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ON A CLASS OF NONLINEAR SCHRÖDINGER EQUATIONS ON FINITE GRAPHS

Part of: Miscellaneous topics - Partial differential equations Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  20 February 2020

SHOUDONG MAN Open the ORCID record for SHOUDONG MAN [Opens in a new window]
SHOUDONG MAN*
Affiliation:
Department of Mathematics,Tianjin University of Finance and Economics, Tianjin300222, PR China email [email protected], [email protected]
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Abstract

Suppose that $G=(V,E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\unicode[STIX]{x1D6E5}$ be the usual graph Laplacian. Consider the nonlinear Schrödinger equation of the form

$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u-\unicode[STIX]{x1D6FC}u=f(x,u),\quad u\in W^{1,2}(V),\end{eqnarray}$$
on the graph $G$, where $f(x,u):V\times \mathbb{R}\rightarrow \mathbb{R}$ is a nonlinear real-valued function and $\unicode[STIX]{x1D6FC}$ is a parameter. We prove an integral inequality on $G$ under the assumption that $G$ satisfies the curvature-dimension type inequality $CD(m,\unicode[STIX]{x1D709})$. Then by using the Poincaré–Sobolev inequality, the Trudinger–Moser inequality and the integral inequality on $G$, we prove that there is a nontrivial solution to the nonlinear Schrödinger equation if $\unicode[STIX]{x1D6FC}<2\unicode[STIX]{x1D706}_{1}^{2}/m(\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D709})$, where $\unicode[STIX]{x1D706}_{1}$ is the first positive eigenvalue of the graph Laplacian.

Keywords

Laplacian on graphscurvature-dimension type inequalitySchrödinger equation on finite graphs

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35A15: Variational methods 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 3 , June 2020 , pp. 477 - 487
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by the National Natural Science Foundation of China (Grant No. 11601368).

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