Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2010, Article ID 134078, 10 pages
http://dx.doi.org/10.1155/2010/134078
Research Article

Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia

Received 18 June 2010; Accepted 25 September 2010

Academic Editor: A. Mikelic

Copyright © 2010 Faten Toumi. 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

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem in subject to some Dirichlet conditions, where is a regular domain in with compact boundary. The nonlinearity is nonnegative continuous and the potential belongs to some Kato class . So we prove the existence of a positive continuous solution depending on by the use of a potential theory approach.

1. Introduction

In this paper, we study the existence of positive bounded solution of semilinear elliptic problem where is a -domain in with compact boundary, and are fixed nonnegative constants such that , and when is bounded. The parameter is nonnegative, and the function is nontrivial nonnegative and continuous on .

Numerous works treated semilinear elliptic equations of the type For the case of nonpositive function , many results of existence of positive solutions are established in smooth domains or in , for instance, see [15] and the references therein.

In the case where changes sign, many works can be cited, namely, the work of Glover and McKenna [6], whose used techniques of probabilistic potential theory for solving semilinear elliptic and parabolic differential equations in . Ma and Song [7] adapted the same techniques of those of Glover and McKenna to elliptic equations in bounded domains. More precisely, the hypotheses in [6, 7] require in particular that and on each compact, there is a positive constant such that .

In [8], Chen et al. used an implicit probabilistic representation together with Schauder's fixed point theorem to obtain positive solutions of the problem (). In fact, the authors considered a Lipschitz domain in , with compact boundary and imposed to the function to satisfy on , where is nonnegative Borel measurable function defined on and the potentials are nonnegative Green-tight functions in . In particular, the authors showed the existence of solutions of () bounded below by a positive harmonic function.

In [9], Athreya studied () with the singular nonlinearity , , in a simply connected bounded -domain in . He showed the existence of solutions bounded below by a given positive harmonic function , under the boundary condition , where is a constant depending on , , and .

Recently, Hirata [20] gave a Chen-Williams-Zhao type theorem for more general regular domains . More precisely, the author imposed to the function to satisfy where the potential belongs to a class of functions containing Green-tight ones. We remark that the class of functions introduced by Hirata coincides with the classical Kato class introduced for smooth domains in [10, 11].

In this paper, we will consider . We impose to the potential to be in a new Kato class (see Definition 1.1 below), which contains the Green-tight functions and the classical Kato class used by Hirata. More precisely, we will prove using potential theory's tools, the existence of positive solution for (). Moreover, we will give global behaviour for the solution.

So, in the remainder of this introduction, we will give some results related on potential theory, and we will prove others. In the second section, we will give the main theorem and some examples of applications.

Let us recall that is the set of Borel measurable functions in and is the set of continuous ones vanishing at . The exponent means that only the nonnegative functions are considered.

We denote by the bounded continuous solution of the Dirichlet problem where the function is nontrivial nonnegative continuous on . In the remainder of this paper, we denote , and we remark that when is bounded.

Let us recall some notations and notions concerning essentially the potential theory.(i)For , we denote by the potential defined in by where is the Green function of the Laplace operator on with Dirichlet conditions.(ii)We recall that if and , then we have in (in the sense of distributions), see [10, page 52].(iii)Let be the Brownian motion in and be the probability measure on the Brownian continuous paths starting at . For , we define the kernel by where is the expectation on and . If is such that , the kernel satisfies (see [10, 12]) So for, each such that , we have (iv)We recall that a function is called completely monotone if , for each . Moreover, if is completely monotone on , then by [13, Theorem  12a], there exists a nonnegative measure on such that So, using this fact and the Hölder inequality, we deduce that if is completely monotone from to , then is a convex function.(v)Let be such that . From (1.5), it is easy to see that for each , the function is completely monotone on .

Now, we recall some properties relating to the Kato class .

Definition 1.1 (see [14, 15]). A Borel measurable function in belongs to the class if satisfies

where and is the Euclidean distance between and .

Remark 1.2. When is a bounded domain, then we can replace by and the condition (1.9) is superfluous.

Proposition 1.3 (See [14, 15]). Let be a nonnegative function in . Then one has (i)(ii)the potential .

Proposition 1.4 (see [16, 17]). Let be a nonnegative function belonging to . Then, one has (i)(ii) for any nonnegative superharmonic function in , one has

Proposition 1.5. Let be a nonnegative superharmonic function in and be a nonnegative function in . Then, for each such that , one has

Proof. Let be a nonnegative superharmonic function, then by [18, Theorem  2.1, page 164], there exists a sequence of nonnegative measurable functions in such that the sequence defined in by increases to .
Let such that . Then, there exists such that , for .
Now, for a fixed , we consider the function . Since the function is completely monotone on , then is convex on . Therefore, which means Hence, it follows from Proposition 1.4 (i) that Consequently, from (1.6), we obtain that By letting , we deduce the result.

2. Main Result

In this section, we will give an existence result for the problem (). Assume the following assumptions.  The function is nonnegative and belongs to .   The function is a nonnegative, continuous on and satisfies , such that, , .   .

Remark 2.1. Let be in , then for , the function satisfies . In particular, if is nonincreasing, then holds.

Consider the function , where is the constant associated to the potential defined by (1.10). It is obvious to see that is bijective from to .

Theorem 2.2. Assume that the hypotheses ()–() are satisfied. Then, for each , the problem () has a positive continuous bounded solution satisfying

Remark 2.3. We remark that if satisfies the hypothesis and , we take , in this case for each , the problem () has a positive bounded solution satisfying

Now, let us give some examples of applications of the above theorem.

Example 2.4. Assume that is satisfied. Let . Then, for each , the following problem admits a positive continuous bounded solution. Indeed, for each , one verifies that for , the function satisfies .

Example 2.5. Let . Assume and . Consider the following: Then, the function is in and decreasing. By Remark 2.1, the hypothesis is satisfied for . So that for each , (2.4) has a positive continuous bounded solution satisfying

Example 2.6. Let be a -bounded domain and suppose that the hypothesis is satisfied. Let be a nonnegative function in such that and suppose that . Then, has a positive continuous solution.
Let us verify the assumptions and . From [16, Proposition 2.3], the function , and so the hypothesis is satisfied. From [16, Proposition 2.7(iii)], there exists a constant such that we have for each Now, since is nontrivial continuous function at , then there exists , such that one has on Thus, and so the assumption is satisfied.

Example 2.7. Let be the exterior of the unit ball in . Suppose that the hypothesis is satisfied. Let such that . Then, has a positive continuous solution.
From [14], the function and so the assumption is satisfied. Moreover, from [14, Proposition 3.5], there exists a constant such that one has Now, from [19, page 258], there exists a constant such that one has on Thus, and so the assumption is satisfied.

In the next, we will give the proof of Theorem 2.2.

Proof. Let and put . Let , then from , there exists , such that the function is a nondecreasing function on . Let be the constant given by , and let where . Put . Consider the nonempty bounded convex set given by Let be the operator defined on by We claim that the operator maps to itself. Indeed, by and using the monotony of the function , we have for On the other hand, by using Proposition 1.5 and , we have Hence, .
Next, we prove that the operator is nondecreasing on . Let such that then by hypothesis , we obtain Now, consider the sequence defined by and for .
It is obvious to see that and . Thus, using the fact that is invariant under and the monotony of , we deduce that Hence, the sequence converges to a function .
Therefore, from the monotone convergence theorem and the fact that is continuous, the sequence converges to . So, or equivalently Applying the operator to both sides of the above equality and using (1.6) and (1.7), we conclude that satisfies Finally, let us verify that is a solution of the problem (). Using the fact that and is bounded on , we obtain . So, Proposition 1.3 (ii) yields which implies with the continuity of the harmonic continuous function that is continuous on . This completes the proof.

Acknowledgment

The author express her sincere gratitude to Professor Habib Msâagli for his guidance and the useful discussions. Thanks go to the referees for valuable comments and useful remarks on the paper.

References

  1. A. L. Edelson, “Entire solutions of singular elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 139, no. 2, pp. 523–532, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. A. V. Lair and A. W. Shaker, “Entire solution of a singular semilinear elliptic problem,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 498–505, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. A. C. Lazer and P. J. McKenna, “On a singular nonlinear elliptic boundary-value problem,” Proceedings of the American Mathematical Society, vol. 111, no. 3, pp. 721–730, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. H. Mâagli and M. Zribi, “Existence and estimates of solutions for singular nonlinear elliptic problems,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 522–542, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. N. Zeddini, “Positive solutions for a singular nonlinear problem on a bounded domain in 2,” Potential Analysis, vol. 18, no. 2, pp. 97–118, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. Glover and P. J. McKenna, “Solving semilinear partial differential equations with probabilistic potential theory,” Transactions of the American Mathematical Society, vol. 290, no. 2, pp. 665–681, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. Z. M. Ma and R. M. Song, “Probabilistic methods in Schrödinger equations,” in Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), E. Cinlar, K. L. Chung, and R. K. Getoor, Eds., vol. 18 of Progress in Probability, pp. 135–164, Birkhäuser Boston, Boston, Mass, USA, 1990. View at Google Scholar · View at Zentralblatt MATH
  8. Z. Q. Chen, R. J. Williams, and Z. Zhao, “On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions,” Mathematische Annalen, vol. 298, no. 3, pp. 543–556, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Athreya, “On a singular semilinear elliptic boundary value problem and the boundary Harnack principle,” Potential Analysis, vol. 17, no. 3, pp. 293–301, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, vol. 312 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1995.
  11. Z. X. Zhao, “On the existence of positive solutions of nonlinear elliptic equations—a probabilistic potential theory approach,” Duke Mathematical Journal, vol. 69, no. 2, pp. 247–258, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. H. Maâgli, “Perturbation semi-linéaire des résolvantes et des semi-groupes,” Potential Analysis, vol. 3, pp. 61–87, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. D. V. Widder, The Laplace Transform, vol. 6 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1941.
  14. I. Bachar, H. Maâgli, and N. Zeddini, “Estimates on the Green function and existence of positive solutions of nonlinear singular elliptic equations,” Communications in Contemporary Mathematics, vol. 5, no. 3, pp. 401–434, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. H. Mâagli and M. Zribi, “On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domains of n,” Positivity, vol. 9, no. 4, pp. 667–686, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. F. Toumi, “Existence of positive solutions for nonlinear boundary value problems in bounded domains of n,” Abstract and Applied Analysis, vol. 2006, Article ID 95480, 18 pages, 2006. View at Google Scholar · View at Zentralblatt MATH
  17. F. Toumi and N. Zeddini, “Existence of positive solutions for nonlinear boundary-value problems in unbounded domains of n,” Electronic Journal of Differential Equations, vol. 2005, 14 pages, 2005. View at Google Scholar
  18. S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory. Probability and Mathematical Statistic, Academic Press, New York, NY, USA, 1978.
  19. D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer Monographs in Mathematics, Springer, London, UK, 2001. View at Zentralblatt MATH
  20. K. Hirata, “On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 885–891, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH