Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

Abstract

In this paper, we study the following fractional Schrödinger‐Poisson system involving competing potential functions ϵ2sfalse(−normalΔfalse)su+Vfalse(xfalse)u+φu=Kfalse(xfalse)ffalse(ufalse)+Qfalse(xfalse)false|u|2s∗−2u,in0.1emR3,ϵ2tfalse(−normalΔfalse)tφ=u2,in0.1emR3, where ϵ > 0 is a small parameter, f is a function of C1 class, superlinear and subcritical nonlinearity, 2s∗=63−2s, s>34, t ∈ (0,1), V(x), K(x), and Q(x) are positive continuous functions. Under some suitable assumptions on V, K, and Q, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small ϵ > 0, of which it is concentrating on the set of minimal points of V(x) and the sets of maximal points of K(x) and Q(x).

Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent

“…To complete the proof, we only need to prove the decay properties of u ε . Similar argument to the proof of Lemma 5.6 in [46], we can obtain that…”

“…which is a contradiction. Thus, up to a subsequence, using Brezis-Lieb Lemma, we conclude that v n → v in H s [46], we have that u ∈ C 2,α (R 3 ) for some α ∈ (0, 1). The remain proof is to show u is positive.…”

“…Another main reason is that the fractional Laplacian (−∆) s (s ∈ (0, 1)) is a non-local operator comparing with the classical Laplacian −∆ which is a local one, the methods which were previously developed, maybe not be applied directly. We refer the interesting readers to see the recent progresses such as [9,13,15,17,20,19,32,33,34,41,39,42,44,45,46,48] and the references therein.…”

“…Using the similar methods, the authors in [30] studied the system (1.1) and established the multiplicity and concentration behavior of solutions. In [46], we also studied the concentration of positive ground state solution via the Nehari manifold for the following system…”

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

“…However similar results on the fractional Schrödinger-Poisson systems are not as rich as the Schrödinger-Poisson system (1.2), especially there are very few results on the existence and concentration results with steep potential well. Very recently, K. Teng and R. Agarwal [41] considered the semiclassic case for the following fractional Schrödinger-Poisson system…”