Asymptotic behavior and existence of solutions for singular elliptic equations

Abstract

We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$-\Delta u=\frac{f(x)}{u^{\gamma }}\text{ in }\Omega ,$$ where $\Omega$ is an open, bounded subset of $\RN$ and $f$ is a bounded function. We deal with the existence of a limit equation under two different assumptions on $f$: either strictly positive on every compactly contained subset of $\Omega$ or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to $$-\Delta v+\frac{|\nabla v{|}^2}{v}=f\text{ in }\Omega .$$

31 pages

References in corpus (4)

• Inverse Problem for a Curved Quantum Guide
• Existence and nonexistence of solutions for singular quadratic quasilinear equations
• Finite and Infinite energy solutions of singular elliptic problems: Existence and Uniqueness
• Qualitative properties of singular solutions to semilinear elliptic problems