Obstacle Problems for Elliptic Operators with Solution-Dependent Shifts: Existence and Uniqueness via a Three-Term Decomposition

We prove the existence and uniqueness of weak solutions to an obstacle problem for a nonlinear elliptic operator in divergence form. The variational inequality under consideration involves an integral over the domain of the Frobenius inner product of the operator S(z,∇u−O(u)) with the gradient difference ∇(v−u), plus the Euclidean inner product of u and v−u, which is required to be nonnegative for all admissible functions v. The admissible set consists of functions in the Sobolev space W1,2(Ω;Rm) with prescribed Dirichlet boundary trace and lying above a given obstacle ψ almost everywhere. The obstacle condition v≥ψ a.e. models a lower bound constraint (e.g., a membrane or a displacement limit) that the admissible functions must respect, while the boundary value δ prescribes the Dirichlet data. The principal part contains a solution-dependent shift O(u), which is Lipschitz continuous, while S is assumed to be globally Lipschitz and strongly monotone with respect to equal shifts, with quadratic growth and coercivity. This structural framework can be interpreted in terms of symmetry: the strong monotonicity condition expresses a quantitative symmetry property of S with respect to equal shifts, and the shift O(u) introduces a symmetry-breaking coupling. The smallness condition ensures that this asymmetry remains under control. However, we do not pursue a full group-invariance or Lie-symmetry analysis; the symmetry perspective is used here as a heuristic and interpretative tool. The main difficulty lies in the mismatch of shifts when comparing two admissible functions. This is resolved by a three-term decomposition of the monotonicity estimate, combined with Young’s inequality and Poincaré’s inequality, under the smallness condition that the product of the Lipschitz constant of S, the Lipschitz constant of O, and the Poincaré constant is bounded above by one quarter of the strong monotonicity modulus. Existence follows from the Kinderlehrer–Stampacchia theorem; uniqueness is obtained from the same decomposition. The result unifies and extends previous contributions that treated either the lower-order term or the shift coupling separately, and it does so within a unified quadratic framework that avoids the technical overhead of variable exponents and Young measures.