  We deal with Dirichlet problems of the form $$ \Delta u+f(u)=0 \mbox{ in
}\Omega,\qquad u=0\ \mbox{ on }\partial \Omega $$ where $\Omega$ is a bounded
domain of $\mathbb{R}^n$, $n\ge 3$, and $f$ has supercritical growth from the
viewpoint of Sobolev embedding. In particular, we consider the case where
$\Omega$ is a tubular domain $T_\varepsilon(\Gamma_k)$ with thickness
$\varepsilon>0$ and centre $\Gamma_k$, a $k$-dimensional, smooth, compact
submanifold of $\mathbb{R}^n$. Our main result concerns the case where $k=1$
and $\Gamma_k$ is contractible in itself. In this case we prove that the
problem does not have nontrivial solutions for $\varepsilon>0$ small enough.
When $k\ge 2$ or $\Gamma_k$ is noncontractible in itself we obtain weaker
nonexistence results. Some examples show that all these results are sharp for
what concerns the assumptions on $k$ and $f$.
