Two solutions for a singular elliptic equation by variational methods
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 1, pp. 143-165.

We find two nontrivial solutions of the equation -Δu=(-1 u β +λu p )χ {u>0} in Ω with Dirichlet boundary condition, where 0<β<1 and 0<p<1. In the first approach we consider a sequence of ϵ-problems with 1/u β replaced by u q /(u+ϵ) q+β with 0<q<p<1. When the parameter λ>0 is large enough, we find two critical points of the corresponding ϵ-functional which, at the limit as ϵ0, give rise to two distinct nonnegative solutions of the original problem. Another approach is based on perturbations of the domain Ω, we then find a unique positive solution for λ large enough. We derive gradient estimates to guarantee convergence of approximate solutions u ϵ to a true solution u of the problem.

Published online:
Classification: 34B16, 35J20, 35B65
Montenegro, Marcelo 1; Silva, Elves A. B. 2

1 Universidade Estadual de Campinas IMECC, Departamento de Matemática Rua Sergio Buarque de Holanda, 651 Campinas, SP, Brazil, CEP 13083-970
2 Universidade de Brasília Departamento de Matemática Brasília, DF, Brazil, CEP 70910-900
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Montenegro, Marcelo; Silva, Elves A. B. Two solutions for a singular elliptic equation by variational methods. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 1, pp. 143-165. http://archive.numdam.org/item/ASNSP_2012_5_11_1_143_0/

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