\langle {A(u),v} \right\rangle = {\lambda _1}\int \limits _\Omega {{{\left| u \right| }^{p - 2}}uv\mathrm{d}x} + \int \limits _\Omega {f(x,u)v\mathrm{d}x - \int \limits _\Omega {k(x)v\mathrm{d}x} } \end{aligned}$$

(1.7)

for all $v\in X(\Omega )$.

In order to establish our main theorem, we introduce the following hypotheses: $\left( H_{1}\right)$ [1,2,3,4,5]

• (i) $k(x)\not = 0$ a.e. $x\in \Omega$, $k(x)\in L^{p'}(\Omega ),$$\frac{1}{p}+\frac{1}{p'}=1.$

• (ii) $f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a Carathéodory function, $f(x,0)=0$ and there exists a function $\tau (x)\in L^{p'}(\Omega )$ such that

  $$\begin{aligned} |f(x,s)|\le \tau (x)\text { for a.e.} x\in \Omega , \text { all }s\in {\mathbb {R}}. \end{aligned}$$

Denotes by

$$\begin{aligned} {f^{ + \infty }}(x) = \mathop {\lim }\limits _{s \rightarrow + \infty } f(x,s)\quad , \quad {f_{ - \infty }}(x) = \mathop {\lim }\limits _{s \rightarrow - \infty } f(x,s){\text { a.e. }}x \in \Omega . \end{aligned}$$

(1.8)

$$\begin{aligned} {F^{ + \infty }}(x) = \mathop {\lim }\limits _{\tau \rightarrow + \infty } \frac{1}{\tau }\int \limits _0^\tau {f(x,y{\varphi _1})\mathrm{d}y,\qquad {\text { a.e. }} x \in \Omega .} \end{aligned}$$

(1.9)

$$\begin{aligned} {F_{ - \infty }}(x) = \mathop {\lim }\limits _{\tau \rightarrow + \infty } \frac{1}{\tau }\int \limits _0^\tau {f(x,-y{\varphi _1})\mathrm{d}y,\qquad {\text { a.e. }} x \in \Omega .} \end{aligned}$$

(1.10)

$\left( H_{21}\right)$

• (i)
  $$\begin{aligned} f_{-\infty }(x)<k(x)<f^{+\infty }(x),\text { a.e. } x\in \Omega . \end{aligned}$$

  (1.11)
• (ii)
  $$\begin{aligned} \int \limits _\Omega {{F^{ + \infty }}(x){\varphi _1}(x)\mathrm{d}x}< \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} < \int \limits _\Omega {{F_{ - \infty }}(x){\varphi _1}(x)\mathrm{d}x}. \end{aligned}$$

  (1.12)

$\left( H_{22}\right)$

• (i)
  $$\begin{aligned} f^{+\infty }(x)<k(x)<f_{-\infty }(x),\text { a.e. } x\in \Omega . \end{aligned}$$

  (1.13)
• (ii)
  $$\begin{aligned} \int \limits _\Omega {{F_{ - \infty }}(x){\varphi _1}(x)\mathrm{d}x}< \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} < \int \limits _\Omega {{F^{ + \infty }}(x){\varphi _1}(x)\mathrm{d}x}. \end{aligned}$$

  (1.14)

Our main result is given by the following theorem

Theorem 1.1

Problem (1.1) admits a nonzero weak solution in $X(\Omega )$ if one of following two conditions

• (i) $\left( H_1 \right)$ and $\left( H_{21} \right)$,

or

• (ii) $\left( H_1 \right)$ and $\left( H_{22} \right)$ holds.

Proof of the Theorem 1.1 is based on variational techniques via the Minimum principle and the saddle point theorem.

Theorem 1.2

(Minimum principle (see [20, 21])) Let $\mathfrak {F}\in C^1(Y),$ where Y is a Banach space.

Assume that

• (i) ${\mathfrak {F}}$ is bounded from below, $c=\inf {\mathfrak {F}}$.

• (ii) ${\mathfrak {F}}$ satisfies the Palais–Smale condition in Y.

Then there exists $u_0 \in Y$ such that $\mathfrak {F}(u_0)=c.$

Theorem 1.3

(saddle point theorem-P.H.Rabinowitz (see [21, 22])) Let $X=E\oplus Y$ be a Banach space with Y closed in X and $\dim E<+\infty$. For $\rho >0$ define

$$\begin{aligned} M=\{u\in E: \Vert u\Vert <\rho \}\quad ,\quad M_0=\{u\in E:\Vert u\Vert =\rho \}. \end{aligned}$$

Let $F\in C^1(X,R)$ be such that

$$\begin{aligned} b=\inf _{u\in Y}F(u)>a=\max _{u\in M_0}F(u). \end{aligned}$$

If F satisfies the $(P-S)_c$ condition with

$$\begin{aligned} c=\inf _{\gamma \in \Gamma }\max _{u\in M} F(\gamma (u)), \end{aligned}$$

where

$$\begin{aligned} \Gamma =\{\gamma \in C(M,X):\gamma |_{M_0}=1\}, \end{aligned}$$

then c is a critical value of F.

2 Proof of Theorem 1.1(i) (Minimum principle and Existence of Weak Solutions)

We define the Euler–Lagrange functional associated with the problem (1.1) as

$$\begin{aligned} \begin{aligned} I(u)&= \frac{1}{p}{\left\| u \right\| ^p} - \frac{{\lambda _1}}{p}\int \limits _\Omega {{{\left| u \right| }^p}\mathrm{d}x} - \int \limits _\Omega {F(x,u)\mathrm{d}x} + \int \limits _\Omega {k(x)u\mathrm{d}x}\\&=J(u)+T(u),\quad u\in X(\Omega ), \\ \end{aligned} \end{aligned}$$

(2.1)

where

$$\begin{aligned} J(u) = \frac{1}{p}{\left\| u \right\| ^p}=\frac{1}{p}\left\langle {A(u),u} \right\rangle , \quad u\in X(\Omega ). \end{aligned}$$

(2.2)

$$\begin{aligned} T(u)=- \frac{{\lambda _1}}{p}\int \limits _\Omega {{{\left| u \right| }^p}\mathrm{d}x} - \int \limits _\Omega {F(x,u)\mathrm{d}x} + \int \limits _\Omega {k(x)u\mathrm{d}x}, \quad u\in X(\Omega ) \end{aligned}$$

$$\begin{aligned} F(x,u)=\int \limits _0^u f(x,s)\mathrm{d}s. \end{aligned}$$

We deduce that $I\in C^1(X(\Omega ))$ (see [6]) and the derivative of I is defined by

$$\begin{aligned} \left\langle {I'(u),v} \right\rangle= & {} \left\langle {A(u),v} \right\rangle - {\lambda _1}\int \limits _\Omega {{{\left| u \right| }^{p - 2}}uv\mathrm{d}x} - \int \limits _\Omega {f(x,u)v\mathrm{d}x} \nonumber \\&+ \int \limits _\Omega {k(x)v\mathrm{d}x}, \forall u,v\in X(\Omega ). \end{aligned}$$

(2.3)

Therefore the critical points of I are weak solutions of the problem (1.1).

Proposition 2.1

Assuming the hypotheses $(H_1)$ and $(H_{21})$ are fulfilled, then the functional $I: X(\Omega ) \rightarrow {\mathbb {R}}$ given by (2.1) satisfies the (P–S) condition in $X(\Omega )$.

Proof

Let $\{u_m\}$ be a Palais–Smale sequence in $X(\Omega )$, i.e.,

$$\begin{aligned} |I(u_m)|\le M \text { with a positive constant } M. \end{aligned}$$

(2.4)

$$\begin{aligned} I'(u_m)\rightarrow 0 \text { in } X(\Omega )^* \text { as } m\rightarrow +\infty . \end{aligned}$$

(2.5)

First, we shall prove that the sequence $\{u_m\}$ is bounded in $X(\Omega )$. We suppose by contradiction that the sequence $\{u_m\}$ is not bounded in $X(\Omega )$. Without loss of generality, we assume that

$$\begin{aligned} \Vert u_m\Vert \rightarrow +\infty \text { as } m\rightarrow +\infty . \end{aligned}$$

Let $\widehat{u}_m=\frac{u_m}{\Vert u_m\Vert }$. Thus the sequence $\{\widehat{u}_m\}$ is bounded in $X(\Omega )$. Then there exists a subsequence $\{\widehat{u}_{m_k}\}$ which converges weakly to $\widehat{u}$ in $X(\Omega )$. Since the embedding $X(\Omega )$ into $L^p(\Omega )$ is compact, $\{\widehat{u}_{m_k}\}$ converges strongly to $\widehat{u}$ in $L^p(\Omega )$.

From (2.4), we have

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \left\{ {\frac{1}{p}{{\left\| {{{\widehat{u}}_{{m_k}}}} \right\| }^p} - \frac{{{\lambda _1}}}{p}\int \limits _\Omega {{{\left| {{{\widehat{u}}_{{m_k}}}} \right| }^p}\mathrm{d}x} - \int \limits _\Omega {\frac{{F\left( {x,{u_{{m_k}}}} \right) }}{{{{\left\| {{u_{{m_k}}}} \right\| }^p}}}\mathrm{d}x + } \int \limits _\Omega {\frac{{k(x){{\widehat{u}}_{{m_k}}}}}{{{{\left\| {{u_{{m_k}}}} \right\| }^{p - 1}}}}\mathrm{d}x} } \right\} \le 0. \end{aligned}$$

(2.6)

By hypotheses $(H_1)$, we have:

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \int \limits _\Omega {\frac{{F\left( {x,{u_{{m_k}}}} \right) }}{{{{\left\| {{u_{{m_k}}}} \right\| }^p}}}\mathrm{d}x = 0}. \end{aligned}$$

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \int \limits _\Omega {\frac{{k(x){{\widehat{u}}_{{m_k}}}}}{{{{\left\| {{u_{{m_k}}}} \right\| }^{p - 1}}}}\mathrm{d}x}= 0. \end{aligned}$$

Moreover

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \int \limits _\Omega {{{\left| {{{\widehat{u}}_{{m_k}}}} \right| }^p}\mathrm{d}x} = \int \limits _\Omega {{{\left| {\widehat{u}} \right| }^p}\mathrm{d}x} = \left\| {\widehat{u}} \right\| _p^p. \end{aligned}$$

Then, from (2.6) we obtain that

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \sup {{{\Vert {{{\widehat{u}}_{{m_k}}}} \Vert }^p}\mathrm{d}x} \le {\lambda _1}\left\| {\widehat{u}} \right\| _p^p. \end{aligned}$$

(2.7)

From (2.7) and since the functional $J(u)=\frac{\Vert u\Vert ^p}{p}$ is sequentially weakly lower semicontinous in $X(\Omega )$, we come to a conclusion that

$$\begin{aligned} \begin{array}{l} \frac{{{\lambda _1}}}{p}\left\| {\widehat{u}} \right\| _p^p = \frac{{{\lambda _1}}}{p}\int \limits _\Omega {{{\left| {\widehat{u}} \right| }^p}\mathrm{d}x} \le J\left( {\widehat{u}} \right) \le \mathop {\lim }\limits _{k \rightarrow + \infty } \inf J\left( {{{\widehat{u}}_{{m_k}}}} \right) \\ \qquad \le \mathop {\lim }\limits _{k \rightarrow + \infty } \sup J\left( {{{\widehat{u}}_{{m_k}}}} \right) = \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \frac{1}{p}{\left\| {{{\widehat{u}}_{{m_k}}}} \right\| ^p} \le \frac{{{\lambda _1}}}{p}\left\| {\widehat{u}} \right\| _p^p. \end{array} \end{aligned}$$

Hence,

$$\begin{aligned} J\left( {\widehat{u}} \right) = \frac{1}{p}{\left\| {\widehat{u}} \right\| ^p} = \frac{{{\lambda _1}}}{p}\left\| {\widehat{u}} \right\| _p^p. \end{aligned}$$

(2.8)

By definition of $\lambda _1$, from (2.8) we deduce that $\widehat{u}=\pm \varphi _1$, where $\varphi _1(x)$ is $\lambda _1$ eigenfunction of the eigenvalue problem (1.5).

We shall consider following two cases.

First, we assume that $\widehat{u}_{m_k}\rightarrow \varphi _1$ in $L^p(\Omega )$ as $k\rightarrow +\infty$; hence, $u_{m_k}(x)\rightarrow +\infty$ a.e. $x\in \Omega$ and $\widehat{u}_{m_k}(x) \rightarrow \varphi _1(x)$ a.e. $x\in \Omega$.

From (2.4), we have

$$\begin{aligned}&- pM \le - {\left\| {{u_{{m_k}}}} \right\| ^p} + {\lambda _1}\int \limits _\Omega {{{\left| {{u_{{m_k}}}} \right| }^p}\mathrm{d}x} + p\int \limits _\Omega F\left( {x,{u_{{m_k}}}} \right) \mathrm{d}x \nonumber \\&\quad - p\int \limits _\Omega {k(x){u_{{m_k}}}(x)\mathrm{d}x} \le pM. \end{aligned}$$

(2.9)

and from (2.5), there exists a sequence $\{\epsilon _k\}$, $\epsilon _k>0$, $\epsilon _k \rightarrow 0$ as $k\rightarrow +\infty$ such as

$$\begin{aligned} \left| {\left\langle {I'\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}}} \right\rangle } \right| \le {\varepsilon _k}\left\| {{u_{{m_k}}}} \right\| ,\qquad \left( {k = 1,2,\ldots .} \right) \end{aligned}$$

that is

$$\begin{aligned} \begin{aligned} - {\varepsilon _k}\left\| {{u_{{m_k}}}} \right\| \le {\left\| {{u_{{m_k}}}} \right\| ^p}&- {\lambda _1}\int \limits _\Omega {{{\left| {{u_{{m_k}}}} \right| }^p}\mathrm{d}x} - \int \limits _\Omega {f\left( {x,{u_{{m_k}}}} \right) {u_{{m_k}}}\mathrm{d}x }\\&+ {\int \limits _\Omega {k(x){u_{{m_k}}}(x)\mathrm{d}x} \le {\varepsilon _k}\left\| {{u_{{m_k}}}} \right\| .} \end{aligned} \end{aligned}$$

(2.10)

By summing (2.9) and (2.10), we have

$$\begin{aligned} \begin{aligned} - pM - {\varepsilon _k}\left\| {{u_{{m_k}}}} \right\|&\le p\int \limits _\Omega {F\left( {x,{u_{{m_k}}}} \right) \mathrm{d}x} - \int \limits _\Omega {f\left( {x,{u_{{m_k}}}} \right) {u_{{m_k}}}\mathrm{d}x}\\&\quad +{ \left( {1 - p} \right) \int \limits _\Omega {k(x){u_{{m_k}}}(x)\mathrm{d}x} \le pM + {\varepsilon _k}\left\| {{u_{{m_k}}}} \right\| .} \end{aligned} \end{aligned}$$

(2.11)

After dividing (2.11) by $\Vert u_{m_k}\Vert$, remark that

$$\begin{aligned} \begin{array}{l} \mathop {\lim }\limits _{k \rightarrow + \infty } \int \limits _\Omega {f\left( {x,{u_{{m_k}}}} \right) {{\widehat{u}}_{{m_k}}}(x)\mathrm{d}x = \int \limits _\Omega {{f^{ + \infty }}(x){\varphi _1}(x)\mathrm{d}x,} } \\ \mathop {\lim }\limits _{k \rightarrow + \infty } \int \limits _\Omega {k\left( x \right) {{\widehat{u}}_{{m_k}}}(x)\mathrm{d}x = \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} } \end{array} \end{aligned}$$

and due to the Lebesgue Theorem, we have

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \int \limits _\Omega {\left( {p\frac{{F\left( {x,{u_{{m_k}}}} \right) }}{{\left\| {{u_{{m_k}}}} \right\| }} - {f^{ + \infty }}(x){\varphi _1}(x)} \right) \mathrm{d}x = \left( {p - 1} \right) \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x}.} \end{aligned}$$

(2.12)

Denote $l_k=\Vert u_{m_k}\Vert \rightarrow +\infty$ as $k\rightarrow +\infty ,$ by hypotheses $(H_1)$, and we have

$$\begin{aligned}\begin{aligned} \left| {\int \limits _\Omega {\frac{1}{{{l_k}}}\left( {\int \limits _0^{{u_{{m_k}}}} {f\left( {x,s} \right) \mathrm{d}s} - \int \limits _0^{{l_k}{\varphi _1}} {f\left( {x,s} \right) \mathrm{d}s} } \right) } } \right|&\le \int \limits _\Omega {\frac{1}{{{l_k}}}} \left| {{u_{{m_k}}} - {l_k}{\varphi _1}} \right| \tau \left( x \right) \mathrm{d}x\\&\le \left\| \tau \right\| _{p^{'}}{\left\| {{{\widehat{u}}_{{m_k}}} - {\varphi _1}} \right\| _p} \rightarrow 0 \text { as } k\rightarrow +\infty . \end{aligned} \end{aligned}$$

This implies

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \int \limits _\Omega {\frac{{F\left( {x,{u_{{m_k}}}} \right) }}{{\left\| {{u_{{m_k}}}} \right\| }}} \mathrm{d}x = \mathop {\lim }\limits _{k \rightarrow + \infty } \int \limits _\Omega {\left( {\frac{1}{{{l_k}}}\int \limits _0^{{l_k}{\varphi _1}} {f\left( {x,s} \right) \mathrm{d}s} } \right) } \mathrm{d}x. \end{aligned}$$

By changing $s=y\varphi _1,\mathrm{d}s=\varphi _1 \mathrm{d}y$, we deduce that

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \frac{1}{{{l_k}}}\int \limits _0^{{l_k}{\varphi _1}} {f\left( {x,s} \right) \mathrm{d}s} = \mathop {\lim }\limits _{k \rightarrow + \infty } \frac{1}{{{l_k}}}\int \limits _0^{{l_k}} {f\left( {x,y{\varphi _1}} \right) {\varphi _1}\mathrm{d}y = {F^{ + \infty }}} (x){\varphi _1}(x), \end{aligned}$$

where $F^{+\infty }(x)$ is given by (1.9). Hence

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \int \limits _\Omega {\frac{{F\left( {x,{u_{{m_k}}}} \right) }}{{\left\| {{u_{{m_k}}}} \right\| }}} \mathrm{d}x = \int \limits _\Omega {{F^{ + \infty }}\left( x \right) } {\varphi _1}(x)\mathrm{d}x. \end{aligned}$$

(2.13)

Therefore from (2.12), (2.13), we obtain that

$$\begin{aligned} \int \limits _\Omega {\left( {p{F^{ + \infty }}(x) - {f^{ + \infty }}(x)} \right) {\varphi _1}(x)\mathrm{d}x = \left( {p - 1} \right) \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x}.} \end{aligned}$$

(2.14)

On the other hand, from the hypotheses (1.11) we have

$$\begin{aligned} f^{+\infty } (x)-k(x)\ge 0 \text { a.e. } x\in \Omega . \end{aligned}$$

Hence (2.14) implies that

$$\begin{aligned}\begin{aligned} \int \limits _\Omega p{F^{ + \infty }}(x){\varphi _1}(x)\mathrm{d}x&= p\int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} + \int \limits _\Omega {\left( {{f^{ + \infty }}(x) - k(x)} \right) } {\varphi _1}(x)\mathrm{d}x \\&\ge p\int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} \end{aligned} \end{aligned}$$

which contradicts $(H_{21})$. $\square$

In the case when $\widehat{u}_{m_k}\rightarrow -\varphi _1(x)$ as $k\rightarrow +\infty$, by similar arguments we also have

$$\begin{aligned} \int \limits _\Omega \left( {p{F_{ - \infty }}(x) - {f_{ - \infty }}(x)} \right) {\varphi _1}(x)\mathrm{d}x = \left( {p - 1} \right) \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x}. \end{aligned}$$

(2.15)

By hypotheses, (1.11), (2.15) imply that

$$\begin{aligned}\begin{aligned} \int \limits _\Omega p{F_{ - \infty }}(x){\varphi _1}(x)\mathrm{d}x&= p\int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} + \int \limits _\Omega {\left( {{f_{ - \infty }}(x) - k(x)} \right) } {\varphi _1}(x)\mathrm{d}x\\&\le p\int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x,} \end{aligned} \end{aligned}$$

which contradicts $(H_{21})$.

This implies that the (P–S) sequence $\{u_m\}$ is bounded in $X(\Omega )$. Then there exists a subsequence $\{u_{m_k}\}$ which converges weakly to $u_0$ in $X(\Omega )$. We will prove that the subsequence converges strongly to $u_0$ in $X(\Omega )$.

Indeed, since $u_{m_k}\rightharpoonup u_0$ in $X(\Omega )$ and the embedding $X(\Omega )$ into $L^p(\Omega )$ is compact, $\{u_{m_k}\}$ converges strongly to $u_0$ tin $L^p(\Omega )$.

Firstly we remark that by $(H_1)$

$$\begin{aligned}\begin{aligned} \left| {\left\langle {T'\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}} - {u_0}} \right\rangle } \right|&\le {\lambda _1}\left\| {{u_{{m_k}}}} \right\| _p^{p - 1}{\left\| {{u_{{m_k}}} - {u_0}} \right\| _p}\\&\quad + {\left\| \tau \right\| _{p'}}{\left\| {{u_{{m_k}}} - {u_0}} \right\| _p} + {\left\| k \right\| _{p'}}{\left\| {{u_{{m_k}}} - {u_0}} \right\| _p} \\&\le \left( {{\lambda _1}\left\| {{u_{{m_k}}}} \right\| _p^{p - 1} + {{\left\| \tau \right\| }_{p'}} + {{\left\| k \right\| }_{p'}}} \right) {\left\| {{u_{{m_k}}} - {u_0}} \right\| _p} . \end{aligned} \end{aligned}$$

Since $\{u_{m_k}\}$ is bounded in $L^p(\Omega )$, $\left\| {{u_{{m_k}}} - {u_0}} \right\| _p\rightarrow 0$ as $k\rightarrow +\infty$, we obtain that

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \left\langle {T'\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}} - {u_0}} \right\rangle = 0. \end{aligned}$$

(2.16)

Combining (2.16) and the fact

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \left\langle {I'\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}} - {u_0}} \right\rangle = 0 \end{aligned}$$

we get

$$\begin{aligned}\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \left\langle {J'\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}} - {u_0}} \right\rangle&=\mathop {\lim }\limits _{k \rightarrow + \infty } \left\langle {I'\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}} - {u_0}} \right\rangle \\&\quad -\mathop {\lim }\limits _{k \rightarrow + \infty } \left\langle {T'\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}} - {u_0}} \right\rangle = 0. \end{aligned} \end{aligned}$$

That is

$$\begin{aligned} \mathop {\lim }\limits _{k \rightarrow + \infty } \left\langle {A\left( {{u_{{m_k}}}} \right) ,{u_{{m_k}}} - {u_0}} \right\rangle = 0. \end{aligned}$$

(2.17)

From (2.17), by the (S)-property of the operator A (see Lemma 1.1), we deduce that the subsequence $\{u_{m_k}\}$ converges strongly to $u_0$ in $X(\Omega )$. Therefore the functional I satisfies the Palais–Smale condition in $X(\Omega )$. Proposition 2.1 is proved.

Proposition 2.2

The functional I given by (2.1) is coercive on $X(\Omega )$ provided $(H_{21})$ holds.

Proof

Firstly we noted that, in the proof of the Proposition 2.1, we have proved that if $\{I(u_m)\}$ is a sequence bounded from above with a sequence $\{u_m\}$ in $X(\Omega )$ such that $\Vert u_m\Vert \rightarrow +\infty$ as $m\rightarrow +\infty$, then (up to a subsequence),

$$\begin{aligned} \widehat{u}_m=\frac{u_m}{\Vert u_m\Vert }\rightarrow \pm \varphi _1(x) \text { in } X(\Omega ) \text { as } m\rightarrow +\infty . \end{aligned}$$

Using this fact, we will prove that the functional I is coercive in $X(\Omega )$ if $(H_{21})$ satisfied.

Indeed, suppose by contradiction that I is not coercive, it is possible to choose a sequence $\{u_m\}$ in $X(\Omega )$ such that $\Vert u_m\Vert \rightarrow +\infty$ as $m\rightarrow +\infty$, $I(u_m) \le$ const and

$$\begin{aligned} \widehat{u}_m=\frac{u_m}{\Vert u_m\Vert }\rightarrow \pm \varphi _1(x) \text { in } X(\Omega ) \text { as } m\rightarrow +\infty . \end{aligned}$$

Remark that by (1.6) we deduce that

$$\begin{aligned} - \int \limits _\Omega {F\left( {x,{u_m}} \right) } \mathrm{d}x + \int \limits _\Omega {k(x){u_m}(x)\mathrm{d}x \le I\left( {{u_m}} \right) } \le \text { const}, \quad m = 1,2,\ldots \end{aligned}$$

(2.18)

We now consider following two cases

Case 1: Assume that $\widehat{u}_m \rightarrow \varphi _1$ as $m\rightarrow +\infty$.

Dividing (2.18) by $\Vert u_m\Vert$, we get

$$\begin{aligned}\begin{aligned} - \int \limits _\Omega&{{F^{ + \infty }}(x){\varphi _1}(x)\mathrm{d}x} + \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x}\\&= \mathop {\lim }\limits _{k \rightarrow + \infty } \sup \left( { - \int \limits _\Omega {\frac{{F\left( {x,{u_m}} \right) }}{{\left\| {{u_m}} \right\| }}} \mathrm{d}x + \int \limits _\Omega {k(x){{\widehat{u}}_m}(x)\mathrm{d}x} } \right) \le \mathop {\lim }\limits _{m \rightarrow + \infty } \sup \frac{\text {const}}{{\left\| {{u_m}} \right\| }} = 0 \end{aligned} \end{aligned}$$

which gives

$$\begin{aligned} \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} \le \int \limits _\Omega {{F^{ + \infty }}(x){\varphi _1}(x)\mathrm{d}x}. \end{aligned}$$

which contradicts $(H_{21}).$

Case 2: Assume that $\widehat{u}_m \rightarrow -\varphi _1$ as $m\rightarrow +\infty$.

By similar computation above, we get

$$\begin{aligned} \int \limits _\Omega {{F_{ - \infty }}(x){\varphi _1}(x)\mathrm{d}x} - \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x}\le 0, \end{aligned}$$

that is

$$\begin{aligned} \int \limits _\Omega {{F_{ - \infty }}(x){\varphi _1}(x)\mathrm{d}x} \le \int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} \end{aligned}$$

which contradicts $(H_{21}).$

It implies that I is coercive in $X(\Omega )$. The Proposition 2.2 is proved. $\square$

Proof of Theorem 1.1(i):

The coerciveness (see Proposition 2.2) and the Palais–Smale condition (see Proposition 2.1) are enough to prove that the functional I attains its proper infimum at some $u_0$ in $X(\Omega )$ (see Theorem 1.3) so that the problem (1.1) has at least a weak solution $u_0\in X(\Omega )$. It is clear that $u_0$ is a nontrivial solution of the problem (1.1). $\square$

3 Proof of the Theorem 1.1(ii): (saddle point theorem and Existence of Weak Solutions)

First, we remark that by similar arguments as in the proof of proposition 2.1, with hypotheses $(H_{22})$, we can prove that the functional I given by (2.1) satisfies the (P–S) condition in $X(\Omega )$.

Splitting $X(\Omega )$ as the sum: $X(\Omega )=E\oplus Y$, where

$$\begin{aligned} \begin{aligned} E&=\{t\varphi _1, t\in {\mathbb {R}}\}.\\ Y&=\{v\in X(\Omega ): \int _{\Omega }\varphi _1^{p-1}v\mathrm{d}x=0\}, \end{aligned} \end{aligned}$$

(3.1)

where $\varphi _1$ is normalized eigenfunction associated with the eigenvalue $\lambda _1$ of the problem (1.5), $\varphi _1>0, x\in \Omega , \Vert \varphi _1\Vert =1$.

For $u=t\varphi _1+v, v\in Y$; then, we have

$$\begin{aligned} \int \limits _\Omega {u\varphi _1^{p - 1}\mathrm{d}x} = t\int \limits _\Omega {{{\left| {{\varphi _1}} \right| }^p}\mathrm{d}x} + \int \limits _\Omega {v\varphi _1^{p - 1}\mathrm{d}x}. \end{aligned}$$

Since $v\in Y, \int _{\Omega }{v\varphi _1^{p - 1}\mathrm{d}x}=0$ and by definition of $\lambda _1,$

$$\begin{aligned} \int _{\Omega }{\varphi _1^{p}\mathrm{d}x}=\frac{1}{\lambda _1}\Vert \varphi _1\Vert ^p=\frac{1}{\lambda _1}. \end{aligned}$$

Hence $t=\lambda _1\int _{\Omega }{u\varphi _1^{p - 1}\mathrm{d}x}$.

On the other hand, for any $u\in X(\Omega )$, take $t=\lambda _1\int _{\Omega }{u\varphi _1^{p - 1}\mathrm{d}x}$, $v=u-t\varphi _1$. It is clear that $v\in Y$. Thus $u=t\varphi _1+v, v\in Y$.

Lemma 3.1

There exists $\overline{\lambda }> \lambda _1$ such that

$$\begin{aligned} \left( Av,v\right) =\Vert v\Vert ^p >\overline{\lambda }\int _{\Omega } |v|^p \mathrm{d}x, \text { for all } v\in Y. \end{aligned}$$

Proof

Let

$$\begin{aligned} {\lambda } =\inf \{\left( Av,v\right) : \int _{\Omega } |v|^p \mathrm{d}x=1, v\in Y\}. \end{aligned}$$

We shall prove that value ${\lambda }$ is attained in Y.

Let $\{v_m\}$ in Y be a minimizing sequence, i.e.,

$$\begin{aligned} \int _{\Omega } |v_m|^p\mathrm{d}x=1, m=1,2,\ldots \end{aligned}$$

$$\begin{aligned} \mathop {\lim }\limits _{m \rightarrow + \infty } \left( {A{v_m},{v_m}} \right) = \mathop {\lim }\limits _{m \rightarrow + \infty } {\left\| {{v_m}} \right\| ^p} = {\lambda }. \end{aligned}$$

This implies that the sequence $\{v_m\}$ is bounded in $X(\Omega )$. Hence there exists a subsequence $\{v_{m_k}\}$ such as

$$\begin{aligned}&v_{m_k} \rightharpoonup v_0 \text { in } X(\Omega ),\\&v_{m_k} \rightarrow v_0 \text { in } L^p(\Omega ) \end{aligned}$$

provided the embedding $X(\Omega )$ into $L^p(\Omega )$ is compact.

Observe that

$$\begin{aligned} \left| {\int \limits _\Omega {\varphi _1^{p - 1}\left( {{v_{{m_k}}} - {v_0}} \right) \mathrm{d}x} } \right| \le \left\| {{\varphi _1}} \right\| _p^{p - 1}{\left\| {{v_{{m_k}}} - {v_o}} \right\| _p} \rightarrow 0 \text { as } m\rightarrow +\infty . \end{aligned}$$

Hence

$$\begin{aligned} 0 = \mathop {\lim }\limits _{k \rightarrow + \infty } \int \limits _\Omega {{v_{{m_k}}}(x)} \varphi _1^{p - 1}(x)\mathrm{d}x = \int \limits _\Omega {{v_0}(x)} \varphi _1^{p - 1}(x)\mathrm{d}x \end{aligned}$$

this implies that $v_0\in Y$.

Besides

$$\begin{aligned} 1 = \mathop {\lim }\limits _{k \rightarrow + \infty } \int \limits _\Omega {{{\left| {{v_{{m_k}}}(x)} \right| }^p}} \mathrm{d}x = \int \limits _\Omega {{{\left| {{v_0}(x)} \right| }^p}} \mathrm{d}x, \end{aligned}$$

so $v_0\not = 0.$

By the lower weak semicontinous of the functional $v\mapsto \Vert v\Vert ^p , v\in X(\Omega ),$ we have

$$\begin{aligned}{\lambda } \le \left( {A{v_0},{v_0}} \right) = {\left\| {{v_0}} \right\| ^p} \le \mathop {\lim }\limits _{k \rightarrow + \infty } \inf {\left\| {{v_{{m_k}}}} \right\| ^p} \le \mathop {\lim }\limits _{k \rightarrow + \infty } {\left\| {{v_{{m_k}}}} \right\| ^p} = \lambda . \end{aligned}$$

hence $\lambda =\Vert v_0\Vert ^p$. It means that the value $\lambda$ is attained at $v_0$.

By the variational characterization of the eigenvalue $\lambda _1$, it is clear that $\lambda \ge \lambda _1$.

If $\lambda =\lambda _1$, by simplicity of $\lambda _1$, there exists $t\in {\mathbb {R}}$ such that $v_0=t\varphi _1$.

But since $v_0\in Y$, we have

$$\begin{aligned} 0=\int _{\Omega }\varphi _1^{p-1}v_0\mathrm{d}x=t\int _{\Omega }\varphi _1^p\mathrm{d}x=t\Vert \varphi _1\Vert ^p_p, \end{aligned}$$

hence $t=0$ and then $v_0=0$ which a contradiction due to $v_0\not = 0$.

This implies that $\overline{\lambda }=\lambda >\lambda _1$ and the proof of the Lemma 2.1 is complete. $\square$

Proposition 3.1

The function I given by (2.1) is coercive on Y provided hypotheses $(H_1)$ and $(H_{22})$ hold.

Proof

Observe that by Holder inequality, Lemma 3.1 and hypotheses $(H_1)$ we have for any $v\in Y$:

$$\begin{aligned} \begin{aligned} \left| {I\left( v \right) } \right|&\ge \frac{1}{p}{\left\| v \right\| ^p} - \frac{{{\lambda _1}}}{p}\int \limits _\Omega {{{\left| v \right| }^p}\mathrm{d}x - } \int \limits _\Omega {\left| {F\left( {x,v} \right) } \right| \mathrm{d}x} - \int \limits _\Omega {k(x)\left| v \right| \mathrm{d}x} \\&\ge \frac{1}{p}\left( {1 - \frac{{{\lambda _1}}}{{\overline{\lambda }}}} \right) {\left\| v \right\| ^p} - \left( {{{\left\| \tau \right\| }_{p'}} + {{\left\| k \right\| }_{p'}}} \right) {\left\| v \right\| _p} \\&\ge \frac{1}{p}\left( {1 - \frac{{{\lambda _1}}}{{\overline{\lambda }}}} \right) {\left\| v \right\| ^p} - M\left( {{{\left\| \tau \right\| }_{p'}} + {{\left\| k \right\| }_{p'}}} \right) {\left\| v \right\| }, \\ \end{aligned} \end{aligned}$$

(3.2)

with M is positive.

From (3.2), since $p\ge 2$ , $1-\frac{\lambda _1}{\overline{\lambda }}>0$ it follows $|I(v)|\rightarrow +\infty \text { as } \Vert v\Vert \rightarrow +\infty$. So that the functional I is coercive on Y and Proposition 3.1 is proved. $\square$

From Proposition 3.1, it implies that

$$\begin{aligned}{B_Y} = \mathop {\min }\limits _{v \in Y} I(v) > - \infty . \end{aligned}$$

Remark that for every $t\in R$, we have

$$\begin{aligned}\frac{1}{p}{\left\| {t{\varphi _1}} \right\| ^p} - \frac{{{\lambda _1}}}{p}\int \limits _\Omega {{{\left| {t{\varphi _1}} \right| }^p}\mathrm{d}x = 0}, \end{aligned}$$

as follows from the definition of $\lambda _1$ and $\varphi _1$. Thus

$$\begin{aligned} \begin{aligned} I\left( {t{\varphi _1}} \right)&= t\int \limits _\Omega {k(x){\varphi _1}(x)\mathrm{d}x} - \int \limits _\Omega {F\left( {x,t{\varphi _1}} \right) \mathrm{d}x} \\&= t\int \limits _\Omega {\left( {k(x){\varphi _1}(x) - \frac{{F\left( {x,t{\varphi _1}} \right) }}{t}} \right) } \mathrm{d}x ,\\ \end{aligned} \end{aligned}$$

(3.3)

where

$$\begin{aligned} \frac{{F\left( {x,t{\varphi _1}} \right) }}{t} = \frac{1}{t}\int \limits _0^{t{\varphi _1}} {f\left( {x,s} \right) \mathrm{d}s}. \end{aligned}$$

Note that

$$\begin{aligned} \begin{aligned} \mathop {\lim }\limits _{t \rightarrow + \infty } \frac{{F\left( {x,t{\varphi _1}} \right) }}{t}&= \mathop {\lim }\limits _{t \rightarrow + \infty } \frac{1}{t}\int \limits _0^{t{\varphi _1}} {f\left( {x,s} \right) \mathrm{d}s} \\&= \mathop {\lim }\limits _{t \rightarrow + \infty } \frac{1}{t}\left( {\int \limits _0^t {f\left( {x,y{\varphi _1}} \right) \mathrm{d}y} } \right) {\varphi _1} = {F^{ + \infty }}(x){\varphi _1} \end{aligned} \end{aligned}$$

(3.4)

and

$$\begin{aligned} \begin{aligned} \mathop {\lim }\limits _{t \rightarrow - \infty } \frac{{F\left( {x,t{\varphi _1}} \right) }}{t}&= \mathop {\lim }\limits _{t \rightarrow - \infty }- \frac{1}{|t|}\int \limits _0^{t{\varphi _1}} {f\left( {x,s} \right) \mathrm{d}s} \\&= \mathop {\lim }\limits _{t \rightarrow - \infty } \frac{1}{|t|}\left( {\int \limits _0^{|t|} {f\left( {x,-y{\varphi _1}} \right) \mathrm{d}y} } \right) {\varphi _1} = {F_{ - \infty }}(x){\varphi _1}. \end{aligned} \end{aligned}$$

(3.5)

Hence by the hypotheses $(H_{22})$, from (3.3), (3.4), (3.5), we have

$$\begin{aligned} \begin{aligned} \mathop {\lim }\limits _{t \rightarrow + \infty } I\left( {t{\varphi _1}} \right)&= \mathop {\lim }\limits _{t \rightarrow + \infty } t\int \limits _\Omega {\left( {k(x){\varphi _1}(x) - \frac{{F\left( {x,t{\varphi _1}} \right) }}{t}} \right) } \mathrm{d}x \\&= \mathop {\lim }\limits _{t \rightarrow + \infty } t\int \limits _\Omega {\left( {k(x){\varphi _1}(x) - {F^{ + \infty }}(x){\varphi _1}} \right) } \mathrm{d}x = - \infty \end{aligned} \end{aligned}$$

(3.6)

and

$$\begin{aligned} \begin{aligned} \mathop {\lim }\limits _{t \rightarrow - \infty } I\left( {t{\varphi _1}} \right)&= \mathop {\lim }\limits _{t \rightarrow - \infty } t\int \limits _\Omega {\left( {k(x){\varphi _1}(x) - \frac{{F\left( {x,t{\varphi _1}} \right) }}{t}} \right) } \mathrm{d}x \\&= \mathop {\lim }\limits _{t \rightarrow - \infty } t\int \limits _\Omega {\left( {k(x){\varphi _1}(x) - {F_{ - \infty }}(x){\varphi _1}} \right) } \mathrm{d}x = - \infty . \end{aligned} \end{aligned}$$

(3.7)

Thus there exists $R>0$ such that for any $t: |t|=R$ we have

$$\begin{aligned} I(t\varphi _1)<B_Y\le I(v) \text { for all } v\in Y.\end{aligned}$$

From this, we can finish the proof of Theorem 1.1 (ii).

Proof of Theorem 1.1(ii):

By Proposition 3.1, applying the saddle point theorem (see Theorem 1.4) we deduce that the functional I attains its proper infimum at some $u_0\in X(\Omega )$, so that the problem (1.1) has at least a weak solution $u_0\in X(\Omega )$ and it is clear that $u_0\not =0$.

The Theorem 1.2 is proved. $\square$