Positive solutions for a two-point nonlinear boundary value problem with applications to semilinear elliptic equations

Abstract

Sharp existence results are given for positive solutions to the boundary value problem $v''+g(t,v) =0$, $v(0)= v(t_0)=0$, where $g(t,v)$ is allowed to be singular at both $t=0$ and $t=t_0$. These results are applied to radial solutions of the semilinear elliptic problem $\Delta u +f(r,u)= 0$ on a ball. Examples and corollaries illustrate the wide class of equations to which the results apply. The proofs use the Mountain Pass Lemma in conjunction with some delicate estimates, which allow the treatment of singular equations.