Existence and convergence of solutions for nonlinear biharmonic equations on graphs

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 Lambda(2)u - Lambda u + (lambda a + 1)u = vertical bar u vertical bar(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 -> +infinity, the solutions u(lambda) converge to a solution of the equation {Delta(2)u - Delta u + u = vertical bar u vertical bar(p-2)u, in Omega, u=0, on partial derivative Omega, where Omega ={x is an element of V: a(x) = 0} is the potential well and partial derivative Omega denotes theboundary of Omega. (C) 2019 Elsevier Inc. All rights reserved.