Abstract and Applied Analysis

Volume 2012 (2012), Article ID 615816, 17 pages

doi:10.1155/2012/615816

Research Article

On Elliptic Equations in Orlicz Spaces Involving Natural Growth Term and Measure Data

Mohammed Kbiri Alaoui

Department of Mathematics, King Khalid University, P.O. Box 9004, Abha 162529, Saudi Arabia

Received 11 July 2012; Accepted 10 September 2012

Academic Editor: Yong Zhou

Copyright © 2012 Mohammed Kbiri Alaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper deals with the existence of solutions of elliptic equations in the framework of Orlicz spaces with right-hand side measure and natural growth term.

1. Introduction

We deal with boundary value problems

where

is a bounded domain of , with the segment property. is a Carathéodory function (i.e., measurable with respect to in for every in , and continuous with respect to in for almost every in ) such that there exist two -functions and for all , , the following hypotheses are true

where belongs to , , to , and , to .

For the sake of simplicity, we suppose in (1.2) that .

This paper is devoted to study the Dirichlet problem for some nonlinear elliptic equations whose simplest model is

This kind of problems has been widely studied. Many authors have proved results for second-order elliptic problems with lower-order terms depending on the gradient; these works include, for instance, [1–6]. After the classical example by Kazdan and Kramer (see [7]), which shows that (1.8) cannot always have solutions, two different kind of questions have been considered. On the one hand, in some papers, the existence of solutions when the source is small in a suitable norm are proved. On the other hand, conditions on which the function have been considered in order to get a solution for all in a given Lebesgue space. This is the way chosen in [1, 5, 6] under the hypothesis .

In [8], the authors present some results concerning existence, nonexistence, multiplicity, and regularity of positive solutions for two elliptic quasilinear problems with Dirichlet data in a bounded domain. The first problem is similar to (1.8) and the other is

where , , , required some specified conditions. The first one, of unknown , involves a gradient term with natural growth. The second one, of unknown , presents a source term of order 0. They gave and established a precise connection between problems in and . Also, they proved a result of existence for the problem in with general bounded Radon measures data and obtained some results for the problem in by using the connection between these two problems. Other authors have established this connection between the two problems (1.8) and (1.9); one can see, for example, [9].

Many researchers have investigated the possibility to find solutions of (1.8) under the sign condition , in which case the term is said to be an absorption term. In [5], the author treated the problem (1.8) without using the above sign condition but by supposing the summability of the function . The principal tools used is the Lebesgue decomposition theorem (see [10]) and the cut functions with respect to the measure . The result given is optimal.

In the spirit of the work [5], our purpose in this paper is to prove existence results in the setting of the Orlicz Sobolev space when the operator does not satisfy the classical polynomial growth.

2. Preliminaries

Let be an -function, that is, is continuous, convex, with for , as and as . The -function conjugate to is defined by .

Let and be two -functions. means that grows essentially less rapidly than ; that is, for each ,

The -function is said to satisfy the condition if for some : for all ; when this inequality holds only for , is said to satisfy the condition near infinity.

Let be an open subset of . The Orlicz class (resp., the Orlicz space ) is defined as the set of (equivalence classes of) real-valued measurable functions on such that (resp. for some .

Note that is a Banach space under the norm and is a convex subset of . The closure in of the set of bounded measurable functions with compact support in is denoted by . In general and the dual of can be identified with by means of the pairing , and the dual norm on is equivalent to .

We now turn to the Orlicz-Sobolev space. (resp. ) is the space of all functions such that and its distributional derivatives up to order lie in (resp. ). This is a Banach space under the norm . Thus and can be identified with subspaces of the product of copies of . Denoting this product by , we will use the weak topologies and . The space is defined as the (norm) closure of the Schwartz space in and the space as the closure of in . We say that converges to for the modular convergence in if for some , for all . This implies convergence for . If satisfies the condition on (near infinity only when has finite measure), then modular convergence coincides with norm convergence.

Fore more details about the Orlicz spaces and their properties one can see [11, 12].

For , we define the truncation at height by: .

3. Main Result

3.1. Useful Results

First, we give the following definitions and results which will be used in our main result.

The -capacity of any set with respect to is defined in the following classical way. The -capacity of any compact set is first defined as

where is the characteristic function of ; we will use the convention that . The -capacity of any open subset is then defined by

Finally, the -capacity of any subset is defined by

Definition 3.1. We say that is a weak solution of the problem if

We define as the space of all Radon measures on with bounded total variation, and as the space of all bounded, continuous functions on , so that is defined for and .

We say that a sequence of measures in converges to a measure in if

for every . If this convergence holds only for all the continuous functions with compact support in , then we have the usual weak convergence in .

Lemma 3.2. Under the hypotheses (1.2)–(1.6), and , there exists at least one positive weak solution of the problem

For the proof see [12].

Let be a fixed -function, we define as the set of functions satisfying the following conditions:

(i) is a convex function,(ii), (iii)there exists an -function such that and near infinity.

Lemma 3.3. Let be a sequence of solutions of the problem

where is a sequence of functions bounded in . Then is bounded in for all -functions .

For the proof one can see that the technique used in [13] and adapted to the elliptic case gives the result, but for the simplicity we give a sketched proof.

Proof. Let denote by , the measure of the unit ball of , and .

Let be a truncation defined by

for all .

Using as a test function, we obtain after tending to zero

Following the same way as in [14], we have for ,

We obtain

So,

Then the sequence is bounded in .

Lemma 3.4. Let be a nonnegative Radon measure which is concentrated in a set of zero p-capacity. Then there exists a sequence of functions such that

For the proof see [15].

Remark 3.5. By the above lemma, we have that converge to zero both strongly in , a.e. in , and in the weak * topology of .

3.2. Existence Result

In what follows, we suppose that the set is nonempty.

Theorem 3.6. Let , under the hypotheses (1.2)–(1.6) and , there exists at least one positive weak solution of the problem

Remark 3.7. The condition is very important to ensure the existence of cut functions for the measure . The case is easily treated, since we come back to the variational case.

(1.9) The condition is supposed to guarantee that we are not in the variational case and the study has a sense (see [16]).

Remark 3.8. The conditions (i), (ii), and (iii) permit us to determine the regularity of the solutions of and improve the one given by Porretta in [5]. If , one can find a solution such that , for some .

Remark 3.9 (see [5]). The condition is optimal in the sense introduced by Porretta in [5].

Indeed, if is the Dirac mass, there is no solution which can be obtained by approximation. In particular, in the reaction case (), if is approximated by a sequence of smooth functions, the sequence of approximating solutions converges to a solution of if , while it blows up everywhere if .

Finally, let us say a few words on how positive constant will be denoted hereafter. If no otherwise specified, we will write to denote any positive constant (possibly different) which only depends on the data, that is on quantities that are fixed in the assumptions (, , and so on…); in any case such constants never depend on the different indexes having a limit. In the sequel and throughout the paper, we will omit for simplicity the dependence on in the function and denote all quantities (possibly different) such that

and this will be in the order in which the parameters we use will tend to infinity, that is, first , then , and finally . Similarly, we will write only , or to mean that the limits are made only on the specified parameters. Moreover, for the sake of simplicity, in what follows, the convergence, even if not explicitly stressed, may be understood to be taken possibly up to a suitable subsequence extraction.

3.2.1. A Sequence of Approximating Problems

Consider the approximate problem

where is a smooth sequence of functions such that and in .

The existence of solutions of the above problem was ensured by Lemma 3.2.

3.2.2. A Priori Estimates

Lemma 3.10. There exists a subsequence of (also denoted ); there exists a measurable function such that and weakly in , strongly in , and a.e. in .

Proof. Let as test function in . One has

then

so is bounded in .

There exist a subsequence also denoted and a measurable function such that

By an easy argument we can see that, there exists a measurable function such that

3.2.3. Almost Everywhere Convergence of Gradients

Lemma 3.11. The subsequence obtained in Lemma 3.10 satisfies

Proof. Let us recall that since the -capacity , there exists a sequence satisfying the Lemma 3.4 with and .

Step 1. In this step we will show the following:

(i), (ii).

Let , such that . Let be a truncation defined by

Let , , .

We denote, respectively, by , the indicator function of .

Let such that with the modular convergence in (see [11]).

Let , as test function in the approximate problem. Then

We have

Then,

About : it is obvious since

that is

About : since is bounded in and , there exists a measurable function such that (up to a subsequence also denoted )

Then .

Since in modular convergence, then

So,

About : since on the set , then

and as for , one has

since when .

About : let as test function, in one hand we obtain

So,

On the other hand

Let us come back to our main estimation (i). We have

Then we deduce that

Since we have , and using (1.2), we deduce that

For , one has

We have is bounded in in and so in since .

Since we have

we easily deduce (i).

For (ii) we proceed as in (i) by using the fact that .

Then we conclude that

Step 2. In this step we prove that a.e. in .

Since is bounded in , then is finite a.e. in .

Hence there exists a measurable set such that and in .

Let , then by using Fatou lemma, we get

So, a.e. in and a.e. in , and for a subsequence still denoted by , we obtain from (3.38) that

Combining with (1.3), we get

Therefore, we have

Lemma 3.12. For all ,

For the proof we can adopt the same way as in [13].

3.2.4. The Convergence of the Problems and the Completion of the Proof of Theorem 3.6

In one hand by using as test function in , we can deduce as in [5] that converge strongly in to .

In the other hand, by Lemma 3.3, we deduce that is bounded in for every . Then is bounded in and the passage to the limit is an easy task.

Corollary 3.13. Under the hypotheses (1.2)–(1.7), is a convex combination of Dirac measures ( and ) and the problem admits at least one weak solution.

The proof is a simple adaptation of the one of Theorem 3.6 by taking in in the place of , where is the cut functions corresponding to the measure .

Remark 3.14. The technique used in the proof of Theorem 3.6 allows us to prove that the problem

has a weak solution.

3.3. General Case

Before giving the general case of Theorem 3.6, let's recall that the set of finite convex combination of Dirac measures is dense in the set of measures probability.

In the following theorem we denote by the sequence defined by for some , and with .

Theorem 3.15. Let be a sequence of Radon measures convergent to a Radon measure , and let , a sequence of weak solutions of . Then there exists such that

for every and every . Moreover, there exists a measurable function such that and is a weak solution of .

Proof. A priori estimate: Let be a sequence of mollifiers functions. Let as test function in

then, by using an easy argument, the properties of convolution, and tending to zero, we get

So is bounded in and as above there exists a measurable function such that

Almost everywhere convergence of gradients: Let us consider as test function in , we have

We prove easily, since , that

Also, .

By using the above estimation, we obtain a similar equation to (3.20) and we follow the same technique used in the step of almost everywhere convergence of gradients in Theorem 3.6 to prove the existence result.

Corollary 3.16. Under the hypotheses (1.2)–(1.7), is positive Radon measure and the problem admits at least one weak solution.

Since is bounded and by using the approximation of which is a measure of probability, the existence of solutions can be obtained as consequence of the last theorem.

Remark 3.17. Let us recall that if we suppose that the -functions and satisfy the condition, we can prove with the same way as in Theorem 3.6 that the problem has a weak solution for all singular measure in the sense that it is concentrated in some Borel set with zero -capacity.

Remark 3.18. One can see that the technique used in this paper can be adopted to prove the existence of solutions of the following problem

where satisfies, for almost every in , for every in , for every and in ,

Here is a measurable function such that

where is a nonnegative function in , , and is a positive real.

Acknowledgments

This research is supported by the Deanship of Research, King Khalid University, Project no. KKU-SCI-11-011. The author would like to thank Dr. Meskine for his fruitful discussion.

References

L. Boccardo, S. Segura de León, and C. Trombetti, “Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term,” Journal de Mathématiques Pures et Appliquées, vol. 80, no. 9, pp. 919–940, 2001.

V. Ferone and F. Murat, “Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small,” in Équations aux Dérivées Partielles et Applications, pp. 497–515, Gauthier-Villars, Paris, Farnce, 1998.

N. Grenon and C. Trombetti, “Existence results for a class of nonlinear elliptic problems with -growth in the gradient,” Nonlinear Analysis: Theory, Methods & Applications, vol. 52, no. 3, pp. 931–942, 2003.

C. Maderna, C. D. Pagani, and S. Salsa, “Quasilinear elliptic equations with quadratic growth in the gradient,” Journal of Differential Equations, vol. 97, no. 1, pp. 54–70, 1992.

A. Porretta, “Nonlinear equations with natural growth terms and measure data,” in Proceedings of the Fez Conference on Partial Differential Equations, Electronic Journal of Differential Equations, vol. 9, pp. 183–202, Southwest Texas State University, San Marcos, Tex, USA, 2002.

S. Segura de León, “Existence and uniqueness for data of some elliptic equations with natural growth,” Advances in Differential Equations, vol. 8, no. 11, pp. 1377–1408, 2003.

J. L. Kazdan and R. J. Kramer, “Invariant criteria for existence of solutions to second-order quasilinear elliptic equations,” Communications on Pure and Applied Mathematics, vol. 31, no. 5, pp. 619–645, 1978.

H. A. Hamid and M. F. Bidaut-Veron, “On the connection between two quasilinear elliptic problems with source terms of order 0 or 1,” Communications in Contemporary Mathematics, vol. 12, no. 5, pp. 727–788, 2010.

V. Ferone and B. Messano, “Comparison and existence results for classes of nonlinear elliptic equations with general growth in the gradient,” Advanced Nonlinear Studies, vol. 7, no. 1, pp. 31–46, 2007.

M. Fukushima, K.-i. Sato, and S. Taniguchi, “On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures,” Osaka Journal of Mathematics, vol. 28, no. 3, pp. 517–535, 1991.

J.-P. Gossez, “Some approximation properties in Orlicz-Sobolev spaces,” Studia Mathematica, vol. 74, no. 1, pp. 17–24, 1982.

J.-P. Gossez, “Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients,” Transactions of the American Mathematical Society, vol. 190, pp. 163–205, 1974.

M. K. Alaoui, “Parabolic problems with data measure,” Boundary Value Problems, vol. 2011, article 46, 2011.

G. Talenti, “Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces,” Annali di Matematica Pura ed Applicata, vol. 120, pp. 159–184, 1979.

G. Dal Maso, F. Murat, L. Orsina, and A. Prignet, “Renormalized solutions of elliptic equations with general measure data,” Annali della Scuola Normale Superiore di Pisa, vol. 28, no. 4, pp. 741–808, 1999.

A. Cianchi, “Optimal Orlicz-Sobolev embeddings,” Revista Matemática Iberoamericana, vol. 20, no. 2, pp. 427–474, 2004.