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.
In this paper, we are concerned with the existence of an unbounded branch of the set of solutions for nonlinear elliptic equations of -Laplacian type subject to the Dirichlet boundary condition
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].
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 .
For any , we define
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 Orlicz-Musielak spaces treated by Musielak in .
The variable exponent Sobolev space is defined by
where the norm is
Definition 2.1 The exponent is said to be log-Hölder continuous if there is a constant C such that
for every with .
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]).
The space is a separable, uniformly convex Banach space, and its conjugate space is , where . For any and , we have
Lemma 2.3 ()
(1) (=1; <1) if and only if (=1; <1), respectively;
(2) If , then ;
(3) If , then .
Lemma 2.4 ()
Let be such that for almost all . If with , then
(1) If , then ;
(2) If , then .
Lemma 2.5 ()
Let satisfy the log-Hölder continuity condition (2.2). Then, for , the -Poincaré inequality
holds, where the positive constantCdepends onpand Ω.
Lemma 2.6 ()
Let be an open, bounded set with Lipschitz boundary and with satisfy the log-Hölder continuity condition (2.2). If with satisfies
for all , then we have
and the imbedding is compact if .
3 Properties of the integral operators
In this section, we shall give some properties of the integral operators corresponding to problem (B) by applying the basic properties of the spaces and which were given in the previous section.
Throughout this paper, let satisfy the log-Hölder continuity condition (2.2) and with the norm
which is equivalent to norm (2.1) due to Lemma 2.5.
(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 Ω 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 :
for almost all . If , then condition (3.1) holds for almost all , and if , then assume for almost all instead
Let denote the usual of X and its dual or the Euclidean scalar product on , respectively. Under hypotheses (HJ1) and (HJ2), we define an operator by
for all .
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
holds for all , wherecis the positive constant from (HJ3).
Proof Let with . Let for and set . Observe that
for all . We assume that . Setting , it follows from (3.1) that
where , we have by (3.5) and (3.6) that
Without loss of generality, we may suppose that . Then we obtain, for all ,
Now assume that . As before, we obtain from (3.1) and (3.2) that
for . Using the fact that , we get
This completes the proof. □
From Proposition 3.1, we can obtain the following result.
Theorem 3.2Assume that (HJ1)-(HJ3) hold. Then the operator is a continuous, bounded, strictly monotone and coercive onX.
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 any u in X with , it follows from (HJ3) that
for some positive constant C. Thus we get that
as and therefore the operator J is coercive on X.
Next we will show that the operator J is strictly monotone on X. Set
(Of course, if the sets and are nonempty, then by the continuity of .) It is clear that
To get strict monotonicity of the operator J, without loss of generality, we divide the proof into two cases.
Case 1. Let u, v be in X with for . By Proposition 3.1, we have
for almost all . Integrating the above inequality over Ω and using Lemma 2.3, we assert that
For almost all , by Proposition 3.1, we have
where . From Hölder’s inequality in Lemma 2.2, we obtain that
The first term on the right-hand side in (3.9) is calculated by Lemma 2.4 as follows:
for any . Since , we deduce that
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
for some positive constant . From the previous inequalities (3.11) and (3.12), we have that
where is a positive constant. Consequently, we obtain by (3.7) and (3.13) that
for some positive constants and .
Case 2. Let u, v be in X with for and . For almost all , the following inequality holds:
From the above relation (3.14) and Lemmas 2.2 and 2.4, we obtain that
where α is either or . Since , we assert by (3.15) and Proposition 3.1 that
for almost all . For almost all , Proposition 3.1 yields the following estimate:
Consequently, it follows from (3.16) and (3.17) that
for some constant . This completes the proof. □
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.
Lemma 3.3If (HJ1), (HJ2) and (HJ3) hold, then is a homeomorphism onto .
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
where . 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. □
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
and an operator defined by
for all .
To discuss the asymptotic behavior of J, we require the following hypothesis.
(HJ4) For each , there is a function such that for almost all the following holds:
for all with .
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
Proof Given , choose an such that for almost all the following holds:
for all with . We have by (HJ2) that for almost all the estimate
holds for all with . Set
Then belongs to and for almost all , the estimate
holds for all . From Hölder’s inequality, we have that
for all , and hence for each , we obtain by Minkowski’s inequality and the fact that that
for some positive constants and . From
the conclusion follows, because the right-hand side of the inequality tends to ε as . This completes the proof. □
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:
for almost all . Assume that
(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
for almost all and all .
(F3) There exist a function and a locally bounded function with such that
for almost all and all .
Under assumptions (F1) and (F2), we can define an operator by
and an operator by
for all .
Lemma 3.5If (F1) and (F2) hold, then is continuous and compact. Moreover, the operator is continuous and compact.
Proof A linear operator defined by
is clearly bounded because for some positive constant C. Set . Define the superposition operator by
If I is a bounded interval in ℝ and and are chosen from (F2), then Φ is bounded because
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
for any . From the relation , it follows that F is continuous and compact. In particular, if we set , then G is continuous and compact. This completes the proof. □
We observe the behavior of at infinity.
Lemma 3.6Under assumptions (F1) and (F3), the operator has the following property:
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
for all , where C are some positive constants. It follows from Hölder’s inequality that
for all . Therefore, we get
Recall that a real number μ is called an eigenvalue of (E) if the equation
has a solution in X that is different from the origin.
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
Choose an unbounded sequence in X with such that
Set for . Then we have
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.
Definition 4.1 A weak solution of (B) is a pair in such that
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 that is a homeomorphism and is a continuous and compact operator such that the composition is odd. Let be a continuous and compact operator. If the set
is bounded, then the set
has an unbounded connected set such that intersects .
Proof Since is odd, Borsuk’s theorem implies that the condition
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 set such that every point inCis a weak solution of the above problem (B) and intersects .
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
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. □
Finally, we give an example which illustrates an application of our bifurcation result.
Example 4.4 Let , and . Assume that and there is a real number δ in such that
for almost all . Let
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
Proof Putting , we claim that
If , the inequality is clear. Now let . It follows from Young’s inequality that
Set and choose such that for almost everywhere . Then
Thus (HJ1) and (HJ2) are satisfied. Removing some null set from Ω if necessary, we may suppose that the hypotheses are satisfied for all . If we put
we observe that the first relation in (3.1) holds, because and . Moreover, a straightforward calculation shows that for all ,