Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order

Abstract

In this paper we consider the equation $\Delta^2 u = \lambda |u|^{q-2} u + |u|^{2^*-2} u\equiv f(u)$ in a smooth bounded domain $\Omega\subset\mathbf{R}{N}$ with boundary conditions either $u|_{\partial\Omega} =\frac{\partial u}{\partial n}|_{\partial \Omega}=0$ or $u|_{\partial\Omega}=\Delta u|_{\partial \Omega}=0$, where $N>4$, $ 1< q <2, \,\lambda>0$ and $2^* = 2N/(N-4)$ We prove the existence of $\lambda_0$ such that for $0<\lambda<\lambda_0$ the above problems have infinitely many solutions. For the problem with the second boundary conditions, we prove the existence of a positive solution also in the supercritical case, i.e., when we have an exponent larger than $2^*$. Moreover, in the critical case, we show the existence of at least two positive solutions.