We are concerned with the following nonlinear problem: in Ω, on ∂Ω, which is subject to a Neumann boundary condition, provided that μ is not an eigenvalue of the -Laplacian. The aim of this paper is to study the structure of the set of solutions for the degenerate -Laplacian Neumann problems by applying a bifurcation result for nonlinear operator equations.
MSC: 35B32, 35D30, 35J70, 47J10, 47J15.
Keywords:-Laplacian; weighted variable exponent Lebesgue-Sobolev spaces; Neumann boundary condition; eigenvalue
In recent years, there has been much interest in studying differential equations and variational problems involving -growth conditions since they can model physical phenomena which arise in the study of elastic mechanics, electro-rheological fluid dynamics and image processing, etc. We refer the readers to [1-5] and references therein. In the case of a constant, called the p-Laplacian, there are a lot of papers, for instance, [6-14] and references therein.
when μ is not an eigenvalue of the divergence form
where Ω is a bounded domain in with the Lipschitz boundary ∂Ω, denotes the outer normal derivative of u with respect to ∂Ω, the variable exponent is a continuous function, , w is a weighted function in Ω and satisfies a Carathéodory condition.
Since the inceptive study of bifurcation theory by Krasnoselskii , Rabinowitz  claimed that the bifurcation occurring in the Krasnoselskii theorem is actually a global phenomenon. As regards the p-Laplacian and generalized operators, the nonlinear eigenvalue and bifurcation problems have been widely studied by many researchers in various approaches in the spirit of Rabinowitz ; see also [6-9,13,17].
The authors in [6,7] obtained the bifurcation phenomenon for the nonlinear Dirichlet problem which bifurcates from the first eigenvalue of the p-Laplacian. As in [6,7], Khalil and Ouanan  got the result for the nonlinear Neumann problem of the form
which is based on the fact  that the first eigenvalue of the p-Laplacian is simple and isolated under suitable conditions on m.
While many researchers considered global branches bifurcating from the first eigenvalue of the p-Laplacian, Väth  came at it from another viewpoint to establish the existence of a global branch of solutions for the p-Laplacian with Dirichlet boundary condition by applying nonlinear spectral theory for homogeneous operators. From this point of view, for the case that is a constant function, the existence of a global branch of solutions for the problem (B) was attained in  (for generalization to equations involving nonhomogeneous operators, see also ) when μ is not eigenvalue of (E).
Compared to the p-Laplacian equation, an analysis for the -Laplacian equation has to be carried out more carefully because it has complicated nonlinearities (it is nonhomogeneous) and includes a weighted function. As mentioned before, the fact that the principal eigenvalue for nonlinear eigenvalue problems related to the p-Laplacian under either Dirichlet boundary condition or Neumann boundary condition is isolated plays a key role in obtaining the bifurcation result from the principal eigenvalue of the p-Laplacian. However, unlike the p-Laplacian case, under some conditions on , the first eigenvalue for the -Laplacian Neumann problems is not isolated (see ), that is, the infimum of all eigenvalues of the problem might be zero (see  for Dirichlet boundary condition). Thus we cannot investigate the existence of global branches bifurcating from the principal eigenvalue of the -Laplacian. For this reason, the global behavior of solutions for nonlinear problems involving the -Laplacian had been considered in . To the best of our knowledge, there are no papers concerned with the bifurcation theory for the -Laplacian Neumann problems with weighted functions.
This paper is organized as follows. We first state some basic results for the weighted variable exponent Lebesgue-Sobolev spaces which were given in . Next we give some properties of the corresponding integral operators. Finally we show the existence of a global bifurcation for a Neumann problem involving the -Laplacian by using a bifurcation result in an abstract setting.
In this section, we state some elementary properties for the (weighted) variable exponent Lebesgue-Sobolev spaces which will be used in the next sections. The basic properties of the variable exponent Lebesgue-Sobolev spaces, that is, when can be found from .
endowed with the Luxemburg norm
where the norm is
It is significant that smooth functions are not dense in without additional assumptions on the exponent . This feature was observed by Zhikov  in connection with the Lavrentiev phenomenon. However, if the exponent is log-Hölder continuous, i.e., there is a constant C such that
for every with , then 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 ).
Lemma 2.1 ()
Lemma 2.2 ()
Lemma 2.3 ()
We assume that w is a measurable positive and a.e. finite function in Ω satisfying that
The reasons that we assume (w1) and (w2) can be found in .
Lemma 2.4 ()
We shall frequently make use of the following (compact) imbedding theorem for the weighted variable exponent Lebesgue-Sobolev space in the next sections.
Lemma 2.5 ()
3 Properties of the integral operators
In this section, we give the definitions and some properties of the integral operators corresponding to the problem (B), by applying the basic properties of the spaces and X which are given in the previous section.
The following estimate, which can be found in , plays a key role in obtaining the homeomorphism of the operator J.
From Lemma 3.1, we can obtain the following topological result, which will be needed in the main result. Compared to the case of being constant (see ), the following result is hard to prove because it has complicated nonlinearities.
Then the operators , are bounded and continuous. In fact, for any , let in X as . Then there exist a subsequence and functions v, in for such that as , and for all and for almost all . Without loss of generality, we assume that for . Then we have
and the integrands at the right-hand sides in (3.2) and (3.3) are dominated by some integrable functions. Since in X as , we can deduce that and as for almost all . Therefore, the Lebesgue dominated convergence theorem tells us that in and in as , that is, , are continuous on X. Also it is easy to show that these operators are bounded on X.
for some positive constant C. Thus we get
By using Lemma 3.1 and (3.5), we find that J is strictly monotone on X. The Browder-Minty theorem hence implies that the inverse operator exists and is bounded; see Theorem 26.A in .
Hence using Lemma 3.1, we assert that
for any positive numbers a, b, r, and s, it follows that
Applying Lemma 2.3 and Minkowski’s inequality,
From now on we deal with the properties for the superposition operator induced by the function f in (B). We assume that the variable exponents are subject to the following restrictions:
(F3) f satisfies the following inequality:
For our aim, we need some properties of the operators F and G. In contrast with , we give a direct proofs for the continuity and compactness of F and G without using a continuity result on superposition operators.
Then for fixed , the operator is bounded and continuous. In fact, for any , let in X as . Then there exist a subsequence and functions v, in for such that and as , and and for all and for almost all . Suppose that we can choose such that implies that . For , we have
and (F2) implies that the integrand at the right-hand side is dominated by an integrable function. Since the function f satisfies a Carathéodory condition, we obtain as for almost all . Therefore, the Lebesgue dominated convergence theorem tells us that in as . We conclude that in as and thus is continuous on X. The boundedness of follows from (F2), Minkowski’s inequality, and the imbedding continuously (see Theorem 2.11 in ) as follows:
Minkowski’s inequality and (3.12) imply in view of (F3) that
for all and for all . This shows that for any bounded subset , the family is equicontinuous at each . Hence it follows from the continuity of that Ψ is continuous on , on observing the following relation:
Moreover, Ψ is bounded. Indeed, if and are bounded, we have to verify that is bounded. We may assume that is compact. By the equicontinuity and the compactness of , we can find finitely many numbers such that for every there is an integer with
Recall that the embedding is continuous and compact (see e.g.) and so the adjoint operator given by
is also compact. As F can be written as a composition of with Ψ, we conclude that F is continuous and compact on . The operator G is continuous and compact because G can be regarded as a special case of F. This completes the proof. □
Proof Let . Choose a positive constant R such that for all . Since b is locally bounded, there is a nonnegative constant such that for all . Let with . Set . Without loss of generality, we may suppose that
Recall that a real number μ is called an eigenvalue of (E) if the equation
The following lemma is a consequence about nonlinear spectral theory and its proof can be found in . For the case that is a constant, this assertion has been obtained by using the Furi-Martelli-Vignoli spectrum; see Theorem 4 of  or Lemma 27 of .
Lemma 3.5Suppose that assumptions (w1) and (w2) are fulfilled. Ifμis not an eigenvalue of (E), then we have
4 Bifurcation result
In this section, we are ready to prove the main result. We give the definition of weak solutions for our problem.
where J, F and G are defined by (3.1), (3.13) and (3.14), respectively.
The following result, taken from Theorem 2.2 of , is a key tool to obtain our bifurcation result.
Lemma 4.2LetXbe a Banach space andYbe a normed space. Suppose thatis a homeomorphism andis a continuous and compact operator such that the compositionis odd. Letbe a continuous and compact operator. If the set
is bounded, then the set
Finally we establish the existence of an unbounded branch of the set of solutions for Neumann problem (B) thereby using Lemma 4.2.
Theorem 4.3Let conditions (w1), (w2), and (F1)-(F4) be satisfied. Ifμis not an eigenvalue of (E), then there is an unbounded connected setsuch that every pointinCis a weak solution of the above problem (B) andintersects.
is bounded. By Lemma 4.2, the set
In particular the following example illustrates an application of our bifurcation result.
Example 4.4 Suppose that assumptions (w1) and (w2) are fulfilled and . If μ is not an eigenvalue of (E), then there is an unbounded connected set C such that every point in C is a weak solution of the following nonlinear problem:
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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