Existence of solutions for a fourth order problem at resonance

Abstract

In this work, we are interested at the existence of nontrivial solutions of two fourth order problems governed by the weighted p-biharmonic  operator. The first is the following$$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda_1 m(x)|u|^{p-2}u+f(x,u)-h \mbox{ in }\Omega,\,\, u=\Delta u=0 \mbox{ on }\partial\Omega,$$where $\lambda_1$ is the first eigenvalue for the eigenvalue problem$ \Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x) |u|^{p-2}u\mbox{in }\Omega, \,\, u=\Delta u=0 \mbox{ on } \partial\Omega.$ \\In the seconde problem, we replace  $\lambda_1$ by $\lambda$ suchthat $\lambda_1<\lambda<\bar{\lambda}$, where $\bar{\lambda}$ is given bellow.