Existence and convergence of solutions for nonlinear biharmonic equations on graphs

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 $${\Delta }^{2}u-\Delta u+(\lambda a+1)u=|u{|}^{p-2}u$$ on $G=(V,E)$. Under some suitable assumptions, we prove that for any $\lambda >1$ and $p>2$, the equation admits a ground state solution $u_{\lambda }$. Moreover, we prove that as $\lambda \to +\infty$, the solutions $u_{\lambda }$ converge to a solution of the equation \begin{align*} \begin{cases} Δ^{2}u -Δu+u = |u|^{p-2}u, &\text{in}\ \ Ω, u=0, &\text{on}\ \ \partialΩ, \end{cases} \end{align*} where $\Omega =\{x\in V:a(x)=0\}$ is the potential well and $\partial \Omega$ denotes the the boundary of $\Omega$.

21 pages