Ground state solutions for planar coupled system involving nonlinear Schrödinger equations with critical exponential growth

Abstract

We consider the following two coupled nonlinear Schrödinger system: −normalΔu+u=f1false(x,ufalse)+λfalse(xfalse)v,x∈ℝ2,−normalΔv+v=f2false(x,vfalse)+λfalse(xfalse)u,x∈ℝ2, where the coupling parameter satisfies 0 < λ(x) ≤ λ0 < 1 and the reactions f1, f2 have critical exponential growth in the sense of Trudinger–Moser inequality. Using non‐Nehari manifold method together with the Lions's concentration compactness and the Trudinger‐Moser inequality, we show that the above system has a Nehari‐type ground state solution and a nontrivial solution. Our results improve and extend the previous results.

The existence and non-existence of ground state solutions to Schrödinger systems with general potentials

“…In addition, there are many results about semiclassical state solutions, sign-changing solutions, normalized solutions and ground state solutions of elliptic systems, see for example, [1,7,8,10,11,15,20,[22][23][24][25][26][27][28][29] and the references therein.…”