Semi-classical states for elliptic system near saddle points of potentials

Abstract

In this paper, we study the following Hamiltonian elliptic system −ε2Δu+u=Hv(x,u,v) in RN,−ε2Δv+v=Hu(x,u,v) in RN,u(x)→0, v(x)→0 as |x|→∞, where ɛ > 0 is a small parameter, H is a super-quadratic sub-critical Hamiltonian. Our investigation focuses on the cases that and , where K(x) possesses saddle points. By introducing a new penalization associated with the barycenters of functions and applying the Leray–Schauder degree theory, we adopt the local variational arguments in Byeon and Jeanjean (2007 Arch. Ration. Mech. Anal. 185 185–200); Byeon and Tanaka (2013 J. Eur. Math. Soc. 15 1859–99; 2014 Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations (Providence, RI: American Mathematical Society)) and obtain the existence of semi-classical states for the Hamiltonian elliptic system, which concentrate around the saddle points of V as ɛ → 0.