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 equationUnder 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 → +∞, the solutions u λ converge to a solution of the equationwhere Ω = {x ∈ V : a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.

Existence and convergence of solutions for nonlinear elliptic systems on graphs

“…To define the weak solution, we need formulas of integration by parts on graphs, which are also fundamental when we use methods from calculus of variations. The proofs of the next two lemmas can be found in [17] and we omit them here.…”

“…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.…”

“…The set of functions with compact support is C c (V ) := {u : V → R | {x ∈ V : u (x) = 0} is of finite cardinality}. In [17], they have proved that W 1,2 (V ) under the norm…”

Superlinear elliptic inequalities on weighted graphs

“…Besides the above nonexistence result (or called Liouville theorems) for elliptic equation on graphs, a lot of attention has been paid to different types of elliptic equations on graphs, see [1,3,4,7,8,9,12,13,14,19]. there are also many literature devoted to the parabolic equations on graphs, see [16,17,18].…”