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arXiv · 2203.05146

A class of semilinear elliptic equations on lattice graphs

B. Hua·R. Li·L. Wang
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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.

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B. Hua, R. Li, L. Wang. 2022-03-10. A class of semilinear elliptic equations on lattice graphs. https://arxiv.org/abs/2203.05146

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