Two solutions for a singular elliptic equation by variational methods

作者：Montenegro Marcelo^*; Silva Elves A B

来源：Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2012, 11(1): 143-165.

DOI：10.2422/2036-2145.201003_002

We find two nontrivial solutions of the equation -Delta u = (- 1/u(beta) + lambda u(p)) chi({u %26gt; 0}) in Omega with Dirichlet boundary condition, where 0 %26lt; beta %26lt; 1 and 0 %26lt; p %26lt; 1. In the first approach we consider a sequence of epsilon-problems with 1/u(beta) replaced by u(q)/(u + epsilon)(q-beta) with 0 %26lt; q %26lt; p %26lt; 1. When the parameter lambda %26gt; 0 is large enough, we find two critical points of the corresponding epsilon-functional which, at the limit as epsilon -%26gt; 0, give rise to two distinct nonnegative solutions of the original problem. Another approach is based on perturbations of the domain Omega, we then find a unique positive solution for lambda large enough. We derive gradient estimates to guarantee convergence of approximate solutions u(epsilon) to a true solution u of the problem.

• 出版日期2012