Abstract
Keywords:
Laplacian; variable exponent LebesgueSobolev spaces; weak solution; eigenvalue1 Introduction
The variable exponent problems appear in a lot of applications, for example, elastic mechanics, electrorheological 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 [14] and references therein.
In this paper, we are concerned with the eigenvalue problem of a class of equations of Laplacian type
where the function is of type with continuous nonconstant function and satisfies a Carathéodory condition. Recently, the authors in [5] 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 [511]. In particular, by using the LjusternikSchnirelmann critical point theory, Fan et al.[8] established the existence of the sequence of eigenvalues of the 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 Laplacian is a natural generalization of the pLaplacian, where is a constant. There are a bunch of papers, for instance, [1418] and references therein. But the Laplace operator possesses more complicated nonlinearities than the pLaplace operator, for example, it is nonhomogeneous, so a more complicated analysis has to be carefully carried out. Some properties of the pLaplacian 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 [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 LebesgueSobolev 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 LebesgueSobolev spaces, which will be used in the next section. The basic properties of the variable exponent LebesgueSobolev spaces can be found from [19,20].
To make a selfcontained paper, we first recall some definitions and basic properties of the variable exponent Lebesgue spaces and the variable exponent LebesgueSobolev spaces .
Set
For any , we introduce the variable exponent Lebesgue space
endowed with the Luxemburg norm
The dual space of is , where . The variable exponent Lebesgue spaces are a special case of OrliczMusielak spaces treated by Musielak in [21].
The variable exponent Sobolev space is defined by
where the norm is
Definition 2.1 The exponent is said to be logHölder continuous if there is a constant C such that
Smooth functions are not dense in the variable exponent Sobolev spaces, without additional assumptions on the exponent . 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 satisfies the logHölder 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]).
The spaceis a separable, uniformly convex Banach space, and its conjugate space is, where. For anyand, we have
Lemma 2.3[19]
Denote
Then
(1) (=1; <1) if and only if (=1; <1), respectively;
Lemma 2.4[11]
Letbe such thatfor almost all. Ifwith, then
Lemma 2.5[23]
Letbe an open, bounded set with Lipschitz boundary, and letwithsatisfy the logHölder continuity condition (2.2). Ifwithsatisfies
then we have
and the imbedding is compact if.
Lemma 2.6[25]
Suppose thatis a Lipschitz function with. Letandfor 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.
Throughout this paper, let satisfy the logHölder continuity condition (2.2) and with the norm
which is equivalent to norm (2.1).
Definition 3.1 We say that is a weak solution of the problem (E) if
Denote
(we allow the case that one of these sets is empty). Then it is obvious that . We assume that:
(HJ1) 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
(HJ3) There exists a positive constant c such that the following conditions are satisfied for almost all :
for almost all . In case , assume that condition (3.1) holds for almost all , and in case , assume that for almost all instead
(HJ4) For all and all , the estimate holds
Let us put
Then [5], and its Gateaux derivative is
Let be a realvalued function. We assume that the function f satisfies the Carathéodory condition in the sense that is measurable for all and is continuous for almost all . Denote
where q is given in (H1) and . We assume that
(F1) For all , , and there is a nonnegative measurable function m with such that
Denoting , it follows from (F1) that
Then it is easy to check that , and its Gateaux derivative is
for any . Let us consider the following quantity:
For the case of and , where satisfies a suitable condition, Benouhiba [6] 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:
and
Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
for some positive constant C. Let u in X with . Then it follows from (HJ3), (HJ4), (3.8) and Lemma 2.3(3) that
Next, we show that relation (3.7) holds. From (H2), there exists a positive constant δ such that , and thus we have
Let be a measurable function such that
Then we have and . Let with . Then it follows from (F1′) and Lemma 2.2 that
Therefore, without loss of generality, we may suppose that . From the inequality above, by using Lemma 2.3, Lemma 2.2 and Lemma 2.4 in order, we have
By Young’s inequality, we get
Using (3.11), we get that
holds for almost all . Hence it follows from Lemma 2.6 that
for some positive constant C. Therefore, we obtain that
From (3.10), with the inequality above, we conclude that relation (3.7) holds. □
Lemma 3.3Assume that (HJ1)(HJ3) and (H1) hold. Then Φ is weakly lower semicontinuous, i.e., inXimplies that.
Proof Suppose that in X as . Since (HJ3) implies that is strictly monotone on X, we have that Φ is convex, and so,
for any n. Then we get that
The proof is complete. □
Lemma 3.4Assume that (H1) and (F1) hold. For anyand all, the following estimate holds:
Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
for some positive constant C. □
Lemma 3.5Assume that (H1) and (F1) hold. For almost alland all, the following estimate holds:
Proof Since , estimate (3.15) is obtained from (F1′) and Young’s inequality. □
Lemma 3.6Assume that (H1) and (F1) hold. Then Ψ is weaklystrongly continuous, i.e., inXimplies that.
Proof Let be a sequence in X such that in X. Then is bounded in X. By Lemma 3.4, for each , there is a positive constant such that
holds for each . It follows from Lemma 3.5 that the Nemytskij operator
is continuous from into ; see Theorem 1.1 in [26]. This together with Lemma 2.5 yields 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 3.7Assume that (HJ1)(HJ4), (H1), (H2), and (F1) hold. Thenis a positive eigenvalue of the problem (E). Moreover, the problem (E) has a nontrivial weak solution for any.
Proof It is trivial by (3.5) that . Suppose to the contrary that . Let be a sequence in such that
As in (3.9), we have
for some positive constant C. Since , we obtain that as . Hence it follows from Lemma 3.2 that
which contradicts with the hypothesis. Hence we get . The analogous argument as that in the proof of Theorem 4.5 in [5] proves that 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 . 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 semicontinuous 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
Consequently, we conclude that . This completes the proof. □
Now, we consider an example to demonstrate our main result in this section.
Example 3.8 Let with satisfy the logHölder continuity condition (2.2). Suppose that , and there is a positive constant such that for almost all . Let us consider
In this case, put
for all . Denote the quantities
If conditions (H1)(H2) hold, then we have
(ii) is a positive eigenvalue of the problem (E_{0}),
(iii) the problem (E_{0}) has a nontrivial weak solution for any ,
(iv) λ is not an eigenvalue of (E_{0}) for .
Proof It is clear that conditions (HJ1)(HJ4) and (F1) hold. From the definitions of and , we know that
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 (E_{0}). Then there is an element such that
By the definition of , we get that
a contradiction. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
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 (NRF2012R1A1A2042187) and a 2013 Research Grant from Sangmyung University.
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