Refined boundary layer asymptotics for elliptic equations with multiplicative nonlocal effects

We investigate singularly perturbed elliptic problems with multiplicative nonlocal diffusion terms subject to Robin boundary conditions. The diffusion depends on a global quantity of the solution, which introduces a nonlocal coupling between the global behavior of the solution and the boundary asymptotics. As the perturbation parameter tends to zero, we establish precise asymptotic expansions of the solutions that capture the structure of boundary layers coupled with the multiplicative nonlocal diffusion effect. Moreover, the interaction between the nonlocal diffusion and the boundary geometry manifests as refined higher-order terms wherein geometric quantities, such as the mean curvature, appear explicitly; our analysis thus quantifies the influence of global coupling on the boundary layer structure, extending classical singular perturbation theory to multiplicative nonlocal frameworks.