Nodal solutions of a p-Laplacian equation

Abstract

We prove that the $p$-Laplacian problem $-\Delta_p u = f(x, u)$, with $u \in W^{1, p}_0 (\Omega)$ on a bounded domain $\Omega \subset R^N$, with $p > 1$ arbitrary, has a nodal solution provided that $f : \Omega \times R \to R$ is subcritical, and $f(x, t) / |t|^{p - 2}$ is superlinear. Infinitely many nodal solutions are obtained if, in addition, $f(x, -t) = -f(x, t)$.