Nonexistence of solutions for Dirichlet problems with supercritical growth in tubular domains

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 $mathbb{R}^n$, $nge 3$, and $f$ has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where $Omega$ is a tubular domain $T_arepsilon(Gamma_k)$ with thickness $arepsilon>0$ 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 $arepsilon>0$ small enough. When $kge 2$ 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 $f$.