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.

中文翻译：

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关于有限图上的一类非线性薛定谔方程

假设$G=(V,E)$是一个有顶点集的有限图$V$和边集$E$. 让$\unicode[STIX]{x1D6E5}$是通常的图拉普拉斯算子。考虑以下形式的非线性薛定谔方程$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u-\unicode[STIX]{x1D6FC}u=f(x,u),\quad u\in W^{1,2}(V) ,\end{eqnarray}$$在图表上$G$， 在哪里$f(x,u):V\times \mathbb{R}\rightarrow \mathbb{R}$是一个非线性实值函数，并且$\unicode[STIX]{x1D6FC}$是一个参数。我们证明了一个积分不等式$G$在假设$G$满足曲率维数型不等式$CD(m,\unicode[STIX]{x1D709})$. 然后通过使用 Poincaré-Sobolev 不等式、Trudinger-Moser 不等式和积分不等式$G$，我们证明非线性薛定谔方程有一个非平凡解，如果$\unicode[STIX]{x1D6FC}<2\unicode[STIX]{x1D706}_{1}^{2}/m(\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D709 })$， 在哪里$\unicode[STIX]{x1D706}_{1}$是图拉普拉斯算子的第一个正特征值。