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Ground state solutions for the nonlinear fractional Schrodinger-Poisson system
TL;DR: In this article, the existence of ground state solutions for the nonlinear fractional Dinger-Poisson system was studied and a nontrivial ground state solution was established through using a monotonicity trick and global compactness Lemma.
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Abstract: In this paper, we study the existence of ground state solutions for the nonlinear fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^su+V(x)u+\phi u=|u|^{p-1}u, & \hbox{in $\mathbb{R}^3$,} (-\Delta)^s\phi=u^2,& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $2<p<2_s^{\ast}-1 = \frac{3+2s}{3-2s}$, $s\in(\frac{3}{4},1)$. Under certain assumptions on $V$, a nontrivial ground state solution $(u,\phi)$ is established through using a monotonicity trick and global compactness Lemma. As its supplementary results, we prove some nonexistence results in the case of $1<p\leq 2$ and $p=2_s^{\ast}-1$.
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Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent
TL;DR: In this paper, the existence of ground state solutions for the nonlinear fractional Schrodinger-Poisson system with critical Sobolev exponent was studied and a nontrivial ground state solution was proved under certain assumptions on V ( x ), using the method of Pohozaev-Nehari manifold and the arguments of Brezis-Nirenberg, the monotonic trick and global compactness Lemma.
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Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal L2-critical or L2-supercritical perturbation
TL;DR: In this article, the authors studied the existence and asymptotic properties of solutions to the fractional Schrodinger equation under the normalized constraint ∫RNu2=a2, where N ≥ 2, σ ∈ (0, 1), α ∈(0, N), q∈(2+4σN,2NN−2σ), a, μ > 0, Iα(x) = |x|α−N, and λ∈R appears as a Lagrange multiplier.
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Existence and concentration of positive ground state solutions for nonlinear fractional Schr\"odinger-Poisson system with critical growth
Kaimin Teng,Ravi P. Agarwal +1 more
TL;DR: In this article, a fractional Schrodinger-Poisson system with competing potential functions is studied and a family of positive ground state solutions with polynomial growth for sufficiently small varepsilon>0 is shown.
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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),x∈R3,(−△)tϕ=K(x)u2,x∈R3, 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)dξ|τ|σ=∞ uniformly in x∈R3 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)dξ|τ|4=∞ uniformly in x∈R3 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.
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Multiplicity and concentration of nontrivial solutions for fractional Schrödinger–Poisson system involving critical growth
Kaimin Teng,Yiqun Cheng +1 more
TL;DR: In this paper, the concentration and multiplicity of solutions to the following fractional Schrodinger-Poisson system e 2 s ( − Δ ) s u + V ( x ) u + ϕ u = f ( u ) + u 2 s ∗ − 1 in R 3, where s, t ∈ ( 0, 1 ), s > 3 4, e > 0 is a small parameter, f ∈ C 1 ( R +, R ), and V : R 3 → R is a continuous and bounded function satisfying some local assumptions.
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Ground states for fractional magnetic operators
Pietro d'Avenia,Marco Squassina +1 more
TL;DR: In this article, a class of minimization problems for a nonlocal operator involving an external magnetic potential is studied and the notions are physically justified and consistent with the case of absence of magnetic fields.
Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth
Kaimin Teng,Ravi P. Agarwal +1 more
TL;DR: In this article, a fractional Schrodinger-Poisson system with competing potential functions was studied and a family of positive ground state solutions with polynomial growth for sufficiently small varepsilon>0 was shown.