Affiliations: Departamento de Matemática, IMECC, Universidade Estadual de Campinas, Campinas, SP, Brazil. E-mail: msm@ime.unicamp.br | Instituto de Alta Investigación, Universidad de Tarapacá, Arica, Chile. E-mail: slorca@uta.cl
Note: [] Corresponding author: Marcelo Montenegro, Departamento de Matemática, IMECC, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, Brazil, CEP 13083-859. E-mail: msm@ime.unicamp.br
Abstract: The aim of this paper is study the equation −Δu=(log (u)+λup)χ{u>0} in Ω with Dirichlet boundary condition, where 0<p<(N+2)/(N−2) and p≠1. We regularize the term log (u) for u near 0 by using a function gε(u)=−log ((u2+εu+ε)/(u+ε)) for u≥0 which tends to log (u) as ε→0 pointwisely. When the parameter λ>0 is sufficiently large, the corresponding energy functional to the perturbed equation −Δu+gε(u)=λ(u+)p has nontrivial critical points uε in H01(Ω). Letting ε→0, then uε 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.
Keywords: singular problems, multiple solutions, variational methods, a priori estimates
