Ground state solutions for logarithmic p-Laplacian systems on locally finite graphs

Abstract

In this paper,we study the discrete logarithmic p-Laplacian system $$ \begin{cases} -Δ_p u + a(x)|u|^{p-2}u=\frac{p-2}{p}|u|^{p-4}u v^2\log v^2 +\frac{2}{p}|v|^{p-2}u\log u^2+\frac{2}{p} |v|^{p-2}u,\qquad in\quad V,\\ -Δ_p v + b(x)|v|^{p-2}v=\frac{p-2}{p}|v|^{p-4}v u^2\log u^2 +\frac{2}{p}|u|^{p-2}v\log v^2+\frac{2}{p} |u|^{p-2}v,\qquad in\quad V, \end{cases} $$ on locally finite graphs $G=(V,E)$, where $Δ_p u(x)=div(|\nabla u(x)|^{p-2}\nabla u(x))$ is the discrete p-Laplacian on graphs. Firstly, under certain assumptions on the potential $a(x)$ and $b(x)$, we establish two Sobolev compact embedding theorems in the case when different assumptions on $a(x)$ and $b(x)$, which leads to two different energy functionals, the one is not well-defined, while the other one is $C^1$ continuous. In the former case, we prove that the system admits a ground state solution by Nehari manifold method. In the latter case, we prove that the system admits a mountain-pass solution. Finally, we establish convergence results by analyzing the concentration behavior of ground state solutions.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Lemma 4
• Lemma 5
• proof
• Remark 2
• Definition 1
• Proposition 1