Ground state solutions for asymptotically periodic fractional Schrödinger-Poisson problems with asymptotically cubic or super-cubic nonlinearities

Abstract

In this paper, we consider the following fractional Schrödinger–Poisson problem: (−△)su+V(x)u+K(x)ϕ(x)u=f(x,u),1emx∈double-struckR3,(−△)tϕ=K(x)u2,1emx∈double-struckR3, where s,t∈(0,1],4s+2t>3,V(x),K(x), and f(x,u) are periodic or asymptotically periodic in x. We use the non‐Nehari manifold approach to establish the existence of the Nehari‐type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions lim|τ|→∞∫0τf(x,ξ)normaldξ|τ|σ=∞ uniformly in x∈double-struckR3 with σ:=max{3,4−2t} and f(x,τ)τ3−f(x,kτ)(kτ)3sign(1−k)+θ0V(x)|1−k2|(kτ)2⩾0,∀x∈R3,k>0,τ≠0 with constant θ0∈(0,1), instead of lim|τ|→∞∫0τf(x,ξ)normaldξ|τ|4=∞ uniformly in x∈double-struckR3 and the usual Nehari‐type monotonic condition on f(x,τ)/|τ|3. Our results unify both asymptotically cubic or super‐cubic nonlinearities, which are new even for s=t=1. Copyright © 2017 John Wiley & Sons, Ltd.

Solutions of Perturbed Fractional Schrödinger–Poisson System with Critical Nonlinearity in $\mathbb{R}^{3}$

“…Under certain assumptions on the nonlinearity, a nontrivial nonnegative solution is obtained by perturbation method. In [8], using the Non-Nehari manifold approach, the authors established the existence of the Nehari-type ground state solutions for fractional Schrödinger-Poisson system (1.2). By introducing some new tricks, in [9], Chen and Tang proved that the single problem admits a ground state solution of Pohoźaev type and a least energy solution.…”

Existence of solutions for asymptotically periodic fractional Schrödinger equation with critical growth

“…In the last few decades, there are many of works concerned with asymptotically periodic classical Schrödinger equation, please see previous studies 17‐21 and the references therein. Very recently, some authors investigated the asymptotically periodic fractional Schrödinger equation, and they obtained the existence of solutions by variational methods, see previous studies 13‐16,22 and the references therein. For example, in Zhang et al, 14 the authors studied the following fractional Schrödinger equation …”

Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in RN$\mathbb{R}^{N}$

“…Zhang, Zhang and Mi [26] established the existence of solutions for (1.4) in periodic case and asymptotically periodic case via variational methods. We refer the reader to [27][28][29][30][31][32] and the references therein. Although the fractional Schrödinger equations have been widely studied, to the best of our knowledge, there are few papers concerning on the fractional Kirchhoff-type problems like (1.1) in the literature.…”