Existence of solution for a class of biharmonic equations

Résumé

In this paper, We prove the solvability of the biharmonic problem

$$\begin{cases}\Delta^{2}u=f(x,u)+h ~~~~in~~\Omega, &\hbox{}\\

u=\Delta u=0 ~~~~~~on ~~\partial\Omega,\\\end{cases}$$

for a given function $h\in L^2(\Omega)$, if the limits at infinity of the quotients $f(x,s)/s$ and $2F(x,s)/s$ for a.e.$x\in\Omega$ lie between two consecutive eigenvalues of the biharmonic operator $\Delta^2$, where $F(x,s)$ denotes the primitive $F(x,s)=\int_{0}^{s}{f(x,t)dt}$.