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

We find two nontrivial solutions of the equation $-\Delta u=\left(-\frac{1}{{u}^{\beta }}+\lambda {u}^{p}\right){\chi }_{\left\{u>0\right\}}$ in $\Omega$ with Dirichlet boundary condition, where $0<\beta <1$ and $0. In the first approach we consider a sequence of $ϵ$-problems with $1/{u}^{\beta }$ replaced by ${u}^{q}/{\left(u+ϵ\right)}^{q+\beta }$ with $0. When the parameter $\lambda >0$ is large enough, we find two critical points of the corresponding $ϵ$-functional which, at the limit as $ϵ\to 0$, 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 ${u}_{ϵ}$ to a true solution $u$ of the problem.

Publié le :
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},
pages = {143--165},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {1},
year = {2012},
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, Série 5, Tome 11 (2012) no. 1, pp. 143-165. http://www.numdam.org/item/ASNSP_2012_5_11_1_143_0/

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