On the Existence of a Non-trivial Solution for the $p$-Laplacian Equation with a Jumping Nonlinearity

Abstract

We consider the existence of a non-trivial weak solution for the equation $$ \left\{ \begin{array}{@{}ll} -\Delta_p u= f(x,u) & \text{in } \ \Omega\,, \\ u=0 & \text{on } \ \partial\Omega\,, \end{array}\right. $$ where $f$ satisfies $f(x,u)=a u_+^{p-1} -bu_-^{p-1} + o(|u|^{p-1})$ ($p>1$) at 0 or $\infty$. By using Morse theory and calculating the critical groups, we show the existence of a non-trivial weak solution to the equation under mild auxiliary conditions.