Two solutions for a singular elliptic equation by variational methods. (English) Zbl 1241.35103

Montenegro, Marcelo; Silva, Elves A. B.

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 11, No. 1, 143-165 (2012).

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

MSC:

35J75	Singular elliptic equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations
35J20	Variational methods for second-order elliptic equations
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

nonnegative solutions; gradient estimates