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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 boundeddomain of $\mathbb{R}^n$, $n\ge 3$, and $f$ has supercritical growth from theviewpoint 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, compactsubmanifold 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 theproblem does not have nontrivial solutions for $\varepsilon>0$ small enough.When $k\ge 2$ or $\Gamma_k$ is noncontractible in itself we obtain weakernonexistence results. Some examples show that all these results are sharp forwhat concerns the assumptions on $k$ and $f$.

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