Concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system involving critical exponent

2019 ◽  
Vol 21 (06) ◽  
pp. 1850027 ◽  
Author(s):  
Zhipeng Yang ◽  
Yuanyang Yu ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Critical Exponent ◽  
Ground State ◽  
Local Minimum ◽  
Fractional Laplacian ◽  
Ground State Solution ◽  
Critical Nonlinearity ◽  
Poisson System ◽  
Ground State Solutions ◽  
Concentration Behavior

We are concerned with the existence and concentration behavior of ground state solutions of the fractional Schrödinger–Poisson system with critical nonlinearity [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text] denotes the fractional Laplacian of order [Formula: see text] and satisfies [Formula: see text]. The potential [Formula: see text] is continuous and positive, and has a local minimum. We obtain a positive ground state solution for [Formula: see text] small, and we show that these ground state solutions concentrate around a local minimum of [Formula: see text] as [Formula: see text].

Existence and Concentration of Positive Ground State Solutions for Schrödinger-Poisson Systems

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Jun Wang ◽  
Lixin Tian ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Small Parameter ◽  
Ground State ◽  
Ground State Solution ◽  
Real Parameter ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Subcritical Nonlinearity

AbstractIn this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson systemwhere ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution

The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity

2019 ◽  
Vol 19 (4) ◽  
pp. 779-795
Author(s):  
Guangze Gu ◽  
Xianhua Tang
Keyword(s):  
Continuous Function ◽  
Small Parameter ◽  
Ground State ◽  
Sobolev Inequality ◽  
Ground States ◽  
Ground State Solution ◽  
Positive Continuous Function ◽  
Primitive Function ◽  
Concentration Behavior ◽  
Kirchhoff Type Problems

AbstractIn this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity\left\{\begin{aligned} \displaystyle-&\displaystyle\biggl{(}\varepsilon^{2}a+% \varepsilon b\int_{\mathbb{R}^{3}}\lvert\nabla u\rvert^{2}\mathop{}\!dx\biggr{% )}\Delta u+V(x)u=\varepsilon^{\mu-3}\biggl{(}\int_{\mathbb{R}^{3}}\frac{K(y)F(% u(y))}{\lvert x-y\rvert^{\mu}}\mathop{}\!dy\biggr{)}K(x)f(u),\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right.where {\varepsilon>0} is a small parameter, {a,b>0}, {\mu\in(0,3)}, {V,K} are two positive continuous function and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of the Hardy–Littlewood–Sobolev inequality. We show that the equation admits a positive ground state solution for {\varepsilon>0} sufficiently small. Furthermore, we prove that these ground state solutions concentrate around such points which are both the minima points of the potential V and the maximum points of the potential K as {\varepsilon\to 0}.

scholarly journals Existence of positive ground state solutions of Schrödinger–Poisson system involving negative nonlocal term and critical exponent on bounded domain

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenxuan Zheng ◽  
Wenbin Gan ◽  
Shibo Liu
Keyword(s):  
Critical Exponent ◽  
Bounded Domain ◽  
Ground State ◽  
Mountain Pass Theorem ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Ground State Solutions ◽  
Concentration Compactness Principle ◽  
Compactness Principle

AbstractIn this paper, we prove the existence of positive ground state solutions of the Schrödinger–Poisson system involving a negative nonlocal term and critical exponent on a bounded domain. The main tools are the mountain pass theorem and the concentration compactness principle.

scholarly journals Ground State Solution of Pohožaev Type for Quasilinear Schrödinger Equation Involving Critical Exponent in Orlicz Space

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 779 ◽  
Author(s):  
Jianqing Chen ◽  
Qian Zhang
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Orlicz Space ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
Existence Results ◽  
Quasilinear Schrödinger Equation ◽  
Ground State Solutions

We study the following quasilinear Schrödinger equation involving critical exponent - Δ u + V ( x ) u - Δ ( u 2 ) u = A ( x ) | u | p - 1 u + λ B ( x ) u 3 N + 2 N - 2 , u ( x ) > 0 for x ∈ R N and u ( x ) → 0 as | x | → ∞ . By using a monotonicity trick and global compactness lemma, we prove the existence of positive ground state solutions of Pohožaev type. The nonlinear term | u | p - 1 u for the well-studied case p ∈ [ 3 , 3 N + 2 N - 2 ) , and the less-studied case p ∈ [ 2 , 3 ) , and for the latter case few existence results are available in the literature. Our results generalize partial previous works.

scholarly journals Existence and concentration of positive solutions for a p-fractional Choquard equation

2021 ◽  
Vol 6 (11) ◽  
pp. 12929-12951
Author(s):  
Xudong Shang ◽  
Keyword(s):  
Small Parameter ◽  
Positive Solutions ◽  
Ground State ◽  
Global Minimum ◽  
Ground State Solution ◽  
Global Maximum ◽  
Choquard Equation ◽  
Number Of Solutions ◽  
Concentration Behavior ◽  
Fractional Choquard Equation

<abstract><p>In this work, we study the existence, multiplicity and concentration behavior of positive solutions for the following problem involving the fractional $ p $-Laplacian</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \varepsilon^{ps}(-\Delta )^{s}_{p}u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}(\frac{1}{|x|^{\mu}}\ast K|u|^{q})K(x)|u|^{q-2}u \hskip0.2cm\text{in}\hskip0.1cm \mathbb{R}^{N}, \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 &lt; p &lt; \infty $, $ N &gt; ps $, $ 0 &lt; \mu &lt; ps $, $ p &lt; q &lt; \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N}) $, $ (-\Delta)^{s}_{p} $ is the fractional $ p $-Laplacian and $ \varepsilon &gt; 0 $ is a small parameter. Under certain conditions on $ V $ and $ K $, we prove the existence of a positive ground state solution and express the location of concentration in terms of the potential functions $ V $ and $ K $. In particular, we relate the number of solutions with the topology of the set where $ V $ attains its global minimum and $ K $ attains its global maximum.</p></abstract>

scholarly journals Existence of ground state solutions for an asymptotically 2-linear fractional Schrödinger–Poisson system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Continuous Function ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Linear Condition

AbstractIn this paper, we investigate the following fractional Schrödinger–Poisson system: $$\left \{ \textstyle\begin{array}{l@{\quad}l} (-\Delta)^{s} u + u + \phi u = f(u), & \text{in } \mathbb{R}^{3}, \\ (-\Delta)^{t} \phi= u^{2}, & \text{in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $${(−Δ)su+u+ϕu=f(u),in R3,(−Δ)tϕ=u2,in R3, where $\frac{3}{4} < s < 1$34<s<1, $\frac{1}{2} < t < 1$12<t<1, and f is a continuous function, which is superlinear at zero, with $f(\tau) \tau \ge3 F(\tau) \ge0$f(τ)τ≥3F(τ)≥0, $F(\tau) = \int_{0}^{\tau} f(s) \,ds$F(τ)=∫0τf(s)ds, $\tau \in\mathbb{R}$τ∈R. We prove that the system admits a ground state solution under the asymptotically 2-linear condition. The result here extends the existing study.

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