Positivity of the infimum eigenvalue for equations of -Laplace type in
© Kim and Kim; licensee Springer 2013
Received: 29 June 2013
Accepted: 21 August 2013
Published: 2 October 2013
We study the following elliptic equations with variable exponents
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.
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].
To make a self-contained paper, we first recall some definitions and basic properties of the variable exponent Lebesgue spaces and the variable exponent Lebesgue-Sobolev spaces .
The dual space of is , where . The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in .
for every with .
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]).
Lemma 2.3 
(=1; <1) if and only if (=1; <1), respectively;
if , then ;
if , then .
Lemma 2.4 
if , then ;
if , then .
Lemma 2.5 
and the imbedding is compact if .
Lemma 2.6 
Suppose that is a Lipschitz function with . Let and for almost all . Then there is a continuous embedding .
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).
for all .
(we allow the case that one of these sets is empty). Then it is obvious that . We assume that:
(H1) , , and .
satisfies the following conditions: is measurable on for all and is locally absolutely continuous on for almost all .
- (HJ2)There are a function and a nonnegative constant b such that
for almost all and for all .
- (HJ3)There exists a positive constant c such that the following conditions are satisfied for almost all :(3.1)for almost all . In case , assume that condition (3.1) holds for almost all , and in case , assume that for almost all instead(3.2)
- (HJ4)For all and all , the estimate holds
where q is given in (H1) and . We assume that
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.
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., in X implies that .
The proof is complete. □
where is either or .
for some positive constant C. □
Proof Since , 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., in X implies that .
Using (3.16) and (3.17), we deduce that as . 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 is a positive eigenvalue of the problem (E). Moreover, the problem (E) has a nontrivial weak solution for any .
Consequently, we conclude that . This completes the proof. □
Now, we consider an example to demonstrate our main result in this section.
is a positive eigenvalue of the problem (E0),
the problem (E0) has a nontrivial weak solution for any ,
λ is not an eigenvalue of (E0) for .
a contradiction. □
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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