Nodal solutions to semilinear elliptic equations in a ball

Abstract

In this paper we are concerned with the existence and multiplicity of nodal solutions to the Dirichlet problem associated to the elliptic equation $\Delta u+q(|x|)g(u)=0$ in the unit ball in ${\bf R}^N$. The nonlinearity $g$ has a linear growth at infinity and zero, while the weight function $q$ is nonnegative in $[0,1]$ and strictly positive in some interval $[r_1,r_2]\subset [0,1]$. By means of a topological degree approach, we are able to prove the existence of solutions with prescribed nodal properties, depending on the behaviour of the ratio $g(u)/u$ at infinity and zero.