Nonexistence of Solutions for Dirichlet Problems with Supercritical Growth in Tubular Domains

Abstract We deal with Dirichlet problems of the form { ? ? u + f ( stretchy="false">) columnalign="left"> /> = 0 in mathvariant="normal">? , on ? ? \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where ? is a bounded domain ? n {\mathbb{R}^{n}} , ? 3 {n\geq 3} and f has supercritical growth from viewpoint Sobolev embedding. In particular, we consider case tubular T ? mathvariant="normal">? k {T_{\varepsilon}(\Gamma_{k})} thickness > {{\varepsilon}>0} center {\Gamma_{k}} k -dimensional, smooth, compact submanifold . Our main result concerns 1 {k=1} contractible in itself. this prove that problem does not have nontrivial solutions for small enough. When 2 {k\geq 2} or noncontractible itself obtain weaker nonexistence results. Some examples show all these results are sharp what assumptions on