Coercive Neumann energy with a positive reaction term ($B[u,u]\ge \min (\nu ,\mu )\Vert u{\Vert }_{H^1}^2$)

A symmetric uniformly elliptic divergence-form energy with zeroth-order coefficient at least $\mu >0$ controls the full $H^1$ norm, including constants. Its quadratic energy minus a bounded load has a unique minimizer by the direct method, and vanishing first variation gives the natural Neumann weak equation. Unlike a pure Neumann Laplacian, this problem needs no zero-mean compatibility condition. The exact energy difference from the weak solution is half the bilinear energy of the difference.