On the p-biharmonic equation involving concave-convex nonlinearities and sign-changing weight function

Abstract

In this paper, we study the combined effect of concave and convex nonlinearities on the number of nontrivial solutions for the $p$-biharmonic equation of the form \begin{equation}\left\{ \begin{array}{l} \Delta_{p}^{2}u=\vert u\vert^{q-2}u+\lambda f(x)\vert u\vert^{r-2}u \quad\quad \text{ in}\,\,\ \Omega, \\ u=\nabla u=0\quad\quad\quad\text{ on }\partial \Omega , \end{array} \right.\end{equation} where $\Omega$ is a bounded domain in $R^{N}$, $f\in C(\overline{\Omega})$ be a sign-changing weight function. By means of the Nehari manifold, we prove that there are at least two nontrivial solutions for the problem.

Keywords

• p-biharmonic equations
• nehari manifold
• concave-convex nonlinearities
• sign-changing weight function