Pohozaev obstructions on non-starlike domains

Abstract

This paper investigates the existence of a Pohozaev obstruction on non-starlike domains. The elliptic problems we examine are of the form \(\begin{array}{rl}\displaystyle \rm div\left\{\, g(x, \nabla u) \right\} + \lambda f(x,u) = 0, & x\in\Omega , \\\displaystyle u = 0, & x\in\partial\Omega , \end{array}\) where \(\Omega\) is a bounded domain with smooth boundary. It is well understood that the nonexistence theory can be extended, but not well understood how far. This paper looks at classes of domains which fall outside the class of starlike domains. For the extensions of the starlike domains studied, existence, uniqueness and non-existence are discussed for the elliptic problem above.