Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 32, pp. 1-16.

Title:  Domain geometry and the Pohozaev identity

Authors: Jeff McGough (South Dakota School of Mines, SD, USA)
         Jeff Mortensen (Univ. of Nevada-Reno, USA)
         Chris Rickett (South Dakota School of Mines, SD, USA)
	 Gregg Stubbendieck (South Dakota School of Mines, SD, USA)
 
Abstract: 
 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.
 
Submitted August 26, 2004. Published March 22, 2005.
Math Subject Classifications: 35J20, 35J65.
Key Words: Partial differential equations; variational identities; 
           Pohozaev identities; numerical methods