Ground state solutions for p-Laplacian system with logarithmic coupling terms on locally finite graphs

TL;DR

This paper investigates the existence and behavior of ground state solutions for a class of discrete p-Laplacian systems with logarithmic coupling on locally finite graphs, addressing challenges posed by non-separable singularities.

Contribution

It introduces an original exponent calibration technique to handle logarithmic singularities and establishes existence results using variational methods.

Findings

Existence of ground states under two different hypotheses.

Development of an exponent calibration method for singularities.

Analysis of concentration behavior of solutions.

Abstract

In this paper, we first study a class of discrete $p$-Laplacian systems with logarithmic coupling on locally finite graphs. The system is specifically designed to capture the variational interplay between nonlinear diffusion and logarithmic saturation, and takes the form $$ \begin{cases} -\Delta_p u + a(x)|u|^{p-2}u=\frac{p-2}{p}|u|^{p-4}u v^2\log v^2 +\frac{2}{p}|v|^{p-2}u\log u^2+\frac{2}{p} |v|^{p-2}u,\qquad in\quad V, -\Delta_p v + b(x)|v|^{p-2}v=\frac{p-2}{p}|v|^{p-4}v u^2\log u^2 +\frac{2}{p}|u|^{p-2}v\log v^2+\frac{2}{p} |u|^{p-2}v,\qquad in\quad V, \end{cases} $$ on locally finite graphs $G=(V,E)$ with $p>4$. The logarithmic coupling terms would possibly render the energy functional not well-defined on the natural Sobolev space--a fundamental obstacle that does not rise in scalar equations. We establish existence of ground states under two distinct hypotheses: via the Nehari manifold when the functional lacks regularity, and via the mountain pass theorem otherwise. The core novelty is an original exponent calibration technique, specifically devised to resolve the non-separable logarithmic singularities. Finally, we establish convergence results by analyzing the concentration behavior of the ground state solutions.