### Abstract

We are concerned with the following nonlinear problem

subject to Dirichlet boundary conditions, provided that *μ* is not an eigenvalue of the

**MSC: **
35B32, 35D30, 35J60, 35P30, 37K50, 46E35, 47J10.

##### Keywords:

### 1 Introduction

Rabinowitz [1] showed that the bifurcation occurring in the Krasnoselskii theorem is actually a
global phenomenon by using the topological approach of Krasnoselskii [2]. 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 [10] 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 [11] 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
*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

when *μ* is not an eigenvalue of

Here Ω is a bounded domain in
*∂*Ω, the functions
*p*-Laplacian type operator is obtained in [11] (for generalizations to unbounded domains with weighted functions, see also [16,17]). For the case of a variable function

In the particular case when
*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

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.

### 2 Preliminaries

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

Set

For any

For any

endowed with the Luxemburg norm

The dual space of

The variable exponent Sobolev space

where the norm is

**Definition 2.1** The exponent
*C* such that

for every

Without additional assumptions on the exponent

*The space*
*is a separable*, *uniformly convex Banach space*, *and its conjugate space is*
*where*
*For any*
*and*
*we have*

**Lemma 2.3** ([24])

*Denote*

*Then*

(1)
*if and only if*
*respectively*;

(2) *If*
*then*

(3) *If*
*then*

**Lemma 2.4** ([23])

*Let*
*be such that*
*for almost all*
*If*
*with*
*then*

(1) *If*
*then*

(2) *If*
*then*

**Lemma 2.5** ([20])

*Let*
*satisfy the log*-*Hölder continuity condition* (2.2). *Then*, *for*
*the*
*Poincaré inequality*

*holds*, *where the positive constant**C**depends on**p**and* Ω.

**Lemma 2.6** ([28])

*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

Throughout this paper, let

which is equivalent to norm (2.1) due to Lemma 2.5.

Denote

(We allow the case that one of these sets is empty.) Then it is obvious that

(HJ1)

(HJ2) There are a function
*b* such that

for almost all

(HJ3) There exists a positive constant *c* such that the following conditions are satisfied for almost all

for almost all

Let
*X* and its dual

for all

The following estimate is a starting point for obtaining that the operator *J* is a homeomorphism. When
*ϕ* is independent of *x* is considered in [6]. The proof of the following proposition is essentially the same as that in [31]. For convenience, we give the proof.

**Proposition 3.1***Let* (HJ1) *and* (HJ3) *be satisfied*. *Then the following estimate*

*holds for all*
*where**c**is the positive constant from* (HJ3).

*Proof* Let

for all

and so

Noticing that

where

Without loss of generality, we may suppose that

and

Now assume that

for

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*.

*Proof* In view of (HJ1) and (HJ2), the superposition operator

acts from
*J* follows from the fact that *J* is the composition of the continuous map

Hence the operator *J* is bounded and continuous on *X*.

For any *u* in *X* with

for some positive constant *C*. Thus we get that

as
*J* is coercive on *X*.

Next we will show that the operator *J* is strictly monotone on *X*. Set

and

(Of course, if the sets

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 almost all

For almost all

where

The first term on the right-hand side in (3.9) is calculated by Lemma 2.4 as follows:

for any

If

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

where

for some positive constants

Case 2. Let *u*, *v* be in *X* with

From the above relation (3.14) and Lemmas 2.2 and 2.4, we obtain that

where *α* is either

and so

for almost all

Consequently, it follows from (3.16) and (3.17) that

for some constant

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*

*Proof* From Theorem 3.2, we see that
*h* in

where
*X* and
*u* in *X*. Thus,

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

and an operator

for all

To discuss the asymptotic behavior of *J*, we require the following hypothesis.

(HJ4) For each

for all

Now we can show that the operators *J* and

**Proposition 3.4***Assume that* (HJ1), (HJ2) *and* (HJ4) *are fulfilled*. *Then we have*

*Proof* Given

for all

holds for all

Then

holds for all

for all

for some positive constants

the conclusion follows, because the right-hand side of the inequality tends to *ε* as

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 [18]. We assume that the variable exponents are subject to the following restrictions:

for almost all

(F1)

(F2) For each bounded interval

for almost all

(F3) There exist a function

for almost all

Under assumptions (F1) and (F2), we can define an operator

and an operator

for all

In proving the following result, a key idea is to use a continuity result on the superposition
operators due to Väth [34]. For the case that

**Lemma 3.5***If* (F1) *and* (F2) *hold*, *then*
*is continuous and compact*. *Moreover*, *the operator*
*is continuous and compact*.

*Proof* A linear operator

is clearly bounded because
*C*. Set

If *I* is a bounded interval in ℝ and

Since *Y* is a generalized ideal space and
*Y*. Recalling the fact that the conjugate function of

for any
*F* is continuous and compact. In particular, if we set
*G* is continuous and compact. This completes the proof. □

We observe the behavior of

**Lemma 3.6***Under assumptions* (F1) *and* (F3), *the operator*
*has the following property*:

*Proof* Let
*R* such that
*b* is locally bounded, there is a nonnegative constant

for all
*C* are some positive constants. It follows from Hölder’s inequality that

for all

□

Recall that a real number *μ* is called an *eigenvalue of* (*E*) if the equation

has a solution
*X* that is different from the origin.

Now we consider the following spectral result for nonhomogeneous operators. When

**Lemma 3.7***If**μ**is not an eigenvalue of* (E), *we have*

*Proof* Suppose that

Choose an unbounded sequence
*X* with

Set

Hence it follows from Proposition 3.4 and (3.21) that

By the compactness of *G*, we may assume that

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

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 [11] (see also [10]) as a key tool in obtaining our bifurcation result.

**Lemma 4.2***Let**X**be a Banach space and**Y**be 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 satisfied for all sufficiently large
*I* is the identity operator on *X* and
*X* centered at 0 of radius *r*, respectively. In view of Theorem 2.2 of [11], 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*

*Proof* Apply Lemma 4.2 with
*G* and *F* are continuous and compact, and
*μ* is not an eigenvalue of (E), Lemmas 3.6 and 3.7 imply that for some

for all

is bounded. By Lemma 4.2, the set

contains an unbounded connected set *C* which

Finally, we give an example which illustrates an application of our bifurcation result.

**Example 4.4** Let
*δ* in

for almost all

for all
*μ* is not an eigenvalue of (E) and assumptions (F1)-(F3) are fulfilled, then there is
an unbounded connected set *C* intersecting
*C* is a weak solution of the nonlinear equation

*Proof* Putting

If

and hence

Set

Thus (HJ1) and (HJ2) are satisfied. Removing some null set from Ω if necessary, we
may suppose that the hypotheses are satisfied for all

we observe that the first relation in (3.1) holds, because