Constrained variational problems on perturbed lattice graphs

In this paper, we solve some constrained variational problems on perturbed lattice graphs $G$. The first problem addresses the existence of ground state normalized solutions to Schr\"odinger equations \begin{equation*} \left\{ \begin{aligned} &-\Delta_{G} u+\lambda u=\vert u\vert^{p-2}u,x\in G &\Vert u\Vert_{l^2(G)}^2=a. \end{aligned} \right. \end{equation*} We prove that if the graph is obtained by deleting finite edges in lattice graphs while maintaining connectivity, then there exists a threshold $\alpha_G\in[0,\infty)$ such that there do not exist ground state normalized solution if $0<a<\alpha_G$, and there exists a ground state normalized solution if $a>\alpha_G.$ If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exist $E^{'}$ and $a_1$ such that for all $a>a_1,$ there do not exist ground state normalized solutions. The second problem concerns the existence of an extremal function for the Sobolev inequality. If the graph $G$ is obtained by deleting finite edges in lattice graphs while maintaining connectivity, for the Sobolev super-critical regime, we prove that there exists an extremal function. for the Sobolev critical regime, we prove that there exists $G$ such that extremal can be attained. If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exists $E^{'}$ such that there does not exist an extremal function.