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