Logarithmically Coupled p-Laplacian Systems: A Complete Variational Theory from Compactness to Rigidity

Abstract

This paper forges a new coupled \(p\)-Laplacian system with logarithmic nonlinearities on locally finite graphs and in the critical case \(p=N\), on \(\mathbb R^N\) with a regularised version. The logarithmic coupling renders the energy functional ill-defined on the natural Sobolev space---an obstruction absent in scalar equations. To overcome this non-separable singularity, we develop an \textbf{exponent calibration technique} that converts the logarithmic terms into strictly lower-order power estimates. This method drives existence proofs on two different settings via the Nehari manifold and the mountain pass theorem respectively, and is irreplaceable in the continuous critical setting, where failure of the \(L^\infty\) embedding defeats classical approaches. A second irreplaceable instance occurs in the Palais--Smale decomposition for the regularised problem in \(\mathbb{R}^N\) with \(p=N\), where it is the sole mechanism closing estimates after compactness is lost. Beyond compactness and asymptotics, for the first time in the discrete setting, we uncover an intrinsic rigidity at ground-state level: the Sobolev norm and logarithmic interaction energy compensate exactly, yielding an explicit gradient formula for the ground-state energy. Moreover, the Hessian of the ground-state energy with respect to the potential admits a clean factorisation dictated by the Nehari constraint. Together with existence results in both settings, the Palais--Smale decomposition and quantitative convergence rates close a complete circle from ill-posedness through compactness and rigidity to asymptotics, founding the variational theory for logarithmic coupling systems.