Two solutions for a singular elliptic equation by variational methods

Montenegro, Marcelo; Silva, Elves A. B.

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 $-\Delta u=(-\frac{1}{u^{\beta }}+\lambda u^p){\chi }_{\{u>0\}}$ in $\Omega$ with Dirichlet boundary condition, where $0<\beta <1$ and $0<p<1$. In the first approach we consider a sequence of $ϵ$-problems with $1/u^{\beta }$ replaced by $u^q/{(u+ϵ)}^{q+\beta }$ with $0<q<p<1$. 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.

Classification : 34B16, 35J20, 35B65

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