Portal de Periódicos da CAPES

ON A CLASS OF NONLINEAR SCHRÖDINGER EQUATIONS ON FINITE GRAPHS

2020; Cambridge University Press; Volume: 101; Issue: 3 Linguagem: Inglês

10.1017/s0004972720000143

1755-1633

Shoudong Man,

Spectral Theory in Mathematical Physics

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.