Generalized solutions via Galerkin approximation for Neumann competing anisotropic operators

This paper studies the existence of generalized and strong generalized solutions for a nonlinear elliptic Neumann problem driven by the difference of two anisotropic p- and q-Laplacian operators. The main difficulty arises from a convection term depending on the gradient of the solution, which renders the problem non-variational. By combining a Galerkin approximation scheme with Brouwer's fixed point theorem, we overcome this obstruction and obtain existence results for both competing and cooperative configurations. In the cooperative case, we further prove that generalized solutions coincide with weak solutions under suitable growth assumptions. The analysis relies on compact embeddings in anisotropic Sobolev spaces and the S+-property of the operator.