Existence and convergence of solutions for nonlinear biharmonic equations on graphs
arXiv:1908.03993
The paper establishes Sobolev space tools on locally finite graphs and uses variational methods to prove existence of ground state solutions for a nonlinear biharmonic equation on the graph, then shows these solutions converge to a limit problem as a parameter tends to infinity.
Abstract
In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation $$ Î^{2} u -Îu+(λa+1)u= |u|^{p-2}u $$ on $G=(V,E)$. Under some suitable assumptions, we prove that for any $λ>1$ and $p>2$, the equation admits a ground state solution $u_λ$. Moreover, we prove that as $λ\rightarrow +\infty$, the solutions $u_λ$ converge to a solution of the equation \begin{align*} \begin{cases} Î^{2}u -Îu+u = |u|^{p-2}u, &\text{in}\ \ Ω, u=0, &\text{on}\ \ \partialΩ, \end{cases} \end{align*} where $Ω=\{x\in V: a(x)=0\}$ is the potential well and $\partialΩ$ denotes the the boundary of $Ω$.
21 pages