Existence of positive solutions for singular Dirichlet problems

Abstract

We study the existence of positive solutions for singular Dirichlet problems. By the variational method, we show that if $\lambda_1$ is contained in some interval, then a singular Dirichlet problem has a radially symmetric, positive solution on an annulus $\Omega$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with homogeneous Dirichlet boundary condition. Since we consider a problem with singularity, we have difficulties that either the functional $I$ associated with the problem is not differentiable even in the sense of Gâteaux or the strong maximum principle is not applicable. But we show that a function which attains some minimax value for $I$ is a solution. We also treat the case that $\Omega$ is a ball.