Keywords:-Laplacian; variable exponent Lebesgue-Sobolev spaces; weak solution; eigenvalue
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 -growth conditions has attracted interest and attention in recent years. We refer the readers to [1-4] and references therein.
where the function is of type with continuous nonconstant function and satisfies a Carathéodory condition. Recently, the authors in  obtained the positivity of the infimum of all eigenvalues for the -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 -Laplacian type problem in the whole space .
When , the operator involved in (E) is called the -Laplacian, i.e., . The studies for -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. established the existence of the sequence of eigenvalues of the -Laplacian Dirichlet problem; see  for Neumann problems. Mihăilescu and Rădulescu in  obtained the existence of a continuous family of eigenvalues in a neighborhood of the origin under suitable conditions.
The -Laplacian is a natural generalization of the p-Laplacian, where is a constant. There are a bunch of papers, for instance, [14-18] and references therein. But the -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 -Laplacian. For example, under some conditions, the infimum of all eigenvalues for the -Laplacian might be zero; see . 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  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].
endowed with the Luxemburg norm
The dual space of is , where . The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in .
where the norm is
Smooth functions are not dense in the variable exponent Sobolev spaces, without additional assumptions on the exponent . Zhikov  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 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, , as the completion of with respect to the norm (see [23,24]).
then we have
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 and , which were given in the previous section.
which is equivalent to norm (2.1).
Let us put
Then , and its Gateaux derivative is
For the case of and , where satisfies a suitable condition, Benouhiba  proved that . In this section, we shall generalize the conditions on f and ϕ to satisfy still.
The following lemma plays a key role in obtaining the main result in this section.
Lemma 3.2Assume that assumptions (HJ3)-(HJ4), (H1), and (F1) hold and satisfy
then the functionals Φ and Ψ satisfy the following relations:
Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
By Young’s inequality, we get
Using (3.11), we get that
for some positive constant C. Therefore, we obtain that
From (3.10), with the inequality above, we conclude that relation (3.7) holds. □
for any n. Then we get that
The proof is complete. □
Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
for some positive constant C. □
is continuous from into ; see Theorem 1.1 in . This together with Lemma 2.5 yields that
We are in a position to state the main result about the existence of the positive eigenvalue for the problem (E).
As in (3.9), we have
Finally, we show that the problem (E) has a nontrivial weak solution for any . Notice that u is a weak solution of (E) if and only if u is a critical point of . Assume that is fixed. Let with . With the help of (HJ3) and (HJ4), it follows from proceeding as in the proof of relation (3.13) in Lemma 3.2 that
Since , the inequality above implies that as for , that is, is coercive. Also since the functional is weakly lower semi-continuous by Lemmas 3.3 and 3.6, we deduce that there exists a global minimizer of in X. Since , we verify by definition (3.5) that there is an element ω in such that . Then . So we obtain that
Now, we consider an example to demonstrate our main result in this section.
In this case, put
If conditions (H1)-(H2) hold, then we have
and thus . Also, from the same argument as that in Theorem 3.7, we have , and thus . Applying Theorem 3.7, the conclusions (ii) and (iii) hold. Let . Suppose that λ is an eigenvalue of the problem (E0). Then there is an element such that
a contradiction. □
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final manuscript.
The first author was supported by the Incheon National University Research Grant in 2012, and the second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2012R1A1A2042187) and a 2013 Research Grant from Sangmyung University.
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