A class of semilinear elliptic equations on lattice graphs

TL;DR

This paper investigates the existence of positive solutions for a class of semilinear elliptic equations on lattice graphs, employing variational methods and concentration compactness to handle variable coefficients that tend to constants at infinity.

Contribution

It extends the analysis of semilinear elliptic equations to lattice graphs with variable coefficients, proving existence results using advanced variational techniques.

Findings

Existence of positive solutions for equations with constant coefficients.

Decomposition of bounded Palais-Smale sequences with variable coefficients.

Application of concentration compactness principle on lattice graphs.

Abstract

In this paper, we study the semilinear elliptic equation of the form \begin{eqnarray*} -\Delta u+a(x)|u|^{p-2}u-b(x)|u|^{q-2}u=0 \end{eqnarray*} on lattice graphs $\mathbb{Z}^{N}$, where $N\geq 2$ and $2\leq p<q<+\infty$. By the Br\'{e}zis-Lieb lemma and concentration compactness principle, we prove the existence of positive solutions to the above equation with constant coefficients $\bar{a},\bar{b}$ and the decomposition of bounded Palais-Smale sequences for the functional with variable coefficients, which tend to some constants $\bar{a},\bar{b}$ at infinity, respectively.