Spectral Study about Biharmonic Operator with Landesman-Lazer Condition

We study the existence of weak solutions for the following class of problems $$\left\{ \begin{array}{lcl} \alpha \Delta^{2}u +\beta \Delta u = \mu u +\gamma h(x,u)&\mbox{in}\ \Omega,\\[1mm] B(u) = 0 &\mbox{on}\ \partial\Omega, \end{array} \right.$$ where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $N\geq 1$, $\alpha\geq0$, $-\infty <\beta<\alpha\lambda_1$, $\lambda_1$ is the first eigenvalue of $(-\Delta, H^1_0(\Omega))$, $\mu\in(0,\bar{\mu})$, $\bar{\mu}<\mu_2$, $\mu_2$ is the second eigenvalue of the problem $$(\alpha\Delta^2u+\beta\Delta u,H^1_0(\Omega)\cap H^2(\Omega)),$$ $\gamma\neq0$ is a real parameter and $h:\overline{\Omega}\times\mathbb{R} \to \mathbb{R}$ is a Carath\'eodory function verifying some conditions, the boundary condition $B(u)=0$ on $\partial \Omega$ means that $u=\Delta u=0$ on $\partial\Omega$ when $\alpha>0$ and $u=0$ on $\partial \Omega$ when $\alpha=0$. In this article, we revisit the arguments of Landesman-Lazer, both in the global and local aspects.