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

We deal with Dirichlet problems of the form {Delta u + f(u) = 0 in Omega, u=0 on partial derivative Omega, where Omega is a bounded domain of R-n, n >= 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 center Gamma(k) , a k-dimensional, smooth, compact submanifold of 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 >= 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.