Critical growth biharmonic elliptic problems under Steklov-type boundary conditions

Abstract

We study the fourth-order nonlinear critical problem $\Delta^2 u= u^{2^*-1}$ in a smooth, bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 5$, subject to the boundary conditions $u=\Delta u-d u_\nu=0$ on $\partial \Omega$. We provide estimates for the range of parameters $d \in \mathbb{R}$ for which this problem admits a positive solution. If the domain is the unit ball, we obtain an almost complete description.