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Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

DOI: 10.1155/2010/134078

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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 [1–5] 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


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