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

Abstract

We deal with Dirichlet problems of the form $$ Δu+f(u)=0 \mbox{ in }Ω,\qquad u=0\ \mbox{ on }\partial Ω$$ 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 $T_{\epsilon }({\Gamma }_k)$ with thickness $\epsilon >0$ 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 $\epsilon >0$ 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 $f$.

14 pages