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

Abstract

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 $f$.