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arXiv · 2403.15866

Existence and multiplicity of solutions for the logarithmic Schr\"{o}dinger equation with a potential on lattice graphs

Zhentao He·Chao Ji
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Abstract

In this paper, we consider the existence and multiplicity of solutions for the logarithmic Schr\"{o}dinger equation on lattice graphs $\mathbb{Z}^N$ $$ -\Delta u+V(x) u=u \log u^2, \quad x \in \mathbb{Z}^N, $$ When the potential $V$ is coercive, we obtain infinitely many solutions by adapting some arguments of the Fountain theorem. In the cases of periodic potential, asymptotically periodic potential and bounded potential, we first investigate the existence of ground state solutions via the variation methods, and then we generalize these results from $\mathbb{Z}^N$ to quasi-transitive graphs. Finally, we extend the main results of the paper to the $p$-Laplacian equation with the logarithmic nonlinearity.

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Zhentao He, Chao Ji. 2024-03-23. Existence and multiplicity of solutions for the logarithmic Schr\"{o}dinger equation with a potential on lattice graphs. https://arxiv.org/abs/2403.15866

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