We are concerned with the following nonlinear problem
subject to Dirichlet boundary conditions, provided that μ is not an eigenvalue of the -Laplacian. The purpose of this paper is to study the global behavior of the set of solutions for nonlinear equations of -Laplacian type by applying a bifurcation result for nonlinear operator equations.
MSC: 35B32, 35D30, 35J60, 35P30, 37K50, 46E35, 47J10.
Keywords:-Laplacian; variable exponent Lebesgue-Sobolev spaces; weak solution; eigenvalue
Rabinowitz  showed that the bifurcation occurring in the Krasnoselskii theorem is actually a global phenomenon by using the topological approach of Krasnoselskii . As regards the p-Laplacian and generalized operators, nonlinear eigenvalue and bifurcation problems have been extensively studied by many researchers in various ways of approach; see [3-9]. While most of those results considered global branches bifurcating from the principal eigenvalue of the p-Laplacian, under suitable conditions, Väth  introduced another new approach to establish the existence of a global branch of solutions for the p-Laplacian problems by using nonlinear spectral theory for homogeneous operators. Recently, Kim and Väth  proposed a new approach. They observed the asymptotic behavior of an integral operator corresponding to the nonhomogeneous principal part at infinity and established the existence of an unbounded branch of solutions for equations involving nonhomogeneous operators of p-Laplace type.
In recent years, the study of differential equations and variational problems involving -growth conditions has received considerable attention 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 [12-15] and the references therein.
when μ is not an eigenvalue of
Here Ω is a bounded domain in with Lipschitz boundary ∂Ω, the functions are of type with a continuous function and satisfies a Carathéodory condition. When is a constant function, the existence of an unbounded branch of the set of solutions for equations of p-Laplacian type operator is obtained in  (for generalizations to unbounded domains with weighted functions, see also [16,17]). For the case of a variable function , the authors in  obtained the global bifurcation result for a class of degenerate elliptic equations by observing some properties of the corresponding integral operators in the weighted variable exponent Lebesgue-Sobolev spaces.
In the particular case when , the operator involved in (B) is the -Laplacian. The studies for -Laplacian problems have been extensively considered by many researchers in various ways; see [18-23]. As far as we know, there are no papers concerned with the bifurcation theory for the nonlinear elliptic equations involving variable exponents except . Noting that (B) has more complicated nonlinearities (it is nonhomogeneous) than the p-Laplacian equation, we need some more careful and new estimates. In particular, the fact that the principal eigenvalue for problem (E) is isolated plays a key role in obtaining the bifurcation result from the principal eigenvalue. Unfortunately, under some conditions on , the infimum of all positive eigenvalues for the -Laplacian might be zero; see . This means that there is no principal eigenvalue for some variable exponent . Even if there exists a principal eigenvalue , this may not be isolated because is the infimum of all positive eigenvalues. Thus we cannot investigate the existence of global branches bifurcating from the principal eigenvalue of the -Laplacian. However, based on the work of Väth , global behavior of solutions for nonlinear problems involving the -Laplacian was considered in .
This paper is organized as follows. In Section 2, we state some basic results for the variable exponent Lebesgue-Sobolev spaces. In Section 3, some properties of the corresponding integral operators are presented. We will prove the main result on global bifurcation for problem (B) in Section 4. Finally, we give an example to illustrate our bifurcation result.
In this section, we state some elementary properties for the variable exponent Lebesgue-Sobolev spaces which will be used in the next sections. The basic properties of the variable exponent Lebesgue-Sobolev spaces can be found in [24,25].
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
Without additional assumptions on the exponent , smooth functions are not dense in the variable exponent Sobolev spaces. This was considered by Zhikov  in connection with Lavrentiev phenomenon. The importance of this above notion relies on the following fact: if is log-Hölder continuous, then is dense in the variable exponent Sobolev spaces (see [28,29]).
Lemma 2.3 ()
Lemma 2.4 ()
Lemma 2.5 ()
holds, where the positive constantCdepends onpand Ω.
Lemma 2.6 ()
3 Properties of the integral operators
which is equivalent to norm (2.1) due to Lemma 2.5.
The following estimate is a starting point for obtaining that the operator J is a homeomorphism. When is constant, this is a particular form of Corollary 3.1 in  which is based on Lemma 3.1 in ; see [, Lemma 1]. In fact, the special case that ϕ is independent of x is considered in . The proof of the following proposition is essentially the same as that in . For convenience, we give the proof.
Proposition 3.1Let (HJ1) and (HJ3) be satisfied. Then the following estimate
This completes the proof. □
From Proposition 3.1, we can obtain the following result.
Proof In view of (HJ1) and (HJ2), the superposition operator
acts from into and is continuous; see Corollary 5.2.1 in . Hence the continuity of J follows from the fact that J is the composition of the continuous map , the map Λ and the bounded linear map given by
Hence the operator J is bounded and continuous on X.
for some positive constant C. Thus we get that
Next we will show that the operator J is strictly monotone on X. Set
To get strict monotonicity of the operator J, without loss of generality, we divide the proof into two cases.
The first term on the right-hand side in (3.9) is calculated by Lemma 2.4 as follows:
then it follows from (3.9), (3.10), Lemmas 2.3 and 2.4 that
Hence we deduce that
On the other hand, if
then the analogous argument implies that
From the above relation (3.14) and Lemmas 2.2 and 2.4, we obtain that
Consequently, it follows from (3.16) and (3.17) that
Using the previous result, we show the topological property of the operator J which will be needed in the main result of the next section.
Proof From Theorem 3.2, we see that is strictly monotone and coercive. The Browder-Minty theorem hence implies that the inverse operator exists and is bounded; see Theorem 26.A. in . For each , let be any sequence in that converges to h in . Set and with . We obtain from (3.18) that
The main idea in obtaining our bifurcation result is to study the asymptotic behavior of the integral operator J and then to deduce a spectral result for operators that are not necessarily homogeneous. To do this, we consider a function defined by
To discuss the asymptotic behavior of J, we require the following hypothesis.
Now we can show that the operators J and are asymptotic at infinity, as in Proposition 5.1 of .
Proposition 3.4Assume that (HJ1), (HJ2) and (HJ4) are fulfilled. Then we have
Next we deal with the properties for the superposition operator induced by the function f in (B). In particular, we give the compactness of this operator and the behavior of that at infinity, respectively. The ideas of the proof about these properties are completely the same as in . We assume that the variable exponents are subject to the following restrictions:
Since Y is a generalized ideal space and is a regular ideal space (since satisfies -condition), Theorem 6.4 of  implies that Φ is continuous on Y. Recalling the fact that the conjugate function of is strictly less than , we know by Lemma 2.6 that the embedding is continuous and compact and so is the adjoint operator given by
Proof Let be arbitrary. Choose a positive constant R such that for all . Since b is locally bounded, there is a nonnegative constant such that for all . Let . Set . By assumption (F3), Minkowski’s inequality and the fact that , we obtain
Recall that a real number μ is called an eigenvalue of (E) if the equation
Now we consider the following spectral result for nonhomogeneous operators. When is a constant function, the following assertion has been shown to hold by virtue of the Furi-Martelli-Vignoli spectrum; see Theorem 4 of  or Lemma 27 of .
Lemma 3.7Ifμis not an eigenvalue of (E), we have
Proof Suppose that
Hence it follows from Proposition 3.4 and (3.21) that
By the compactness of G, we may assume that converges to some point . From (3.22) it follows that as . Put . Since is a homeomorphism (see Theorem 3.2 in ), we get that and as and so
We conclude that μ is an eigenvalue of (E). This completes the proof. □
4 Main result
In this section, we are preparing to prove our main result. First we give the definition of weak solutions for our problem.
where J, F and G are defined by (3.3), (3.19) and (3.20), respectively.
The following result about the existence of an unbounded branch of solutions for nonlinear operator equations is taken from Theorem 2.2 of  (see also ) as a key tool in obtaining 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
is satisfied for all sufficiently large , where I is the identity operator on X and is the open ball in X centered at 0 of radius r, respectively. In view of Theorem 2.2 of , the conclusion holds. □
Based on the above lemma, we now can prove the main result on bifurcation result for problem (B).
Theorem 4.3Suppose that conditions (HJ1)-(HJ4) and (F1)-(F3) are 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 Apply Lemma 4.2 with and . From Lemmas 3.3 and 3.5 we know that is a homeomorphism, the operators G and F are continuous and compact, and is odd. Since μ is not an eigenvalue of (E), Lemmas 3.6 and 3.7 imply that for some , there is a positive constant such that
is bounded. By Lemma 4.2, the set
Finally, we give an example which illustrates an application of our bifurcation result.
for all . If μ is not an eigenvalue of (E) and assumptions (F1)-(F3) are fulfilled, then there is an unbounded connected set C intersecting such that every point in C is a weak solution of the nonlinear equation
From an analogous argument in the proof of Corollary 3.2 in , we can show that this expression is bounded from below by a positive constant which is independent of and . Therefore (HJ3) is satisfied. Finally, (HJ4) holds if for each we choose
The author declares that he has no competing interests.
This research was supported by a 2011 Research Grant from Sangmyung University.
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