Sign-changing solutions for $p$-Laplacian equations with jumping nonlinearity and the Fučik spectrum

Abstract

We study the existence of sign-changing solutions for the $p$-Laplacian equation $$ -\Delta_pu +\lambda g(x)|u|^{p-2}u=f(u),\quad x\in \mathbb{R}^N, $$ where $\lambda$ is a positive parameter and the nonlinear term $f$ has jumping nonlinearity at infinity and is superlinear at zero. The Fučik spectrum plays an important role in the proof. We give sufficient conditions for the existence of nontrivial Fučik spectrum.