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, p. 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 : 2018-06-21
Classification:  34B16,  35J20,  35B65
@article{ASNSP_2012_5_11_1_143_0,
     author = {Montenegro, Marcelo and Silva, Elves A. B.},
     title = {Two solutions for a singular elliptic equation~by variational methods},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {1},
     year = {2012},
     pages = {143-165},
     zbl = {1241.35103},
     mrnumber = {2953046},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_1_143_0}
}
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://www.numdam.org/item/ASNSP_2012_5_11_1_143_0/

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