Existence of an unbounded branch of the set of solutions for equations of -Laplace type
© Kim; licensee Springer. 2014
Received: 2 October 2013
Accepted: 9 January 2014
Published: 30 January 2014
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.
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.
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 .
The dual space of is , where . The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in .
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]).
Lemma 2.3 ()
(=1; <1) if and only if (=1; <1), respectively;
If , then ;
If , then .
Lemma 2.4 ()
If , then ;
If , then .
Lemma 2.5 ()
holds, where the positive constant C depends on p and Ω.
Lemma 2.6 ()
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.
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 .
for almost all and for all .
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.
holds for all , where c is the positive constant from (HJ3).
This completes the proof. □
From Proposition 3.1, we can obtain the following result.
Theorem 3.2 Assume that (HJ1)-(HJ3) hold. Then the operator is a continuous, bounded, strictly monotone and coercive on X.
Hence the operator J is bounded and continuous on X.
as and therefore the operator J is coercive on X.
To get strict monotonicity of the operator J, without loss of generality, we divide the proof into two cases.
for some positive constants and .
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 3.3 If (HJ1), (HJ2) and (HJ3) hold, then is a homeomorphism onto .
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. □
for all .
To discuss the asymptotic behavior of J, we require the following hypothesis.
for all with .
Now we can show that the operators J and are asymptotic at infinity, as in Proposition 5.1 of .
the conclusion follows, because the right-hand side of the inequality tends to ε as . This completes the proof. □
for almost all . Assume that
(F1) satisfies the Carathéodory condition in the sense that is measurable for all and is continuous for almost all .
for almost all and all .
for almost all and all .
for all .
Lemma 3.5 If (F1) and (F2) hold, then is continuous and compact. Moreover, the operator is continuous and compact.
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.
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 .
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.
has an unbounded connected set such that intersects .
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 4.3 Suppose that conditions (HJ1)-(HJ4) and (F1)-(F3) are satisfied. If μ is not an eigenvalue of (E), then there is an unbounded connected set such that every point in C is a weak solution of the above problem (B) and intersects .
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.
for almost all . □
This research was supported by a 2011 Research Grant from Sangmyung University.
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