Nonlinear fourth order problems with asymptotically linear nonlinearities

Abstract

We investigate some nonlinear elliptic problems of the form \Delta^2v + \sigma(x) v= h(x,v)\quadin \Omega,\quad v=\Delta v=0 \quadon \partial\Omega, \eqno({\rm P}) where $\Omega$ is a regular bounded domain in $\mathbb{R}^N$, $N\geq2$, $\sigma(x)$ a positive function in $L^{\infty}(\Omega)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

Keywords

• asymptotically linear
• mountain pass theorem
• biharmonic equation
• cerami sequence