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.
Citation
Saïma Khenissy. "Nonexistence and uniqueness for biharmonic problems with supercritical growth and domain geometry." Differential Integral Equations 24 (11/12) 1093 - 1106, November/December 2011. https://doi.org/10.57262/die/1356012878
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