  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 $s\in (0,\bar s)$.
