scholarly journals Existence of Positive Ground State Solution for the Nonlinear Schrödinger-Poisson System with Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Lu Xiao ◽  
Haiqin Qian ◽  
Mengmeng Qu ◽  
Jun Wang
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger

In the present paper we study the existence of positive ground state solutions for the nonautonomous Schrödinger-Poisson system with competing potentials. Under some assumptions for the potentials we prove the existence of positive ground state solution.

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

Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent

2020 ◽  
Vol 20 (3) ◽  
pp. 511-538 ◽  
Author(s):  
Lin Li ◽  
Patrizia Pucci ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Sobolev Exponent

AbstractIn this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent\left\{\begin{aligned} \displaystyle{}{-}\Delta u+V(x)u+q^{2}\phi u&% \displaystyle=\mu|u|^{p-1}u+|u|^{4}u&&\displaystyle\phantom{}\mbox{in }\mathbb% {R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\mbox{in }\mathbb{R}^{3},\end{aligned}\right.where {\mu>0} is a parameter and {2<p<5}. Under certain assumptions on V, we prove the existence of a nontrivial ground state solution, using the method of the Pohozaev–Nehari manifold, the arguments of Brézis–Nirenberg, the monotonicity trick and a global compactness lemma.

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.

scholarly journals Non-local to local transition for ground states of fractional Schrödinger equations on $$\mathbb {R}^N$$

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Bartosz Bieganowski ◽  
Simone Secchi
Keyword(s):  
Ground State ◽  
Ground States ◽  
Ground State Solution ◽  
State Solution ◽  
Local Problem ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Non Local ◽  
Local Transition

Abstract We consider the nonlinear fractional problem $$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$ ( - Δ ) s u + V ( x ) u = f ( x , u ) in R N We show that ground state solutions converge (along a subsequence) in $$L^2_{\mathrm {loc}} (\mathbb {R}^N)$$ L loc 2 ( R N ) , under suitable conditions on f and V, to a ground state solution of the local problem as $$s \rightarrow 1^-$$ s → 1 - .

Existence of semiclassical ground-state solutions for semilinear elliptic systems

2012 ◽  
Vol 142 (4) ◽  
pp. 867-895 ◽  
Author(s):  
Jun Wang ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Ground State ◽  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Limit Problem ◽  
State Solution ◽  
Ground State Solutions ◽  
Semilinear Elliptic ◽  
Small Positive Parameter ◽  
Image Position

This paper is concerned with the following semilinear elliptic equations of the formwhere ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε > 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

scholarly journals Multiple solutions for a quasilinear Schrödinger–Poisson system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang
Keyword(s):  
Nontrivial Solution ◽  
Ground State ◽  
Multiple Solutions ◽  
Continuous Functions ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Poisson System ◽  
Subcritical Growth ◽  
Monotonicity Properties

AbstractIn this article, we consider the following quasilinear Schrödinger–Poisson system $$ \textstyle\begin{cases} -\Delta u+V(x)u-u\Delta (u^{2})+K(x)\phi (x)u=g(x,u), \quad x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, \quad x\in \mathbb{R}^{3}, \end{cases} $$ { − Δ u + V ( x ) u − u Δ ( u 2 ) + K ( x ) ϕ ( x ) u = g ( x , u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ V , K : R 3 → R and $g:\mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ g : R 3 × R → R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.

scholarly journals Ground State Solution for an Autonomous Nonlinear Schrödinger System

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Min Liu ◽  
Jiu Liu
Keyword(s):  
Ground State ◽  
Critical Sobolev Exponent ◽  
Growth Conditions ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Sobolev Exponent ◽  
Schrödinger System

In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where λ , μ , and ν are positive parameters; 2 ∗ = 2 N / N − 2 is the critical Sobolev exponent; and f satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.

scholarly journals Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations

2021 ◽  
Vol 7 (1) ◽  
pp. 1015-1034
Author(s):  
Shulin Zhang ◽  
◽  
Keyword(s):  
Variational Method ◽  
Ground State ◽  
Periodic Potential ◽  
Ground State Solution ◽  
State Solution ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Asymptotically Periodic

<abstract><p>In this paper, we study the existence of a positive ground state solution for a class of generalized quasilinear Schrödinger equations with asymptotically periodic potential. By the variational method, a positive ground state solution is obtained. Compared with the existing results, our results improve and generalize some existing related results.</p></abstract>

scholarly journals SYMMETRY BREAKING FOR GROUND-STATE SOLUTIONS OF HÉNON SYSTEMS IN A BALL

2010 ◽  
Vol 53 (2) ◽  
pp. 245-255 ◽  
Author(s):  
HAIYANG HE
Keyword(s):  
Symmetry Breaking ◽  
Unit Ball ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Asymptotic Estimates ◽  
Ground State Solutions ◽  
Image Position

AbstractWe consider in this paper the problem (1) where Ω is the unit ball in ℝN centred at the origin, 0 ≤ α < pN,β > 0, N ≥ 3. Suppose qϵ → q as ϵ → 0+ and qϵ, q satisfy, respectively, we investigate the asymptotic estimates of the ground-state solutions (uϵ, vϵ) of (1) as β → + ∞ with p, qϵ fixed. We also show the symmetry-breaking phenomenon with α, β fixed and qϵ → q as ϵ → 0+. In addition, the ground-state solution is non-radial provided that ϵ > 0 is small or β is large enough.

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