Existence and convergence of solutions for nonlinear elliptic systems on graphs

Abstract

We consider a kind of nonlinear systems on a locally finite graphs G = (V, E). We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solution which depends on the parameter λ with some suitable assumptions on the potentials. Moreover, we pay attention to the concentration behavior of these solutions and prove that, as λ → ∞, these solutions converge to a ground state solution of a corresponding Dirichlet problem. Finally, we also provide some numerical experiments to illustrate our results.

Extremal functions for the second-order Sobolev inequality on groups of polynomial growth

“…They have certain physical backgrounds, such as traveling waves in a suspension bridge [21] and the static deflection of an elastic plate [1], and have been well studied in continuous setting, see for example [43,13,74,64,66,65,12,59,52,51,68,39,48,25,19,73,75,31,41,54,53,57,29,36,37,38] and references therein. For discrete setting, there are few results of the fourth order nonlinear equations [32,79].…”