Nonexistence and uniqueness for biharmonic problems with supercritical growth and domain geometry

Abstract

We prove nonexistence and uniqueness results of positive solutions for biharmonic supercritical equations $\Delta ^2 u = f(u)$ under Navier boundary conditions on a smooth bounded domain $\Omega \subset \mathbb{R}^N$. The results stand for suitable supercritical nonlinearities $f$ with some geometrical conditions on $ \Omega.$ We define the $h$-starlikeness of $\Omega$ and a classifying number $M(\Omega)$. This allows us to define a generalized critical exponent for these domains which play the role of the classical exponent $\frac{N+4}{N-4}$. Our approach is based on Rellich-Pohozaev type estimates. In particular, we construct some $h $-starlike domains which are topologically nontrivial where our results can apply.