Positivity of the infimum eigenvalue for equations of $p(x)$-Laplace type in ${\mathbb{R}}^N$

Abstract

We study the following elliptic equations with variable exponents

$$-div\left(\varphi (x,|\mathrm{\nabla }u|)\mathrm{\nabla }u\right)=\lambda f(x,u)\text{in }{\mathbb{R}}^N.$$

Under suitable conditions on ϕ and f, we show the existence of positivity of the infimum of all eigenvalues for the problem above, and then give an example to demonstrate our main result.

MSC:35D30, 35J60, 35J92, 35P30, 47J10.

1 Introduction

The variable exponent problems appear in a lot of applications, for example, elastic mechanics, electro-rheological fluid dynamics and image processing, etc. The study of variable mathematical problems involving $p(x)$-growth conditions has attracted interest and attention in recent years. We refer the readers to [1–4] and references therein.

In this paper, we are concerned with the eigenvalue problem of a class of equations of $p(x)$-Laplacian type

$$-div\left(\varphi (x,|\mathrm{\nabla }u|)\mathrm{\nabla }u\right)=\lambda f(x,u)\text{in }{\mathbb{R}}^N,$$

(E)

where the function $\varphi (x,t)$ is of type ${|t|}^{p(x)-2}$ with continuous nonconstant function $p:{\mathbb{R}}^N\to (1,\mathrm{\infty })$ and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ satisfies a Carathéodory condition. Recently, the authors in [5] obtained the positivity of the infimum of all eigenvalues for the $p(x)$-Laplacian type subject to the Dirichlet boundary condition. As far as the authors know, there are no results concerned with the eigenvalue problem for a more general $p(x)$-Laplacian type problem in the whole space ${\mathbb{R}}^N$.

When $\varphi (x,t)={|t|}^{p(x)-2}$, the operator involved in (E) is called the $p(x)$-Laplacian, i.e., ${\mathrm{\Delta }}_{p(x)}u:=div({|\mathrm{\nabla }u|}^{p(x)-2}\mathrm{\nabla }u)$. The studies for $p(x)$-Laplacian problems have been extensively performed by many researchers in various ways; see [5–11]. In particular, by using the Ljusternik-Schnirelmann critical point theory, Fan et al. [8] established the existence of the sequence of eigenvalues of the $p(x)$-Laplacian Dirichlet problem; see [12] for Neumann problems. Mihăilescu and Rădulescu in [13] obtained the existence of a continuous family of eigenvalues in a neighborhood of the origin under suitable conditions.

The $p(x)$-Laplacian is a natural generalization of the p-Laplacian, where $p>1$ is a constant. There are a bunch of papers, for instance, [14–18] and references therein. But the $p(x)$-Laplace operator possesses more complicated nonlinearities than the p-Laplace operator, for example, it is nonhomogeneous, so a more complicated analysis has to be carefully carried out. Some properties of the p-Laplacian eigenvalue problems may not hold for a general $p(x)$-Laplacian. For example, under some conditions, the infimum of all eigenvalues for the $p(x)$-Laplacian might be zero; see [8]. The purpose of this paper is to give suitable conditions on ϕ and f to satisfy the positivity of the infimum of all eigenvalues for (E) still. This result generalizes Benouhiba’s recent result in [6] in some sense.

This paper is organized as follows. In Section 2, we state some basic results for the variable exponent Lebesgue-Sobolev spaces, which are given in [19, 20]. In Section 3, we give sufficient conditions on ϕ and f to obtain the positivity of the infimum eigenvalue for the problem (E) above. Also, we present an example to illustrate our main result.

2 Preliminaries

In this section, we state some elementary properties for the variable exponent Lebesgue-Sobolev spaces, which will be used in the next section. The basic properties of the variable exponent Lebesgue-Sobolev spaces can be found from [19, 20].

To make a self-contained paper, we first recall some definitions and basic properties of the variable exponent Lebesgue spaces $L^{p(x)}({\mathbb{R}}^N)$ and the variable exponent Lebesgue-Sobolev spaces $W^{1,p(x)}({\mathbb{R}}^N)$.

Set

$$C_{+}\left({\mathbb{R}}^N\right)=\{h\in C\left({\mathbb{R}}^N\right):\underset{x\in {\mathbb{R}}^N}{inf}h(x)>1\}.$$

For any $h\in C_{+}({\mathbb{R}}^N)$, we define

$$h_{+}=\underset{x\in {\mathbb{R}}^N}{sup}h(x)\text{and}{h}_{-}=\underset{x\in {\mathbb{R}}^N}{inf}h(x).$$

For any $p\in C_{+}({\mathbb{R}}^N)$, we introduce the variable exponent Lebesgue space

$$L^{p(x)}\left({\mathbb{R}}^N\right):=\{u:u\text{ is a measurable real-valued function, }{\int }_{{\mathbb{R}}^N}{|u(x)|}^{p(x)}dx<\mathrm{\infty }\},$$

endowed with the Luxemburg norm

$${\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}=inf\{\lambda >0:{\int }_{{\mathbb{R}}^N}{\left|\frac{u(x)}{\lambda }\right|}^{p(x)}dx\le 1\}.$$

The dual space of $L^{p(x)}({\mathbb{R}}^N)$ is $L^{p^{\prime }(x)}({\mathbb{R}}^N)$, where $1/p(x)+1/p^{\prime }(x)=1$. The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in [21].

The variable exponent Sobolev space $W^{1,p(x)}({\mathbb{R}}^N)$ is defined by

$$W^{1,p(x)}\left({\mathbb{R}}^N\right)=\{u\in L^{p(x)}\left({\mathbb{R}}^N\right):|\mathrm{\nabla }u|\in L^{p(x)}\left({\mathbb{R}}^N\right)\},$$

where the norm is

$${\parallel u\parallel }_{W^{1,p(x)}({\mathbb{R}}^N)}={\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}+{\parallel \mathrm{\nabla }u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}.$$

(2.1)

Definition 2.1 The exponent $p(\cdot )$ is said to be log-Hölder continuous if there is a constant C such that

$$|p(x)-p(y)|\le \frac{C}{-log|x-y|}$$

(2.2)

for every $x,y\in {\mathbb{R}}^N$ with $|x-y|\le 1/2$.

Smooth functions are not dense in the variable exponent Sobolev spaces, without additional assumptions on the exponent $p(x)$. Zhikov [22] gave some examples of Lavrentiev’s phenomenon for the problems with variable exponents. These examples show that smooth functions are not dense in variable exponent Sobolev spaces. However, when $p(x)$ satisfies the log-Hölder continuity condition, smooth functions are dense in variable exponent Sobolev spaces, and there is no confusion in defining the Sobolev space with zero boundary values, $W_0^{1,p(x)}({\mathbb{R}}^N)$, as the completion of $C_0^{\mathrm{\infty }}({\mathbb{R}}^N)$ with respect to the norm ${\parallel u\parallel }_{W^{1,p(x)}({\mathbb{R}}^N)}$ (see [23, 24]).

Lemma 2.2 [19, 20]

The space $L^{p(x)}({\mathbb{R}}^N)$ is a separable, uniformly convex Banach space, and its conjugate space is $L^{p^{\prime }(x)}({\mathbb{R}}^N)$, where $1/p(x)+1/p^{\prime }(x)=1$. For any $u\in L^{p(x)}({\mathbb{R}}^N)$ and $v\in L^{p^{\prime }(x)}({\mathbb{R}}^N)$, we have

$$\left|{\int }_{{\mathbb{R}}^N}uvdx\right|\le (\frac{1}{p_{-}}+\frac{1}{p_{-}^{\prime }}){\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}{\parallel v\parallel }_{L^{p^{\prime }(x)}({\mathbb{R}}^N)}\le 2{\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}{\parallel v\parallel }_{L^{p^{\prime }(x)}({\mathbb{R}}^N)}.$$

Lemma 2.3 [19]

Denote

$$\rho (u)={\int }_{{\mathbb{R}}^N}{|u|}^{p(x)}dx\mathit{\text{for all }}u\in L^{p(x)}\left({\mathbb{R}}^N\right).$$

Then

• (1) $\rho (u)>1$ (=1; <1) if and only if ${\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}>1$ (=1; <1), respectively;

• (2) if ${\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}>1$, then ${\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}^{p_{-}}\le \rho (u)\le {\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}^{p_{+}}$;

• (3) if ${\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}<1$, then ${\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}^{p_{+}}\le \rho (u)\le {\parallel u\parallel }_{L^{p(x)}({\mathbb{R}}^N)}^{p_{-}}$.

Lemma 2.4 [11]

Let $q\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ be such that $1\le p(x)q(x)\le \mathrm{\infty }$ for almost all $x\in {\mathbb{R}}^N$. If $u\in L^{q(x)}({\mathbb{R}}^N)$ with $u\ne 0$, then

• (1) if ${\parallel u\parallel }_{L^{p(x)q(x)}({\mathbb{R}}^N)}>1$, then ${\parallel u\parallel }_{L^{p(x)q(x)}({\mathbb{R}}^N)}^{q_{-}}\le {\parallel {|u|}^{q(x)}\parallel }_{L^{p(x)}({\mathbb{R}}^N)}\le {\parallel u\parallel }_{L^{p(x)q(x)}({\mathbb{R}}^N)}^{q_{+}}$;

• (2) if ${\parallel u\parallel }_{L^{p(x)q(x)}({\mathbb{R}}^N)}<1$, then ${\parallel u\parallel }_{L^{p(x)q(x)}({\mathbb{R}}^N)}^{q_{+}}\le {\parallel {|u|}^{q(x)}\parallel }_{L^{p(x)}({\mathbb{R}}^N)}\le {\parallel u\parallel }_{L^{p(x)q(x)}({\mathbb{R}}^N)}^{q_{-}}$.

Lemma 2.5 [23]

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be an open, bounded set with Lipschitz boundary, and let $p\in C_{+}(\overline{\mathrm{\Omega }})$ with $1<p_{-}\le p_{+}<N$ satisfy the log-Hölder continuity condition (2.2). If $q\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ with $q_{-}>1$ satisfies

$$q(x)\le p^{\ast }(x):=\frac{Np(x)}{N-p(x)}\mathit{\text{for all }}x\in \mathrm{\Omega },$$

then we have

$$W^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{q(x)}(\mathrm{\Omega }),$$

and the imbedding is compact if ${inf}_{x\in \mathrm{\Omega }}(p^{\ast }(x)-q(x))>0$.

Lemma 2.6 [25]

Suppose that $p:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a Lipschitz function with $1<p_{-}\le p_{+}<N$. Let $q\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and $p(x)\le q(x)\le p^{\ast }(x)$ for almost all $x\in {\mathbb{R}}^N$. Then there is a continuous embedding $W^{1,p(x)}({\mathbb{R}}^N)\hookrightarrow L^{q(x)}({\mathbb{R}}^N)$.

3 Main result

In this section, we shall give the proof of the existence of the positive eigenvalue for the problem (E), by applying the basic properties of the spaces $L^{p(x)}({\mathbb{R}}^N)$ and $W^{1,p(x)}({\mathbb{R}}^N)$, which were given in the previous section.

Throughout this paper, let $p\in C_{+}({\mathbb{R}}^N)$ satisfy the log-Hölder continuity condition (2.2) and $X:=W_0^{1,p(x)}({\mathbb{R}}^N)$ with the norm

$${\parallel u\parallel }_X=inf\{\lambda >0:{\int }_{{\mathbb{R}}^N}|\frac{\mathrm{\nabla }u(x)}{\lambda }{|}^{p(x)}dx\le 1\},$$

which is equivalent to norm (2.1).

Definition 3.1 We say that $u\in X$ is a weak solution of the problem (E) if

$${\int }_{{\mathbb{R}}^N}\varphi (x,|\mathrm{\nabla }u|)\mathrm{\nabla }u(x)\cdot \mathrm{\nabla }\phi (x)dx=\lambda {\int }_{{\mathbb{R}}^N}f(x,u)\phi (x)dx$$

for all $\phi \in X$.

Denote

$${\mathrm{\Omega }}_1=\{x\in {\mathbb{R}}^N:1<p(x)<2\},{\mathrm{\Omega }}_2=\{x\in {\mathbb{R}}^N:p(x)\ge 2\}$$

(we allow the case that one of these sets is empty). Then it is obvious that ${\mathbb{R}}^N={\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2$. We assume that:

(H1) $p,q\in C_{+}({\mathbb{R}}^N)$, $p(x)<N$, and $1<p_{-}\le p_{+}<q_{-}\le q_{+}<p^{\ast }(x)$.

• (HJ1) $\varphi :{\mathbb{R}}^N\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ satisfies the following conditions: $\varphi (\cdot ,\eta )$ is measurable on ${\mathbb{R}}^N$ for all $\eta \ge 0$ and $\varphi (x,\cdot )$ is locally absolutely continuous on $[0,\mathrm{\infty })$ for almost all $x\in {\mathbb{R}}^N$.

• (HJ2) There are a function $a\in L^{p^{\prime }(x)}({\mathbb{R}}^N)$ and a nonnegative constant b such that

  $$\left|\varphi (x,|v|)v\right|\le a(x)+b{|v|}^{p(x)-1}$$

  for almost all $x\in {\mathbb{R}}^N$ and for all $v\in {\mathbb{R}}^N$.

• (HJ3) There exists a positive constant c such that the following conditions are satisfied for almost all $x\in {\mathbb{R}}^N$:

  $$\varphi (x,\eta )\ge c{\eta }^{p(x)-2}\text{and}\eta \frac{\partial \varphi }{\partial \eta }(x,\eta )+\varphi (x,\eta )\ge c{\eta }^{p(x)-2}$$

  (3.1)

  for almost all $\eta \in (0,1)$. In case $x\in {\mathrm{\Omega }}_2$, assume that condition (3.1) holds for almost all $\eta \in (1,\mathrm{\infty })$, and in case $x\in {\mathrm{\Omega }}_1$, assume that for almost all $\eta \in (1,\mathrm{\infty })$ instead

  $$\varphi (x,\eta )\ge c\text{and}\eta \frac{\partial \varphi }{\partial \eta }(x,\eta )+\varphi (x,\eta )\ge c.$$

  (3.2)

• (HJ4) For all $x\in {\mathbb{R}}^N$ and all $\xi \in {\mathbb{R}}^N$, the estimate holds

  $$0\le a(x,\xi )\cdot \xi \le p_{+}{\mathrm{\Phi }}_0(x,|\xi |),$$

  where $a(x,\xi )=\varphi (x,|\xi |)\xi$.

Let us put

$${\mathrm{\Phi }}_0(x,t)={\int }_0^t\varphi (x,\eta )\eta d\eta$$

and define the functional $\mathrm{\Phi }:X\to \mathbb{R}$ by

$$\mathrm{\Phi }(u)={\int }_{{\mathbb{R}}^N}{\mathrm{\Phi }}_0(x,|\mathrm{\nabla }u(x)|)dx.$$

Then $\mathrm{\Phi }\in C^1(X,\mathbb{R})$ [5], and its Gateaux derivative is

$$〈{\mathrm{\Phi }}^{\prime }(u),\phi 〉:={\int }_{{\mathbb{R}}^N}\varphi (x,|\mathrm{\nabla }u(x)|)\mathrm{\nabla }u(x)\cdot \mathrm{\nabla }\phi (x)dx.$$

(3.3)

Let $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ be a real-valued function. We assume that the function f satisfies the Carathéodory condition in the sense that $f(\cdot ,t)$ is measurable for all $t\in \mathbb{R}$ and $f(x,\cdot )$ is continuous for almost all $x\in {\mathbb{R}}^N$. Denote

$$\gamma (x)=\frac{r(x)}{r(x)-q(x)}\text{for almost all }x\in {\mathbb{R}}^N,$$

where q is given in (H1) and $q(x)<r(x)<p^{\ast }(x)$. We assume that

(F1) For all $(x,t)\in {\mathbb{R}}^N\times \mathbb{R}$, $f(x,t)t\ge 0$, and there is a nonnegative measurable function m with $m\in L^{\gamma (x)}({\mathbb{R}}^N)$ such that

$$|f(x,t)|\le m(x){|t|}^{q(x)-1}.$$

Denoting $F(x,t)={\int }_0^{t}f(x,s)ds$, it follows from (F1) that

$$\text{(F1}{}^{\prime }\text{)}0\le F(x,t)\le \frac{m(x)}{q(x)}{|t|}^{q(x)}\text{for all }(x,t)\in {\mathbb{R}}^N\times \mathbb{R}.$$

Define the functional $\mathrm{\Psi },I_{\lambda }:X\to \mathbb{R}$ by

$$\mathrm{\Psi }(u)={\int }_{{\mathbb{R}}^N}F(x,u)dx\text{and}{I}_{\lambda }(u)=\mathrm{\Phi }(u)-\lambda \mathrm{\Psi }(u).$$

Then it is easy to check that $\mathrm{\Psi }\in C^1(X,\mathbb{R})$, and its Gateaux derivative is

$$〈{\mathrm{\Psi }}^{\prime }(u),\phi 〉={\int }_{{\mathbb{R}}^N}f(x,u)\phi (x)dx$$

(3.4)

for any $u,\phi \in X$. Let us consider the following quantity:

$${\lambda }^{\ast }=\underset{u\in X\setminus \{0\}}{inf}\frac{{\int }_{{\mathbb{R}}^N}{\mathrm{\Phi }}_0(x,|\mathrm{\nabla }u|)dx}{{\int }_{{\mathbb{R}}^N}F(x,u)dx}.$$

(3.5)

For the case of $\varphi (x,|t|)={|t|}^{p(x)-2}$ and $f(x,t)=m(x){|t|}^{q(x)-2}t$, where $m(x)$ satisfies a suitable condition, Benouhiba [6] proved that ${\lambda }^{\ast }>0$. In this section, we shall generalize the conditions on f and ϕ to satisfy ${\lambda }^{\ast }>0$ still.

The following lemma plays a key role in obtaining the main result in this section.

Lemma 3.2 Assume that assumptions (HJ3)-(HJ4), (H1), and (F1) hold and satisfy

$$\text{(H2)}{q}_{+}-\frac{1}{2}{p}_{-}<q_{-},$$

then the functionals Φ and Ψ satisfy the following relations:

$$\underset{{\parallel u\parallel }_X\to 0}{lim}\frac{\mathrm{\Phi }(u)}{\mathrm{\Psi }(u)}=\mathrm{\infty }$$

(3.6)

and

$$\underset{{\parallel u\parallel }_X\to \mathrm{\infty }}{lim}\frac{\mathrm{\Phi }(u)}{\mathrm{\Psi }(u)}=\mathrm{\infty }.$$

(3.7)

Proof Applying Lemmas 2.2, 2.4 and 2.6, we get

$$\begin{array}{rcl}|\mathrm{\Psi }(u)|& =& |{\int }_{{\mathbb{R}}^N}F(x,u)dx|\\ \le & {\int }_{{\mathbb{R}}^N}\left|\frac{m(x)}{q(x)}{|u|}^{q(x)}\right|dx\\ \le & \frac{2}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}{\parallel {|u|}^{q(x)}\parallel }_{L^{\frac{r(x)}{q(x)}}({\mathbb{R}}^N)}\\ \le & \frac{2}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}({\parallel u\parallel }_{L^{r(x)}({\mathbb{R}}^N)}^{q_{+}}+{\parallel u\parallel }_{L^{r(x)}({\mathbb{R}}^N)}^{q_{-}})\\ \le & \frac{2C}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}({\parallel u\parallel }_X^{q_{+}}+{\parallel u\parallel }_X^{q_{-}})\end{array}$$

(3.8)

for some positive constant C. Let u in X with ${\parallel u\parallel }_X\le 1$. Then it follows from (HJ3), (HJ4), (3.8) and Lemma 2.3(3) that

$$\left|\frac{\mathrm{\Phi }(u)}{\mathrm{\Psi }(u)}\right|\ge \frac{{\int }_{{\mathbb{R}}^N}{\mathrm{\Phi }}_0(x,|\mathrm{\nabla }u|)dx}{\frac{4C}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}{\parallel u\parallel }_X^{q_{-}}}\ge \frac{\frac{c}{p_{+}}{\parallel u\parallel }_X^{p_{+}}}{\frac{4C}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}{\parallel u\parallel }_X^{q_{-}}}.$$

(3.9)

Since $q_{-}>p_{+}$, we conclude that

$$\frac{\mathrm{\Phi }(u)}{\mathrm{\Psi }(u)}\to \mathrm{\infty }\text{as }{\parallel u\parallel }_X\to 0.$$

Next, we show that relation (3.7) holds. From (H2), there exists a positive constant δ such that $q_{+}-(1/2)p_{-}<\delta <q_{-}$, and thus we have

$$p_{-}>2(q_{+}-\delta )>2(q_{-}-\delta ).$$

(3.10)

Let $\ell (x)$ be a measurable function such that

$$\begin{array}{c}max\{\frac{p(x)\gamma (x)}{p(x)+\delta \gamma (x)},\frac{p^{\ast }(x)}{p^{\ast }(x)+\delta -q(x)}\}\hfill \\ \le \ell (x)\le min\{\frac{p^{\ast }(x)\gamma (x)}{p^{\ast }(x)+\delta \gamma (x)},\frac{p(x)}{p(x)+\delta -q(x)}\}\hfill \end{array}$$

(3.11)

holds for almost all $x\in {\mathbb{R}}^N$ and

$$\delta (\frac{{\ell }_{+}}{{\ell }_{-}}+1)<q_{-}.$$

(3.12)

Then we have $\ell \in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and $1<\ell (x)<\gamma (x)$. Let $u\in X$ with ${\parallel u\parallel }_X>1$. Then it follows from (F1′) and Lemma 2.2 that

$$\begin{array}{rcl}|\mathrm{\Psi }(u)|& \le & \frac{1}{q_{-}}{\int }_{{\mathbb{R}}^N}m(x){|u|}^{\delta }{|u|}^{q(x)-\delta }dx\\ \le & \frac{2}{q_{-}}{\parallel m{|u|}^{\delta }\parallel }_{L^{\ell (x)}({\mathbb{R}}^N)}{\parallel {|u|}^{q(x)-\delta }\parallel }_{L^{{\ell }^{\prime }(x)}({\mathbb{R}}^N)}.\end{array}$$

Therefore, without loss of generality, we may suppose that ${\parallel m{|u|}^{\delta }\parallel }_{L^{\ell (x)}({\mathbb{R}}^N)}>1$. From the inequality above, by using Lemma 2.3, Lemma 2.2 and Lemma 2.4 in order, we have

$$\begin{array}{rcl}|\mathrm{\Psi }(u)|& \le & \frac{2}{q_{-}}{\left({\int }_{{\mathbb{R}}^N}{m}^{\ell (x)}{|u|}^{\delta \ell (x)}\right)}^{\frac{1}{{\ell }_{-}}}{\parallel {|u|}^{q(x)-\delta }\parallel }_{L^{{\ell }^{\prime }(x)}({\mathbb{R}}^N)}\\ \le & \frac{4}{q_{-}}{\parallel m^{\ell (x)}\parallel }_{L^{\frac{\gamma (x)}{\ell (x)}}({\mathbb{R}}^N)}^{\frac{1}{{\ell }_{-}}}{\parallel {|u|}^{\delta \ell (x)}\parallel }_{L^{{(\frac{\gamma (x)}{\ell (x)})}^{\prime }}({\mathbb{R}}^N)}^{\frac{1}{{\ell }_{-}}}{\parallel {|u|}^{q(x)-\delta }\parallel }_{L^{{\ell }^{\prime }(x)}({\mathbb{R}}^N)}\\ \le & \frac{4}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}^{\alpha }({\parallel u\parallel }_{L^{\delta \ell (x){(\frac{\gamma (x)}{\ell (x)})}^{\prime }}({\mathbb{R}}^N)}^{\delta \frac{{\ell }_{+}}{{\ell }_{-}}}+{\parallel u\parallel }_{L^{\delta \ell (x){(\frac{\gamma (x)}{\ell (x)})}^{\prime }}({\mathbb{R}}^N)}^{\delta })\\ \times ({\parallel u\parallel }_{L^{(q(x)-\delta ){\ell }^{\prime }(x)}({\mathbb{R}}^N)}^{q_{+}-\delta }+{\parallel u\parallel }_{L^{(q(x)-\delta ){\ell }^{\prime }(x)}({\mathbb{R}}^N)}^{q_{-}-\delta }),\end{array}$$

where $\alpha =\{\begin{array}{cc}{\ell }_{+}/{\ell }_{-}\hfill & \text{if }{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}>1,\hfill \\ 1\hfill & \text{if }{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}\le 1.\hfill \end{array}$

By Young’s inequality, we get

$$\begin{array}{rcl}|\mathrm{\Psi }(u)|& \le & \frac{4}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}^{\alpha }({\parallel u\parallel }_{L^{\delta \ell (x){(\frac{\gamma (x)}{\ell (x)})}^{\prime }}({\mathbb{R}}^N)}^{2\delta \frac{{\ell }_{+}}{{\ell }_{-}}}+{\parallel u\parallel }_{L^{\delta \ell (x){(\frac{\gamma (x)}{\ell (x)})}^{\prime }}({\mathbb{R}}^N)}^{2\delta }\\ +{\parallel u\parallel }_{L^{(q(x)-\delta ){\ell }^{\prime }(x)}({\mathbb{R}}^N)}^{2(q_{+}-\delta )}+{\parallel u\parallel }_{L^{(q(x)-\delta ){\ell }^{\prime }(x)}({\mathbb{R}}^N)}^{2(q_{-}-\delta )}).\end{array}$$

Using (3.11), we get that

$$p(x)<\delta \ell (x){\left(\frac{\gamma (x)}{\ell (x)}\right)}^{\prime }\le p^{\ast }(x),p(x)<(q(x)-\delta ){\ell }^{\prime }(x)\le p^{\ast }(x)$$

holds for almost all $x\in {\mathbb{R}}^N$. Hence it follows from Lemma 2.6 that

$$|\mathrm{\Psi }(u)|\le \frac{4C}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}^{\alpha }({\parallel u\parallel }_X^{2\delta \frac{{\ell }_{+}}{{\ell }_{-}}}+{\parallel u\parallel }_X^{2(q_{+}-\delta )})$$

(3.13)

for some positive constant C. Therefore, we obtain that

$$\left|\frac{\mathrm{\Phi }(u)}{\mathrm{\Psi }(u)}\right|\ge \frac{\frac{c}{p_{+}}{\parallel u\parallel }_X^{p_{-}}}{\frac{4C}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}^{\alpha }({\parallel u\parallel }_X^{2\delta \frac{{\ell }_{+}}{{\ell }_{-}}}+{\parallel u\parallel }_X^{2(q_{+}-\delta )})}.$$

From (3.10), with the inequality above, we conclude that relation (3.7) holds. □

Lemma 3.3 Assume that (HJ1)-(HJ3) and (H1) hold. Then Φ is weakly lower semi-continuous, i.e., $u_n\rightharpoonup u$ in X implies that $\mathrm{\Phi }(u)\le lim{inf}_{n\to \mathrm{\infty }}\mathrm{\Phi }(u_n)$.

Proof Suppose that $u_n\rightharpoonup u$ in X as $n\to \mathrm{\infty }$. Since (HJ3) implies that ${\mathrm{\Phi }}^{\prime }$ is strictly monotone on X, we have that Φ is convex, and so,

$$\mathrm{\Phi }(u_n)\ge \mathrm{\Phi }(u)+〈{\mathrm{\Phi }}^{\prime }(u),u_n-u〉$$

for any n. Then we get that

$$lim\underset{n\to \mathrm{\infty }}{inf}\mathrm{\Phi }(u_n)\ge \mathrm{\Phi }(u)+lim\underset{n\to \mathrm{\infty }}{inf}〈{\mathrm{\Phi }}^{\prime }(u),u_n-u〉=\mathrm{\Phi }(u).$$

The proof is complete. □

Lemma 3.4 Assume that (H1) and (F1) hold. For any $K\in [0,\mathrm{\infty })$ and all $u\in X$, the following estimate holds:

$${\int }_{|x|\ge K}F(x,u)dx\le \frac{2C}{q_{-}}{({\int }_{|x|\ge K}m(x)dx)}^{\frac{1}{{\gamma }_1}}({\parallel u\parallel }_X^{q_{+}}+{\parallel u\parallel }_X^{q_{-}}),$$

(3.14)

where ${\gamma }_1$ is either ${\gamma }_{+}$ or ${\gamma }_{-}$.

Proof Applying Lemmas 2.2, 2.4 and 2.6, we get

$$\begin{array}{rcl}{\int }_{|x|\ge K}F(x,u)dx& \le & {\int }_{|x|\ge K}\frac{m(x)}{q(x)}{|u|}^{q(x)}dx\\ \le & \frac{2}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}(\{x\in {\mathbb{R}}^N:|x|\ge K\})}{\parallel {|u|}^{q(x)}\parallel }_{L^{\frac{r(x)}{q(x)}}(\{x\in {\mathbb{R}}^N:|x|\ge K\})}\\ \le & \frac{2}{q_{-}}{({\int }_{|x|\ge K}m(x)dx)}^{\frac{1}{{\gamma }_1}}({\parallel u\parallel }_{L^{r(x)}({\mathbb{R}}^N)}^{q_{+}}+{\parallel u\parallel }_{L^{r(x)}({\mathbb{R}}^N)}^{q_{-}})\\ \le & \frac{2C}{q_{-}}{({\int }_{|x|\ge K}m(x)dx)}^{\frac{1}{{\gamma }_1}}({\parallel u\parallel }_X^{q_{+}}+{\parallel u\parallel }_X^{q_{-}})\end{array}$$

for some positive constant C. □

Lemma 3.5 Assume that (H1) and (F1) hold. For almost all $x\in {\mathbb{R}}^N$ and all $t\in \mathbb{R}$, the following estimate holds:

$$F(x,t)\le \frac{1}{q_{-}}(\frac{m{(x)}^{\gamma (x)}}{{\gamma }_{-}}+\frac{{|t|}^{r(x)}}{{({\gamma }_{+})}^{\prime }}).$$

(3.15)

Proof Since $q(x){(\gamma (x))}^{\prime }=r(x)$, estimate (3.15) is obtained from (F1′) and Young’s inequality. □

Lemma 3.6 Assume that (H1) and (F1) hold. Then Ψ is weakly-strongly continuous, i.e., $u_n\rightharpoonup u$ in X implies that $\mathrm{\Psi }(u_n)\to \mathrm{\Psi }(u)$.

Proof Let $\{u_n\}$ be a sequence in X such that $u_n\rightharpoonup u$ in X. Then $\{u_n\}$ is bounded in X. By Lemma 3.4, for each $\epsilon >0$, there is a positive constant $K_{\epsilon }$ such that

$${\int }_{|x|\ge K_{\epsilon }}F(x,u_n)dx\le \epsilon \text{and}{\int }_{|x|\ge K_{\epsilon }}F(x,u)dx\le \epsilon$$

(3.16)

holds for each $n\in \mathbb{N}$. It follows from Lemma 3.5 that the Nemytskij operator

$$u\mapsto F(x,u(x))$$

is continuous from $L^{r(x)}(B_{K_{\epsilon }}(0))$ into $L^1(B_{K_{\epsilon }}(0))$; see Theorem 1.1 in [26]. This together with Lemma 2.5 yields that

$${\int }_{|x|<K_{\epsilon }}F(x,u_n)dx\to {\int }_{|x|<K_{\epsilon }}F(x,u)dx.$$

(3.17)

Using (3.16) and (3.17), we deduce that $\mathrm{\Psi }(u_n)\to \mathrm{\Psi }(u)$ as $n\to \mathrm{\infty }$. The proof is complete. □

We are in a position to state the main result about the existence of the positive eigenvalue for the problem (E).

Theorem 3.7 Assume that (HJ1)-(HJ4), (H1), (H2), and (F1) hold. Then ${\lambda }^{\ast }$ is a positive eigenvalue of the problem (E). Moreover, the problem (E) has a nontrivial weak solution for any $\lambda \ge {\lambda }^{\ast }$.

Proof It is trivial by (3.5) that ${\lambda }^{\ast }\ge 0$. Suppose to the contrary that ${\lambda }^{\ast }=0$. Let $\{u_n\}$ be a sequence in $X\setminus \{0\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\mathrm{\Phi }(u_n)}{\mathrm{\Psi }(u_n)}=0.$$

As in (3.9), we have

$$\left|\frac{\mathrm{\Phi }(u_n)}{\mathrm{\Psi }(u_n)}\right|\ge C{\parallel u_n\parallel }_X^{p_{+}-q_{-}}$$

for some positive constant C. Since $p_{+}<q_{-}$, we obtain that ${\parallel u_n\parallel }_X\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. Hence it follows from Lemma 3.2 that

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\mathrm{\Phi }(u_n)}{\mathrm{\Psi }(u_n)}=\mathrm{\infty },$$

which contradicts with the hypothesis. Hence we get ${\lambda }^{\ast }>0$. The analogous argument as that in the proof of Theorem 4.5 in [5] proves that ${\lambda }^{\ast }$ is an eigenvalue of the problem (E); see also Theorem 3.1 in [6].

Finally, we show that the problem (E) has a nontrivial weak solution for any $\lambda \ge {\lambda }^{\ast }$. Notice that u is a weak solution of (E) if and only if u is a critical point of $I_{\lambda }$. Assume that $\lambda >{\lambda }^{\ast }$ is fixed. Let $u\in X$ with ${\parallel u\parallel }_X>1$. With the help of (HJ3) and (HJ4), it follows from proceeding as in the proof of relation (3.13) in Lemma 3.2 that

$$I_{\lambda }(u)\ge \frac{c}{p_{+}}{\parallel u\parallel }_X^{p_{-}}-\lambda \frac{4C}{q_{-}}{\parallel m\parallel }_{L^{\gamma (x)}({\mathbb{R}}^N)}^{\alpha }({\parallel u\parallel }_X^{2\delta \frac{{\ell }_{+}}{{\ell }_{-}}}+{\parallel u\parallel }_X^{2(q_{+}-\delta )}).$$

Since $p_{-}>2(q_{+}-\delta )>2\delta ({\ell }_{+}/{\ell }_{-})$, the inequality above implies that $I_{\lambda }(u)\to \mathrm{\infty }$ as ${\parallel u\parallel }_X\to \mathrm{\infty }$ for $\lambda >{\lambda }^{\ast }$, that is, $I_{\lambda }$ is coercive. Also since the functional $I_{\lambda }$ is weakly lower semi-continuous by Lemmas 3.3 and 3.6, we deduce that there exists a global minimizer $u_0$ of $I_{\lambda }$ in X. Since $\lambda >{\lambda }^{\ast }$, we verify by definition (3.5) that there is an element ω in $X\setminus \{0\}$ such that $\mathrm{\Phi }(\omega )/\mathrm{\Psi }(\omega )<\lambda$. Then $I_{\lambda }(\omega )<0$. So we obtain that

$$I_{\lambda }(u_0)=\underset{v\in X\setminus \{0\}}{inf}{I}_{\lambda }(v)<0.$$

Consequently, we conclude that $u_0\not\equiv 0$. This completes the proof. □

Now, we consider an example to demonstrate our main result in this section.

Example 3.8 Let $p\in C({\mathbb{R}}^N)$ with $2\le p(x)<N$ satisfy the log-Hölder continuity condition (2.2). Suppose that $a\in L^{2p^{\prime }(x)}({\mathbb{R}}^N)$, and there is a positive constant $a_0$ such that $a(x)\ge a_0$ for almost all $x\in {\mathbb{R}}^N$. Let us consider

In this case, put

$$\varphi (x,|v|)={(a(x)+{|v|}^2)}^{\frac{p(x)-2}{2}}\text{and}{\mathrm{\Phi }}_0(x,|v|)=\frac{1}{p(x)}{(a(x)+{|v|}^2)}^{\frac{p(x)}{2}}$$

for all $v\in {\mathbb{R}}^N$. Denote the quantities

$${\lambda }^{\ast }=\underset{u\in X\setminus \{0\}}{inf}\frac{{\int }_{{\mathbb{R}}^N}\frac{1}{p(x)}{(a(x)+{|\mathrm{\nabla }u|}^2)}^{\frac{p(x)}{2}}dx}{{\int }_{{\mathbb{R}}^N}\frac{m(x)}{p(x)}{|u|}^{q(x)}dx}\text{and}{\lambda }_{\ast }=\underset{u\in X\setminus \{0\}}{inf}\frac{{\int }_{{\mathbb{R}}^N}{(a(x)+{|\mathrm{\nabla }u|}^2)}^{\frac{p(x)}{2}}dx}{{\int }_{{\mathbb{R}}^N}m(x){|u|}^{q(x)}dx}.$$

If conditions (H1)-(H2) hold, then we have

• (i) $0<{\lambda }_{\ast }\le {\lambda }^{\ast }$,

• (ii) ${\lambda }^{\ast }$ is a positive eigenvalue of the problem (E₀),

• (iii) the problem (E₀) has a nontrivial weak solution for any $\lambda \ge {\lambda }^{\ast }$,

• (iv) λ is not an eigenvalue of (E₀) for $\lambda <{\lambda }_{\ast }$.

Proof It is clear that conditions (HJ1)-(HJ4) and (F1) hold. From the definitions of ${\lambda }_{\ast }$ and ${\lambda }^{\ast }$, we know that

$$\frac{q_{-}}{p_{+}}{\lambda }_{\ast }\le {\lambda }^{\ast }\le \frac{q_{+}}{p_{-}}{\lambda }_{\ast },$$

and thus ${\lambda }_{\ast }\le {\lambda }^{\ast }$. Also, from the same argument as that in Theorem 3.7, we have ${\lambda }^{\ast }>0$, and thus ${\lambda }_{\ast }>0$. Applying Theorem 3.7, the conclusions (ii) and (iii) hold. Let $\lambda <{\lambda }_{\ast }$. Suppose that λ is an eigenvalue of the problem (E₀). Then there is an element $v\in X\setminus \{0\}$ such that

$${\int }_{{\mathbb{R}}^N}{(a(x)+{|\mathrm{\nabla }v|}^2)}^{\frac{p(x)}{2}}dx-\lambda {\int }_{{\mathbb{R}}^N}m(x){|v|}^{q(x)}dx=0.$$

By the definition of ${\lambda }_{\ast }$, we get that

$$\begin{array}{rcl}{\lambda }_{\ast }{\int }_{{\mathbb{R}}^N}m(x){|v|}^{q(x)}dx& \le & {\int }_{{\mathbb{R}}^N}{(a(x)+{|\mathrm{\nabla }v|}^2)}^{\frac{p(x)}{2}}dx\\ =& \lambda {\int }_{{\mathbb{R}}^N}m(x){|v|}^{q(x)}dx<{\lambda }_{\ast }{\int }_{{\mathbb{R}}^N}m(x){|v|}^{q(x)}dx,\end{array}$$

a contradiction. □