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Existence results for a variable exponent elliptic problem via topological method
Boundary Value Problems volume 2012, Article number: 99 (2012)
Abstract
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.
1 Introduction
In this paper, we consider the boundary value problem
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 [5]. 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 [22] this method was used for solutions in Hölder spaces, while in [23], 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 [24]), the Brouwer fixed point theorem (see [25]) and the Galerkin method. We notice that our partial results of the present paper are motivated by the papers [26] and [27] 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 [27] 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).
2 Preliminaries
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 [7]). Denote by the set of all measurable real functions defined on Ω.
Write
and
with the norm
and
with the norm
Denote by the closure of in . The spaces , and are separable and reflexive Banach spaces.
Proposition 2.1 (See [7])
Set . For any , then
(1) for , ;
(2) ;
(3) if , then ;
(4) if , then ;
(5) ;
(6) .
In the Poincaré inequality holds, that is, there exists a positive constant such that
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.
Proposition 2.2 (1) [[17], Theorem 4.1] If f satisfies the sub-critical growth condition
where and , , where is the Sobolev critical exponent ( if and if ), then for every weak solution u of (1.1).
(2) [[17], Theorem 4.4] Let be a solution of (1.1). If the function p is log-Hölder continuous on , i.e., there is a positive constant H such that
then for some .
(3) [[21], Theorem 1.2] If in (2), the condition (2.2) is replaced by that p is Hölder continuous on , then for some .
From [20], we know that, for a given , where satisfies (2.1), the problem
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 [18]). 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 [20]) The mapping is continuous. Moreover, the mapping is completely continuous since the embedding is compact.
(2) (See [17]) If p is log-Holder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
(3) (See [21]) If p is Hölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
We note that the method of [26] 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.
Lemma 2.1 Let M be a positive constant and , then there exists one point such that
where K is the solution operator for problem (2.3).
Proof We assume that u is a solution of problem (2.3), then we have . From (2.3), we can also show that
By (2.4) and mean value theorem, for any positive constant C, we can show that there exists one point such that
that is to say,
Let , then . □
3 Existence of a solution via the alternative of Leray-Schauder
Here and in the sequel, E will denote the space
endowed with the sup-norm
Now, we state an existence and localization principle for problem (1.1).
Theorem 3.1 (1) Assume that f satisfies (2.1), and there is a constant , independent of , with
for any solution to
and for each . Then the boundary value problem (1.1) has at least one solution with .
(2) Assume that is log-Hölder continuous and there is a constant , independent of , 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, [24])
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.
Corollary 3.1 (1) Assume that f satisfies (2.1), and there exist nonnegative continuous functions , and a continuous nondecreasing function such that
Suppose, in addition, that there exists a real number such that
where . Then the boundary value problem (1.1) has at least one solution in with .
(2) Assume that is log-Hölder continuous and (3.4) holds. Suppose, in addition, that there exists a real number such that
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).
Proof of Corollary 3.1(3) In order to apply Theorem 3.1(3), we have to show that condition (3.3) holds true for any solution to (3.2). Assume is any solution to (3.2) for some with . Then
Furthermore, for all , by the monotonicity of K and Lemma 2.1, we have
Taking the supremum in the above inequality, we obtain
Therefore, since and . This is a contradiction. □
We note that condition (3.1) can be satisfied under some suitable conditions.
Theorem 3.2 Suppose that , is a continuous function and satisfies
uniformly with respect to . Then the boundary value problem (1.1) has a solution with for some .
Proof From (3.7) it follows that, for all , there exists such that
Then, according to Theorem 1.16 of [7], f induces a Nemytsky operator on , still denoted by f. Setting , according to Proposition 2.3, is compact and (1.1) can be written in the form , . Let us show that there exists such that the homotopy is admissible in . Otherwise, there exist , with , and such that . This is equivalent to , with . Taking as a test function, using (3.8) and the fact that , we get
Case 1. .
In this case, using the Hölder and Poincaré inequality, we deduce
where is the embedding constant of . If we take ε such that , this equation implies that , for some , a contradiction.
Case 2. .
In this case, using the Hölder and Poincaré inequalities, we deduce
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).
Theorem 4.1 Assume that there exist constant and with such that
with
where is the embedding constant of . Then problem (1.1) has at least one weak solution. Besides, any solution u of (1.1) satisfies the estimate
Proof Because is a reflexive and separable Banach space, there exist and such that
and
For convenience, let us put
Then is isometric to . In fact, each is uniquely associated to by the relation . We search for solutions of the approximate problem
To solve this algebraic system, we define the operator ,
We note that is continuous from the continuity of with respect to u. Therefore, we can use the following form of the Brouwer fixed point theorem: if there exists such that whenever , then has a root u satisfying (see, e.g., [25]). From (4.1), Proposition 2.1 and the Poincaré inequality, we have for with
This shows, from (4.2), the existence of , depending only on , a and , such that if . Then system (4.4) has a solution satisfying
From this estimate, going to a subsequence if necessary, there is u such that
Besides, since compactly and the Nemytsky map is continuous from to (see [7]). Then fixing k in (4.4) and letting , we conclude that
From the completeness of , identity (4.5) holds with replaced by any , we get
which shows that u is in fact a solution of problem (1.1). Finally, if u is any solution of problem (1.1), then . Therefore, either or
and (4.3) follows. □
Theorem 4.2 Let the assumptions of Theorem 4.1 hold, with (4.1) replaced by
Then problem (1.1) has exactly one solution.
Proof Taking in (4.7), we get
Setting in (4.8), we get (4.1). Hence, the existence part follows from Theorem 4.1. Now let u and v be two solutions of problem (1.1). Putting , by (4.7) and Theorem 3.1 of [19], we have
We conclude that , and hence . □
References
Ru̇z̆ic̆ka M: Electro-rheological Fluids: Modeling and Mathematical Theory. Springer, Berlin; 2000.
Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 1987, 9: 33-66.
Antontsev SN, Shmarev SI: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 2005, 60: 515-545.
Antontsev SN, Rodrigues JF: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 2006, 52: 19-36. 10.1007/s11565-006-0002-9
Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 2006, 66(4):1383-1406. 10.1137/050624522
Diening L, Hästö P, Nekvinda A: Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings Edited by: Drábek P, Rákosník J. 2004, 38-58.
Fan XL, Zhao D:On the spaces and . J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617
Harjulehto P, Hästö P: An overview of variable exponent Lebesgue and Sobolev spaces. In Future Trends in Geometric Function Theory. Edited by: Herron D. Univ. Jyväskylä, Jyväskylä; 2003:85-93.
Samko S: On a progress in the theory of Lebesgue spaces with variable exponent maximal and singular operators. Integral Transforms Spec. Funct. 2005, 16: 461-482. 10.1080/10652460412331320322
Zhikov VV, Kozlov SM, Oleinik OA: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin; 1994. (Translated from the Russian by GA Yosifian)
Zhikov VV: On some variational problems. Russ. J. Math. Phys. 1997, 5: 105-116.
Dai G:Three symmetric solutions for a differential inclusion system involving the -Laplacian in . Nonlinear Anal. 2009, 71: 1763-1771. 10.1016/j.na.2009.01.012
Dai G:Infinitely many solutions for a Neumann-type differential inclusion problem involving the -Laplacian. Nonlinear Anal. 2009, 70: 2297-2305. 10.1016/j.na.2008.03.009
Dai G:Infinitely many solutions for a hemivariational inequality involving the -Laplacian. Nonlinear Anal. 2009, 71: 186-195. 10.1016/j.na.2008.10.039
Dai G:Three solutions for a Neumann-type differential inclusion problem involving the -Laplacian. Nonlinear Anal. 2009, 70: 3755-3760. 10.1016/j.na.2008.07.031
Dai G:Infinitely many solutions for a differential inclusion problem in involving the -Laplacian. Nonlinear Anal. 2009, 71: 1116-1123. 10.1016/j.na.2008.11.024
Fan XL, Zhao D: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 1996, 36: 295-318.
Fan XL:On the sub-supersolution methods for -Laplacian equations. J. Math. Anal. Appl. 2007, 330: 665-682. 10.1016/j.jmaa.2006.07.093
Fan XL, Zhang QH:Existence of solutions for -Laplacian Dirichlet problems. Nonlinear Anal. 2003, 52: 1843-1852. 10.1016/S0362-546X(02)00150-5
Fan XL, Han XY:Existence and multiplicity of solutions for -Laplacian equations in . Nonlinear Anal. 2004, 59: 173-188.
Fan XL:Global regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ. 2007, 235: 397-417. 10.1016/j.jde.2007.01.008
Gilbarg D, Trudinger N: Elliptic Partial Differential Equations of Second Order. Springer, Berlin; 1983.
O’Regan D, Precup R: Theorems of Leray-Schauder Type and Applications. Gordon and Breach, Amsterdam; 2001.
Zeidler E: Nonlinear Functional Analysis: Part I. Springer, New York; 1985.
Kesavan S: Topics in Functional Analysis and Applications. Wiley, New York; 1989.
Moussaoui T, Precup R: Existence results for semilinear elliptic boundary value problems via topological methods. Appl. Math. Lett. 2009, 22: 126-129. 10.1016/j.aml.2008.03.002
Ngô, QA: Existence results for quasilinear elliptic boundary value problems via topological method. arXiv:0805.0075v1 [math.AP] (2008)
Acknowledgements
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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GD conceived of the study, participated in its design and coordination, and helped to draft the manuscript. SY participated in the design of the study. All authors read and approved the final manuscript.
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Yang, S., Dai, G. Existence results for a variable exponent elliptic problem via topological method. Bound Value Probl 2012, 99 (2012). https://doi.org/10.1186/1687-2770-2012-99
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DOI: https://doi.org/10.1186/1687-2770-2012-99