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

We deal with nonlinear elliptic Dirichlet problems of the form $$ div(|D u|^{p-2}D u )+f(u)=0quadmbox{ in }Omega,qquad u=0 mbox{ on }partialOmega $$ where $Omega$ is a bounded domain in $R^n$, $nge 2$, $p> 1$ and $f$ has supercritical growth from the viewpoint of Sobolev embedding. o 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 $n=2$, $1<2$, $f(u)=|u|^{q-2}u$ with $q>{2pover 2-p}$ and $Omega={( hocos heta, hosin heta) : | heta|{2pover 2-p}$ there exists $ar s>0$ such that the problem has only the trivial solution $uequiv 0$ for all $alphain (0,pi)$ and $sin (0,ar s)$.