Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains

Abstract

We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }Ω,\qquad u\in H^{1,p}_0(Ω) $$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$, $n\ge 2$, $p>1$ and $f$ has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains $\Omega$, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if $n=2$, $1<p<2$, $f(u)=|u{|}^{q-2}u$ with $q>\frac{2p}{2-p}$ and $\Omega =\{(\rho \cos \theta ,\rho \sin \theta ):|\theta |<\alpha ,|\rho -1|<s\}$ with $0<\alpha <\pi$ and $0<s<1$, then for all $q>\frac{2p}{2-p}$ there exists $\bar{s}>0$ such that the problem has only the trivial solution $u\equiv 0$ for all $\alpha \in (0,\pi )$ and $s\in (0,\bar{s})$.