Ground state solutions to some Indefinite Nonlinear Schr\"{o}dinger equations on lattice graphs

arXiv2302.14608MaRDI QIDQ6427912

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Publication date: 28 February 2023

Abstract: In this paper, we consider the Schr"odinger type equation $-Deltau+V(x)u=f(x,u)$ on the lattice graph $mathbb{Z}^N$ with indefinite variational functional, where $-Delta$ is the discrete Laplacian. Specifically, we assume that $V(x)$ and $f(x,u)$ are periodic in $x$, $f$ satisfies some growth condition and 0 lies in a spectral gap of $(-Delta+V)$. We obtain ground state solutions by using the method of generalized Nehari manifold which has been introduced in arXiv:1801.06872.

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