Asymptotically linear fourth-order elliptic problems whose nonlinearity crosses several eigenvalues - Evandro Monteiro (Electronic Journal of Differential Equations, 2011)
In this article we prove the existence of multiple solutions for the fourth-order elliptic problem $$displaylines{ Delta^2u+cDelta u = g(x,u) quadhboxin Omegacr u =Delta u= 0 quadhboxon partial Omega, }$$ where $Omega subset mathbbR^N$ is a bounded domain, $g:OmegaimesmathbbRo mathbbR$ is a function of class $C^1$ such that $g(x,0)=0$ and it is asymptotically linear at infinity. We study the cases when the parameter c is less than the first eigenvalue, and between two consecutive eigenvalues of the Laplacian. To obtain solutions we use the Saddle Point Theorem, the Linking Theorem, and Critical Groups Theory.