Existence results for a variable exponent elliptic problem via topological method
© Yang and Dai; licensee Springer 2012
Received: 8 June 2012
Accepted: 22 August 2012
Published: 7 September 2012
In this paper, existence, localization and uniqueness results of solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder, the Brouwer fixed point theorem and the Galerkin method.
MSC: 35J60, 47H10.
where is a nonempty bounded open set with smooth boundary ∂ Ω, with and is a continuous function.
The operator is said to be the -Laplacian and becomes p-Laplacian when (a constant). The -Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with a variable exponent growth condition has received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to a problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electro-magnetic field [1, 2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal baro-tropic gas through a porous medium [3, 4]. Another field of application of equations with variable exponent growth conditions is image processing . The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [6–11] for an overview of and references on this subject, and to [12–21] for the study of the -Laplacian equations and the corresponding variational problems.
In recent years, many authors have studied the existence of solutions for problem (1.1) from several points of view and with different approaches (see, for example, [18–20]). A useful method for the investigation of solutions to semilinear problems is based on the Leray-Schauder continuation principle, or equivalently, on Schaefers fixed point theorem. For example, in  this method was used for solutions in Hölder spaces, while in , solutions were found in Sobolev spaces.
The aim of this paper is to present new existence, localization and uniqueness results for solutions to problem (1.1) under suitable conditions on the nonlinearity f. Our approach is based on regularity results for the solutions of linear Dirichlet problems, the nonlinear alternative of Leray-Schauder (see ), the Brouwer fixed point theorem (see ) and the Galerkin method. We notice that our partial results of the present paper are motivated by the papers  and  where the authors have obtained some results for semilinear and quasilinear elliptic boundary value problems, respectively. By the Galerkin method, we also establish the results of existence and uniqueness of a solution for problem (1.1). We also would like to point out that the proof of Theorem 3 of  is wrong since is not a linear operator. In this paper, we give a key lemma that can be used to overcome this difficulty.
The rest of this paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we shall use a nonlinear alternative of Leray-Schauder to prove the existence of solutions for problem (1.1). In Section 4, by the Galerkin method, we shall establish the results of existence and uniqueness of a solution for problem (1.1).
In order to discuss problem (1.1), we need some theories on which we call a variable exponent Sobolev space. Firstly, we state some basic properties of spaces which will be used later (for details, see ). Denote by the set of all measurable real functions defined on Ω.
Denote by the closure of in . The spaces , and are separable and reflexive Banach spaces.
Proposition 2.1 (See )
Set . For any , then
(1) for , ;
(3) if , then ;
(4) if , then ;
So, is a norm equivalent to the norm in the space . We shall use the equivalent norm in the following discussion and write for simplicity.
where and , , where is the Sobolev critical exponent ( if and if ), then for every weak solution u of (1.1).
then for some .
(3) [, Theorem 1.2] If in (2), the condition (2.2) is replaced by that p is Hölder continuous on , then for some .
has a unique solution . We denote by the unique solution. K is called the solution operator for problem (2.3). It is well known that the solution operator K is increasing (see Remark 2.1 of ). From the Proposition 2.2 and the embedding theorems, we can obtain the properties of the solution operator K as follows.
Proposition 2.3 (1) (See ) The mapping is continuous. Moreover, the mapping is completely continuous since the embedding is compact.
(2) (See ) If p is log-Holder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
(3) (See ) If p is Hölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
We note that the method of  cannot be directly used in this paper since K is not a linear operator. So, we give a key lemma that will be used in Section 3 to overcome this difficulty.
where K is the solution operator for problem (2.3).
Let , then . □
3 Existence of a solution via the alternative of Leray-Schauder
Now, we state an existence and localization principle for problem (1.1).
and for each . Then the boundary value problem (1.1) has at least one solution with .
for any solution to (3.2) and for each . Then the boundary value problem (1.1) has at least one solution with .
(3) Assume that is Hölder continuous and there is a constant , independent of , with (3.3) for any solution to (3.2) and for each . Then the boundary value problem (1.1) has at least one solution with .
We note that Theorem 3.1 not only guarantees the existence of a solution, but also gives information about its localization. Since the proofs of Theorem 3.1(1)-(3) are identical, we shall just prove Theorem 3.1(3). Firstly, we recall the following well-known results:
Lemma 3.1 (Nonlinear alternative of Leray-Schauder, )
Let denote the closed ball in a Banach space E with radius r, and let be a compact operator. Then either
(i) the equation has a solution in for , or
(ii) there exists an element with satisfying for some .
Proof of Theorem 3.1(3) According to Proposition 2.3, the operator K from to is well defined and compact. We shall apply the nonlinear alternative of Leray-Schauder to E and to the operator , with , where is given by . On the other hand, it is clear that the fixed points of T are the solutions of problem (1.1). Now the conclusion follows from Lemma 3.1 since condition (ii) is excluded by hypothesis. □
Theorem 3.1 immediately yields the following existence and localization results.
where . Then the boundary value problem (1.1) has at least one solution in with .
Then the boundary value problem (1.1) has at least one solution in with .
(3) Assume that is Hölder continuous and (3.4), (3.6) hold. Then the boundary value problem (1.1) has at least one solution in with .
Since the proofs of Corollary 3.1(1)-(3) are identical, we shall just prove Corollary 3.1(3).
Therefore, since and . This is a contradiction. □
We note that condition (3.1) can be satisfied under some suitable conditions.
uniformly with respect to . Then the boundary value problem (1.1) has a solution with for some .
Case 1. .
where is the embedding constant of . If we take ε such that , this equation implies that , for some , a contradiction.
Case 2. .
If we take ε such that , this equation implies that , for some , a contradiction again.
Thus the homotopy is admissible on the ball . Using the homotopy invariance, it follows that , and hence there exists such that , giving rise to a solution of problem (1.1). □
Remark 3.1 We note that the fact that the homotopy is admissible in implies for any solution of (3.2).
Using Theorem 3.1, Theorem 3.2 and Remark 3.1, we easily get:
Corollary 3.2 Assume and f satisfies (3.7), then the boundary value problem (1.1) has at least one solution with for some .
4 Existence and uniqueness via the Galerkin method
In this section, we shall use the Brouwer fixed point theorem and the Galerkin method to prove the existence of a solution for problem (1.1).
and (4.3) follows. □
Then problem (1.1) has exactly one solution.
We conclude that , and hence . □
The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (No. 11261052, No. 11126296).
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