[Paper Review] Existence and convergence of solutions for nonlinear biharmonic equations on graphs

Xiaoli Han, Mengqiu Shao+1 more|arXiv (Cornell University)|Aug 12, 2019

Nonlinear Partial Differential Equations29 references4 citations

TL;DR

This paper establishes the existence of ground state solutions for a nonlinear biharmonic equation on locally finite graphs using variational methods in discrete Sobolev spaces. It proves that as the parameter λ → ∞, solutions converge strongly to a ground state solution on the potential well Ω = {x ∈ V : a(x) = 0}, which solves a limiting Dirichlet problem on a finite subgraph.

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 $$ Δ^{2} u -Δu+(λa+1)u= |u|^{p-2}u $$ 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 $λ ightarrow +\infty$, the solutions $u_λ$ converge to a solution of the equation \begin{align*} \begin{cases} Δ^{2}u -Δu+u = |u|^{p-2}u, & ext{in}\ \ Ω, u=0, & ext{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 $Ω$.

Motivation & Objective

• To establish foundational properties of Sobolev spaces on locally finite graphs for variational analysis of PDEs.
• To prove existence of ground state solutions for a nonlinear biharmonic equation with a potential term on graphs.
• To analyze the asymptotic behavior of solutions as the coupling parameter λ → ∞.
• To show that solutions concentrate on the potential well Ω = {x ∈ V : a(x) = 0} and converge to a solution of a limiting Dirichlet problem on Ω.
• To extend known results from continuous settings (e.g., R^N) to discrete graphs, particularly for fourth-order equations.

Proposed method

• Define a variational framework on graphs using discrete Laplacians and Sobolev-type norms involving Δu and ∇u.
• Introduce the energy functional J_λ(u) = ½∫(|Δu|² + |∇u|² + (λa+1)|u|²)dμ − 1/p∫|u|ᵖdμ for λ > 1 and p > 2.
• Use the Nehari manifold method to find critical points of J_λ corresponding to ground state solutions.
• Prove weak convergence of u_λ in W²²(V) and strong convergence in W²²(Ω) as λ → ∞.
• Apply compactness arguments and energy estimates to show that limit u₀ satisfies the limiting equation on Ω.
• Use the concentration-compactness principle adapted to graphs to analyze the limit behavior and prove convergence in the energy norm.

Experimental results

Research questions

• RQ1 Under what conditions does a nonlinear biharmonic equation on a locally finite graph admit a ground state solution?
• RQ2 How do solutions behave as the parameter λ → ∞ in the equation Δ²u − Δu + (λa + 1)u = |u|ᵖ⁻²u?
• RQ3 Does the solution sequence u_λ converge to a solution of a limiting equation on the potential well Ω = {x ∈ V : a(x) = 0}?
• RQ4 Can the asymptotic concentration of solutions on the potential well be rigorously established on discrete graphs?
• RQ5 Is the variational framework for Sobolev spaces on graphs sufficient to prove existence and convergence results analogous to those in continuous settings?

Key findings

• For any λ > 1 and p > 2, the equation Δ²u − Δu + (λa + 1)u = |u|ᵖ⁻²u admits a ground state solution u_λ on the graph G.
• As λ → ∞, the solutions u_λ converge strongly in W²²(Ω) to a function u₀ that solves the limiting Dirichlet problem: Δ²u − Δu + u = |u|ᵖ⁻²u in Ω, u = 0 on ∂Ω.
• The limit solution u₀ is a ground state solution of the limiting equation on the bounded, connected, and finite subgraph Ω = {x ∈ V : a(x) = 0}.
• The convergence is strong in the energy norm: limₖ→∞‖u_λₖ − u₀‖_{E_λₖ} = 0, with u₀ ∈ W²²(Ω).
• The energy of the solutions J_λ(u_λ) converges to the ground state energy m_Ω of the limiting problem on Ω.
• The potential well Ω is a finite, bounded, and connected subgraph, ensuring compactness and enabling the convergence result.