Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 13 / 3 / d

First, multiplication by $w$ maps $W_0^{1,2}$ into $L^2$ by the Hölder and Sobolev estimates just given. Thus a weak solution $u$ of the perturbed equation has $v=\Delta u=f+wu\in L^2$. Uniqueness for the unperturbed Dirichlet problem gives $u=Tv$. Its equation becomesConversely, if this equation has a solution $v\in L^2$, then $u=Tv$ belongs to $W_0^{1,2}$ and satisfies $\Delta u=v=f+wTv=f+wu$ weakly. These constructions are inverse to one another.

The Fredholm alternative for a compact operator states that, on a Banach space, $I-K$ for compact $K$ is bijective if and only if its kernel is zero; in that case its inverse is bounded. Apply it on $L^2$. Kernels correspond under $T$: a nonzero $v$ with $v=wTv$ gives a nonzero $u=Tv$, since $\Delta Tv=v$; a nonzero homogeneous solution $u$ gives $v=wu$, which cannot be zero because a zero-boundary harmonic weak solution is zero. Therefore triviality of the homogeneous PDE kernel is equivalent to triviality of $\ker (I-K)$ and hence to unique solvability for every $f\in L^2$. The reverse implication also follows immediately by taking $f=0$. No sign assumption on $w$ is required.