Semilinear problems with right-hand sides singular at u = 0 which change sign

Abstract

The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at $u=0$. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at $u=0$, while no restriction on its growth at $u=0$ is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than $1/\left|u\right|$, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is $1/|u{|}^{\gamma }$ with $0<\gamma <1$.