Nonexistence results for elliptic problems with supercritical growth in thin planar domains

We deal with a class of nonlinear elliptic Dirichlet problems with terms having supercritical growth from the viewpoint of the Sobolev embedding. We prove that for every planar set G, which is contractible in itself and consists of a finite number of curves, there exist suitable bounded domains arbitrarily close to G (as thin neighbourhoods of G) such that in these domains these Dirichlet problems do not have any nontrivial solution. Domains of this type may have a shape very different from the starshaped bounded domains where a well known Pohozaev nonexistence result holds. Indeed, our result suggests that, in dimension n = 2, Pohozaev nonexistence result might be extended from the bounded starshaped domains to all bounded contractible domains. Notice that this fact is not true in higher dimensions n = 3. In fact, for example, in a domain as a pierced annulus of R-2 our result guarantees nonexistence of nontrivial solutions while in a pierced annulus of R-n with n = 3 there exist many nontrivial solutions when the size of the perforation is small enough.