Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)The aim of this paper is study the equation -Delta u = (log(u) + lambda u(p))chi({u>0}) in Omega with Dirichlet boundary condition, where 0 < p < N+2/N-2 and p not equal 1. We regularize the term log(u) for u near 0 by using a function g(epsilon)(u) = -log(u(2)+epsilon u+epsilon/u+epsilon) for u >= 0 which tends to log(u) as epsilon -> 0 pointwisely. When the parameter lambda > 0 is sufficiently large, the corresponding energy functional to the perturbed equation -Delta u + g(epsilon)(u) = lambda(u(+))(p) has nontrivial critical points u(epsilon) in H-0(1)(Omega). Letting epsilon -> 0, then u(epsilon) converges to a solution of the original problem, which is nontrivial and nonnegative. For 1 < p < N+2/N-2 there is at least one nontrivial solution. While for 0 < p < 1, there are at least two nontrivial distinct solutions.824167191107Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fondecyt [1120706]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fondecyt [1120706]