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 , whose used techniques of probabilistic potential theory for solving semilinear elliptic and parabolic differential equations in . Ma and Song  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 , 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 , 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  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
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.
Z. M. Ma and R. M. Song, “Probabilistic methods in Schr？dinger 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, Birkh？user Boston, Boston, Mass, USA, 1990.
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.
I. Bachar, H. Maagli, 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.
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.