On a Singular Nonlinear Elliptic Problem

Abstract

Elliptic boundary value problems of the form $Lu - k(x)u^{ - \alpha } $ in $\Omega $ and $u = 0$ on the boundary of $\Omega $ are studied where L is given in the divergence form, $\alpha > 0$ and $k(x)$ is a nonnegative measurable real function. Existence in $C^1 (\bar \Omega )$ and uniqueness of a solution $u \geqq 0$ are established for the equivalent fixed point problem $u(x) = \int _\Omega G(x,s)k(s)[u(s)]^{ - \alpha } ds$, where $G(x,s)$ is the Green’s function for the Dirichlet problem associated to L in $\Omega $. Various inequalities for $G(x,s)$ are proved and a study is made of the action of the integral operator defined by the kernel G on unbounded functions with a prescribed growth near the boundary.