Symmetric elliptic Dirichlet energy with a nonpositive potential ($J(u)=\frac{1}{2}\int (A\nabla u\cdot \nabla u-q{u}^2)+\int fu$)

For a bounded domain, bounded symmetric uniformly elliptic $A$, $q\le 0$, and $f\in L^2$, minimizing $J$ on $\psi +H_0^1$ solves $\mathrm{div}(A\nabla u)+qu=f$. The nonnegative quadratic terms are weakly lower semicontinuous, and Poincaré inequality makes the gradient norm coercive on the zero-boundary part. The first variation gives the weak equation with the forcing sign displayed above. If $q$ is positive, coercivity may fail at a Dirichlet eigenvalue; orthogonality to the homogeneous kernel is then needed for solvability.