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

Abstract: We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u\in H^{1,p}_0(\Omega) $$ 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 $q>{2p\over 2-p}$ and $\Omega={(\rho\cos\theta,\rho\sin\theta)\ :\ |\theta|<\alpha,\ |\rho -1|<s\}$ with $0<\alpha<\pi$ and $0<s\<1$, then for all $q>{2p\over 2-p}$ there exists $\bar s>0$ such that the problem has only the trivial solution $u\equiv 0$ for all $\alpha\in (0,\pi)$ and $s\in (0,\bar s)$.