Existence of least energy sign-changing solution for the nonlinear Schrödinger system with two types of nonlocal terms

Abstract

In the paper, we are concerned with the system of Kirchhoff-Schrödinger-Poisson system under certain assumptions on V 1 , V 2 , K and f . We are interested in the existence of least energy sign-changing solutions to the system on R N . Because two kinds of nonlocal terms φ u and R N |∇u| 2 are involved in the system, the methods are different from the Kirchhoff or the Schrödinger-Poisson system. The two nonlocal terms R N |∇u| 2 and φ u make that the functional J(uthe nonlocal term φ u does not have the convergence property because of the assumption V 2 . In addition, the convergence of these two nonlocal terms are different. In the present paper, we unify the increasing property conditions on sign-changing solution in previous papers. We construct a new homotopy operator and then weaken the assumption that f is C 1 to that of f being only continuous. We prove that the system has a sign-changing solution via a constraint variational method combining with Brouwer's degree theory. MSC: 47J30; 34B15

Infinitely Many High Energy Solutions for Kirchhoff-Schrödinger-Poisson Equation with 4-Superlinear Growth Condition

“…For more generalized form of problem (1.2), one can find the corresponding results in recent papers [9,10]. Motivated mainly by [11,14,15,17,21], the purpose of the present paper is to investigate the existence of infinitely many high energy solutions of problem (1.1). As far as we know, this is the first result concerned with the existence of infinitely many high energy solutions for Kirchhoff-Schrödinger-Poisson type equation.…”

“…Inspired by the works mentioned above, many authors recently considered problem (1.2), see for example [11,14,15,17,21] and the references mentioned therein. Different from (1.3) and (1.5), problem (1.2) has two nonlocal terms, which implies that problem (1.2) is no longer a point-wise identity and brings some additional difficulties.…”