Logarithmically Coupled \(p\)-Laplacian Systems: From Discrete Ground States to Rigidity and Critical Continuous Limits

Abstract

This paper introduces a new coupled \(p\)-Laplacian system with logarithmic nonlinearities, on locally finite graphs and, in the critical case \(p=N>4\), on \(\mathbb R^N\) via a regularised formulation. The logarithmic coupling renders the energy functional ill-defined on the natural Sobolev space. To address this non-separable singularity, we develop an exponent calibration technique that converts the logarithmic terms into strictly lower-order power estimates. This technique underpins the existence proofs in two distinct settings-and is essential in the continuous critical setting, where the failure of the \(L^\infty\)-embedding renders classical methods inapplicable. The same technique is also used to recover compactness, thereby completing the passage from the regularised problem to the original ill-posed equation.In the discrete setting, we further develop a rigidity theory for ground states, which yields an explicit Hessian factorisation dictated by the Nehari constraint. The compactness restoration and limit theorem, the convergence rates, and the existence results in both settings together form a discrete-continuous double loop, from ill-posedness through compactness and rigidity to asymptotics, providing a unified variational framework for logarithmically coupled systems.