Multiplicity results for biharmonic equations with critical growth

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the biharmonic operator \(\Delta ^2\) ${\mathrm{\Delta }}^2$ and involving a critical Sobolev term. In particular, we consider \(\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=\lambda \left| u\right| ^{2^*-2}u+f(x,u) & \text{ in } \Omega \\ u=\Delta u = 0 & \text{ on } \partial \Omega , \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathrm{\Delta }}^{2}u=\lambda {\left|u\right|}^{2^{\ast }-2}u+f(x,u)\hfill & \mathrm{in}\mathrm{\Omega }\hfill \\ u=\mathrm{\Delta }u=0\hfill & \mathrm{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \subset \mathbb {R}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is an open bounded set with continuous boundary, \(n>4\) $n>4$ , \(\lambda \) $\lambda$ is a positive real parameter, \(2^*=2n/(n-4)\) $2^{\ast }=2n/(n-4)$ is the critical Sobolev exponent and f is a Carathéodory function satisfying different subcritical conditions.