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

Author:

Molle Riccardo1,Passaseo Donato2

Affiliation:

1. Dipartimento di Matematica , Università di Roma “Tor Vergata” , Via della Ricerca Scientifica n. 1, 00133 Roma , Italy

2. Dipartimento di Matematica “E. De Giorgi” , Università di Lecce , P.O. Box 193, 73100 Lecce , Italy

Abstract

Abstract We deal with Dirichlet problems of the form { Δ u + f ( u ) = 0 in  Ω , u = 0 on  Ω , \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 Ω is a bounded domain of n {\mathbb{R}^{n}} , n 3 {n\geq 3} , and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain T ε ( Γ k ) {T_{\varepsilon}(\Gamma_{k})} with thickness ε > 0 {{\varepsilon}>0} and center Γ k {\Gamma_{k}} , a k-dimensional, smooth, compact submanifold of n {\mathbb{R}^{n}} . Our main result concerns the case where k = 1 {k=1} and Γ k {\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε > 0 {{\varepsilon}>0} small enough. When k 2 {k\geq 2} or Γ k {\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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

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