Let $U\subset\mathbb{R}^n$ be bounded and open. Let $A_0:H_0^1(U)\to H^{-1}(U)$ be the operator associated to a bounded and coercive uniformly elliptic principal [bilinear form](/page/Bilinear%20Form), so that $A_0$ is an isomorphism by Lax-Milgram. Let $K:H_0^1(U)\to H^{-1}(U)$ be a bounded operator such that

\begin{align*} T=A_0^{-1}K:H_0^1(U)&\to H_0^1(U) \end{align*}

is compact. Set $L=A_0+K$. Equip $H_0^1(U)$ with its usual Hilbert [inner product](/page/Inner%20Product), and let $(I+T)^*$ denote the adjoint of $I+T$ with respect to that inner product. Then the homogeneous solution space $\ker L\subset H_0^1(U)$ is finite-dimensional. If $\ker L=\{0\}$, then for every $f\in H^{-1}(U)$ there is a unique weak solution $u\in H_0^1(U)$ of $Lu=f$. If $\ker L\ne\{0\}$, then $Lu=f$ is solvable if and only if $A_0^{-1}f$ is orthogonal in $H_0^1(U)$ to $\ker((I+T)^*)$; whenever a solution $u_0$ exists, the full solution set is $u_0+\ker L$.