Compatibility conditions for the existence of weak solutions to a singular elliptic equation

This paper deals with the existence of positive solutions to the singular elliptic boundary value problem involving p-Laplace operator -div(|del u|(p-2)del u) = h(x)/u(alpha) + k(x)u(beta), x is an element of Omega; u(x) > 0, x is an element of Omega; u(x) = 0, x is an element of partial derivative Omega; where Omega subset of R-N (N >= 1) is a bounded domain with smooth boundary partial derivative Omega, h is an element of L-1(Omega), h(x) > 0 almost everywhere in Omega, k is an element of L-infinity(Omega) is nonnegative, p > 2, alpha > 1 and beta is an element of (0, p - 1). A compatibility condition on the couple (h(x),alpha) is given for the problem to have at least one solution. More precisely, it is shown that the problem admits a solution if and only if there exists u(0) is an element of H-0(1)(Omega) such that integral(Omega) hu(0)(1-alpha) dx < infinity.