Multiple solutions for some strongly degenerate second order elliptic equations

TL;DR

This paper investigates boundary value problems involving degenerate elliptic operators that vanish on submanifolds, demonstrating the existence of solutions that vanish on the degeneracy set using weighted Sobolev spaces.

Contribution

It introduces a method to establish existence of solutions for degenerate elliptic equations with non-smooth degeneracy sets using weighted Sobolev spaces.

Findings

Existence of solutions that vanish on the degeneracy set.

Solutions are obtained despite the non-smoothness of the degeneracy.

Weighted Sobolev spaces are effective in handling degeneracies.

Abstract

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-\textrm{div} (a(x)\nabla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of $\Omega$ where it is not allowed to be $C^{2}$. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where $a$ vanishes.