Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

Abstract

We consider the nonlinear Schrödinger equation −∆u + (λa(x) + 1)u = |u| p−1 u on a locally finite graph G = (V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution u λ . Moreover, as λ → ∞, the solution u λ converges to a solution of the Dirichlet problem −∆u + u = |u| p−1 u which is defined on the potential well Ω. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.

Existence and convergence of solutions for nonlinear biharmonic equations on graphs

“…Next, we turn to formulas of integration by parts on graphs, which are fundamental when we use methods in calculus of variations. The proofs of the next two lemmas can be found in [31] and we omit them here.…”

“…Moreover, we also study the asymptotic behavior of solutions to our equation and find that their limit is restricted on the potential well which is a finite graph. On graphs, this kind of results was first proved by Zhang and Zhao. In [31], they studied the following second order equation, − ∆u + (λa + 1)u = |u| p−1 u (1) on a locally finite graph G = (V, E), where a(x) is a potential function defined on V . If the potential well Ω = {x ∈ V : a(x) = 0} is a non-empty, connected and bounded domain in V , their results said that, as λ → +∞, the ground state solutions u λ of (1) converge to a ground state solution u 0 of the corresponding Dirichlet equation,…”

“…Noticing the above works, the main purpose of this paper is to investigate whether the asymptotic result in [31] still holds on graphs for nonlinear biharmonic equations. To describe our problems and results, we first introduce some concepts and assumptions.…”

Existence and convergence of solutions for nonlinear elliptic systems on graphs

“…Grigoryan, Lin and Yang [14,15,16] studied several nonlinear elliptic equations on graphs and they pointed out that the required Sobolev spaces on a finite graph is pre-compact, which makes it possible to use the variational method to obtain the existence of solutions. The second author cooperates with others [31,17] by using the Nehari manifold to prove that on a locally finite graph, the nonlinear Schrödinger equation has a nontrivial ground state solution under suitable conditions, and the limit of the solution is limited to a potential well.…”

“…Motivated by [17,31], we aim to study the existence of a nontrivial ground state solution of (1) with a fixed positive parameter λ. Here a ground state solution means it has the least energy among all nontrivial solutions.…”