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

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.

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.

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 $W^{1,p(x)}({\mathbb{R}}^N)$.

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].

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]

Lemma 2.3 [19]

Lemma 2.4 [11]

Lemma 2.5 [23]

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 $W^{1,p(x)}({\mathbb{R}}^N)\hookrightarrow L^{q(x)}({\mathbb{R}}^N)$.

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.

which is equivalent to norm (2.1).

for all $\phi \in X$.

(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 $1<p_{-}\le p_{+}<q_{-}\le q_{+}<p^{\ast }(x)$.

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

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.

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

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 $\mathrm{\Phi }(u)\le lim{inf}_{n\to \mathrm{\infty }}\mathrm{\Phi }(u_n)$.

The proof is complete. □

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

for some positive constant C. □

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 $\mathrm{\Psi }(u_n)\to \mathrm{\Psi }(u)$.

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 $\lambda \ge {\lambda }^{\ast }$.

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].

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

a contradiction. □