Coupled critical growth for anisotropic systems

Introduction: Differential inclusions arising from nonsmooth energy functionals provide a natural setting for variational problems where the governing operators lack differentiability or strict monotonicity. While the isotropic case with variable exponents and the scalar anisotropic case have each been studied separately, genuinely anisotropic systems, in which two inclusions are coupled and each is driven by its own directional growth, have remained largely unexplored. This work proposes a unified variational framework that addresses both obstacles at once and establishes the existence of generalized solutions.

Methods: Working in anisotropic variable-exponent Sobolev spaces, we overcome the lack of monotonicity for the above operators using a discretized Galerkin scheme combined with Clarke's nonsmooth analysis. The growth hypothesis on F is structured in coupled form: the cross-exponents linking the two growth components are built through the conjugates of the proper exponents in the inclusions, so that the generalized Young inequality redistributes each coupling term exactly onto the dominant term of the coupled one.

Results: The central contribution is the treatment of a genuine anisotropic system: two coupled inclusions, each governed by its own directional anisotropy. Each component lives in a distinct anisotropic Sobolev space, with its own critical threshold set by the harmonic mean of its directional exponents. Under an explicit quantitative threshold on the embedding constants, we prove that, in the context of finite-dimensional spaces, the energy functional is coercive and admits a bounded sequence of Galerkin minimizers. Thereby, by passing to the limit, we reach a generalized solution of the system in the competing regime where both ellipticity and monotonicity assumptions are relaxed.

Conclusions: The proposed framework unifies and extends both isotropic problems and scalar anisotropic problems in previous works, hence improving the features of the model problem.