Domain geometry and the Pohozaev identity

In this paper, we investigate the boundary between existence and nonexistence for positive solutions of Dirichlet problem $Delta u + f(u) = 0$, where $f$ has supercritical growth. Pohozaev showed that for convex or polar domains, no positive solutions may be found. Ding and others showed that for domains with non-trivial topology, there are examples of existence of positive solutions. The goal of this paper is to illuminate the transition from non-existence to existence of solutions for the nonlinear eigenvalue problem as the domain moves from simple (convex) to complex (non-trivial topology). To this end, we present the construction of several domains in $R^3$ which are not starlike (polar) but still admit a Pohozaev nonexistence argument for a general class of nonlinearities. One such domain is a long thin tubular domain which is curved and twisted in space. It presents complicated geometry, but simple topology. The construction (and the lemmas leading to it) are new and combined with established theorems narrow the gap between non-existence and existence strengthening the notion that trivial domain topology is the ingredient for non-existence.