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

Abstract

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 $\Gamma$, which is contractible in itself and consists of a finite number of curves, there exist suitable bounded domains arbitrarily close to $\Gamma$ (as thin neighbourhoods of $\Gamma$) 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 $n\ge 3$. In fact, for example, in a domain as a pierced annulus of $\mathbb {R}^2$ our result guarantees nonexistence of nontrivial solutions while in a pierced annulus of $\mathbb {R}^n$ with $n \ge 3$ there exist many nontrivial solutions when the size of the perforation is small enough.