Abstract
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 LebesgueSobolev spaces; Neumann boundary condition; eigenvalue1 Introduction
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, electrorheological fluid dynamics and image processing, etc. We refer the readers to [15] and references therein. In the case of a constant, called the pLaplacian, there are a lot of papers, for instance, [614] and references therein.
In the present paper, we are concerned with the existence of an unbounded branch of the set of solutions for the Laplacian problem with degeneracy subject to the Neumann boundary condition
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 [15], Rabinowitz [16] claimed that the bifurcation occurring in the Krasnoselskii theorem is actually a global phenomenon. As regards the pLaplacian and generalized operators, the nonlinear eigenvalue and bifurcation problems have been widely studied by many researchers in various approaches in the spirit of Rabinowitz [16]; see also [69,13,17].
The authors in [6,7] obtained the bifurcation phenomenon for the nonlinear Dirichlet problem which bifurcates from the first eigenvalue of the pLaplacian. As in [6,7], Khalil and Ouanan [18] got the result for the nonlinear Neumann problem of the form
which is based on the fact [19] that the first eigenvalue of the pLaplacian is simple and isolated under suitable conditions on m.
While many researchers considered global branches bifurcating from the first eigenvalue of the pLaplacian, Väth [20] came at it from another viewpoint to establish the existence of a global branch of solutions for the pLaplacian 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 [11] (for generalization to equations involving nonhomogeneous operators, see also [12]) when μ is not eigenvalue of (E).
Compared to the pLaplacian 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 pLaplacian 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 pLaplacian. However, unlike the pLaplacian case, under some conditions on , the first eigenvalue for the Laplacian Neumann problems is not isolated (see [21]), that is, the infimum of all eigenvalues of the problem might be zero (see [22] 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 [23]. 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 LebesgueSobolev spaces which were given in [23]. 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.
2 Preliminaries
In this section, we state some elementary properties for the (weighted) variable exponent LebesgueSobolev spaces which will be used in the next sections. The basic properties of the variable exponent LebesgueSobolev spaces, that is, when can be found from [24].
To make a selfcontained paper, we recall some definitions and basic properties of the weighted variable exponent Lebesgue spaces and the weighted variable exponent LebesgueSobolev spaces .
Set
Let w is a measurable positive and a.e. finite function in Ω. For any , we introduce the weighted variable exponent Lebesgue space
endowed with the Luxemburg norm
The weighted variable exponent Sobolev space is defined by
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 [25] in connection with the Lavrentiev phenomenon. However, if the exponent is logHölder 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 [26]).
Lemma 2.1 ([24])
The spaceis a separable, uniformly convex Banach space, and its conjugate space iswhere. For anyand, we have
Lemma 2.2 ([23])
Denote
Then
(1) (=1; <1) if and only if (=1; <1), respectively;
Lemma 2.3 ([27])
Letbe such thatfor almost all. Ifwith, then
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 [23].
Lemma 2.4 ([23])
Letand (w1) hold. ThenXis a reflexive and separable Banach space.
We shall frequently make use of the following (compact) imbedding theorem for the weighted variable exponent LebesgueSobolev space in the next sections.
Lemma 2.5 ([23])
Letbe an open, bounded set with Lipschitz boundary andwithsatisfy the logHölder continuity condition (2.2). If assumptions (w1) and (w2) hold andwithsatisfiesfor all, then we have
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.
Throughout this paper, let satisfy the logHölder continuity condition (2.2). We define an operator by
for any where denotes the pairing of X and its dual and the Euclidean scalar product on , respectively.
The following estimate, which can be found in [17], plays a key role in obtaining the homeomorphism of the operator J.
Lemma 3.1For any, the following inequalities hold:
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 [11]), the following result is hard to prove because it has complicated nonlinearities.
Theorem 3.2Let (w1) and (w2) be satisfied. The operatoris homeomorphism ontowith a bounded inverse.
Proof Let and be operators defined by
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
and the integrands at the righthand 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.
Using the continuity for the operators and on X, we finally show that J is continuous on X. From Hölder’s inequality, we have
and the righthand side in (3.4) converges to zero as . Therefore the operator J is continuous on X.
For any u in X with , it follows that
for some positive constant C. Thus we get
as and therefore the operator J is coercive on X.
Denote
Set
and
(Of course, if both the sets and are nonempty, then by the continuity of .) It is clear that
By using Lemma 3.1 and (3.5), we find that J is strictly monotone on X. The BrowderMinty theorem hence implies that the inverse operator exists and is bounded; see Theorem 26.A in [28].
Next we will show that is continuous on . Assume that u and v are any elements in X with . According to Lemma 3.1, we have
and
for almost all and for some positive constants and . Integrating the above inequalities over Ω and using Lemma 2.2, we assert that
for some positive constants , , and . For almost all , the following inequalities hold:
and
where we put and use the shortcuts
Hence using Lemma 3.1, we assert that
for some positive constant . From Hölder’s and Minkowski’s inequalities, and the inequality
for any positive numbers a, b, r, and s, it follows that
Applying Lemma 2.3 and Minkowski’s inequality,
for any where α is either or . In a similar way,
for any where β is either or . It follows from (3.7)(3.10) and Lemma 2.2 that
where γ is either or and is positive constant. So
for some positive constant . Consequently, it follows from (3.6) and (3.11) that
for some positive constants and where . For each , let be any sequence in that converges to h in . Set and with . We obtain from (3.12)
Since is bounded in X and in as , it follows that converges to u in X. Thus, is continuous at each . This completes the proof. □
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:
(F1) satisfies the Carathéodory condition in the sense that is measurable for all and is continuous for almost all .
(F2) For each bounded interval , there are a function and a nonnegative constant such that
(F3) f satisfies the following inequality:
(F4) There exist a function and a locally bounded function with such that
Under assumptions (F1) and (F2), we can define an operator by
For our aim, we need some properties of the operators F and G. In contrast with [23], we give a direct proofs for the continuity and compactness of F and G without using a continuity result on superposition operators.
Theorem 3.3If (w1), (w2), and (F1)(F3) hold, then the operatoris continuous and compact. Also the operatoris continuous and compact.
Proof Let be an operator defined by
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 righthand 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 [23]) as follows:
for all and for some positive constant d.
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
Since is bounded for each , Minkowski’s inequality hence implies that is bounded.
Recall that the embedding is continuous and compact (see e.g.[8]) 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. □
The analog of the following result can be found in [23]. However, our growth condition described in assumption (F4) is slightly different from that of [23].
Lemma 3.4Let assumptions (w1), (w2), (F1) and (F4) be fulfilled. Then the operatorhas the following property:
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
By assumption (F4), Lemma 2.5 and the continuous imbedding , we obtain that
where and are positive constants. It follows from Hölder’s inequality that
for all with , where is a positive constant. Consequently, we get
□
Recall that a real number μ is called an eigenvalue of (E) if the equation
has a solution in X which is different from the origin.
The following lemma is a consequence about nonlinear spectral theory and its proof can be found in [23]. For the case that is a constant, this assertion has been obtained by using the FuriMartelliVignoli spectrum; see Theorem 4 of [29] or Lemma 27 of [20].
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.
Definition 4.1 A weak solution of (B) is a pair in such that
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 [20], 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
has an unbounded connected setsuch thatintersects.
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.
Proof By Theorem 3.2 and Lemma 3.3, is a homeomorphism, the operators G and F are continuous and compact, and is odd. Since μ is not an eigenvalue of (E), we get by Lemma 3.5
This together with Lemma 3.4 implies that for some , there is a positive constant such that
for all with and for all . Therefore, the set
is bounded. By Lemma 4.2, the set
contains an unbounded connected set C which intersects . This completes the proof. □
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:
where and the conjugate function of is strictly less than .
Proof Let . Then it is clear that f satisfies conditions (F1)(F4). Therefore, the conclusion follows from Theorem 4.3. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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