Multiple solutions for some strongly degenerate second order elliptic equations

Abstract

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-\text{div}(a(x)\nabla u)$$ 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.

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