Non-existence of patterns for a class of weighted degenerate operators

A classical result by Casten-Holland and Matano asserts that constants are the only positive and stable solutions to semilinear elliptic PDEs subject to homogeneous Neumann boundary condition in bounded convex domains. In other terms, this result asserts that stable patterns do not exist in convex domains. In this paper we consider a weighted version of the Laplace operator, where the weight may be singular or degenerate at the origin, and prove the nonexistence of patterns, extending the results by Casten-Holland and Matano to general weak solutions (not necessarily stable) and under a suitable assumption on the nonlinearity and the domain. Our results exhibit some intriguing behaviour of the problem according to the weight and the geometry of the domain. Indeed, our main results follow from a geometric assumption on the second fundamental form of the boundary in terms of a parameter which describes the degeneracy of the operator. As a consequence, we provide some examples and show that nonexistence of patterns may occurs also for non convex domains whenever the weight is degenerate.