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.

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.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

2021 ◽  
pp. 1-19
Author(s):  
Jing Zhang ◽  
Lin Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Hardy Potential ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

scholarly journals Existence of Solutions for a Modified Nonlinear Schrödinger System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yujuan Jiao ◽  
Yanli Wang
Keyword(s):  
Perturbation Method ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Schrödinger System

We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  x∈Ω,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  x∈Ω,  u=0,  v=0,  x∈∂Ω, whereα>2,  β>2,  α+β<2·2*,  2*=2N/(N-2)is the critical Sobolev exponent, andΩ⊂ℝN  (N≥3)is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

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 solutions for a semilinear elliptic problem with critical-subcritical growth

2018 ◽  
Vol 9 (1) ◽  
pp. 108-123 ◽  
Author(s):  
Claudianor O. Alves ◽  
Grey Ercole ◽  
M. Daniel Huamán Bolaños
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Ground State Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

Abstract We prove the existence of at least one ground state solution for the semilinear elliptic problem \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=u^{p(x)-1},\quad u% >0,\quad\text{in}\ G\subseteq\mathbb{R}^{N},\ N\geq 3,\\ \displaystyle u&\displaystyle\in D_{0}^{1,2}(G),\end{aligned}\right. where G is either {\mathbb{R}^{N}} or a bounded domain, and {p\colon G\to\mathbb{R}} is a continuous function assuming critical and subcritical values.

scholarly journals Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kazuhiro Kurata ◽  
Yuki Osada
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Ground State ◽  
State Energy ◽  
Wave Interaction ◽  
Vector Solution ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Positive Threshold ◽  
Schrödinger System

<p style='text-indent:20px;'>In this paper, we consider the asymptotic behavior of the ground state and its energy for the nonlinear Schrödinger system with three wave interaction on the parameter <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M2">\begin{document}$ \gamma \to \infty $\end{document}</tex-math></inline-formula>. In addition we prove the existence of the positive threshold <inline-formula><tex-math id="M3">\begin{document}$ \gamma^* $\end{document}</tex-math></inline-formula> such that the ground state is a scalar solution for <inline-formula><tex-math id="M4">\begin{document}$ 0 \le \gamma &lt; \gamma^* $\end{document}</tex-math></inline-formula> and is a vector solution for <inline-formula><tex-math id="M5">\begin{document}$ \gamma &gt; \gamma^* $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Ground state solution of semilinear Schrödinger system with local super-quadratic conditions

2021 ◽  
pp. 1-18
Author(s):  
Jing Chen ◽  
Yiqing Li
Keyword(s):  
Variational Method ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
The Other ◽  
Ground State Solutions ◽  
Approximation Argument ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Schrödinger System

In this paper, we dedicate to studying the following semilinear Schrödinger system equation*-Δu+V1(x)u=Fu(x,u,v)amp;mboxin~RN,r-Δv+V2(x)v=Fv(x,u,v)amp;mboxin~RN,ru,v∈H1(RN),endequation* where the potential Vi are periodic in x,i=1,2, the nonlinearity F is allowed super-quadratic at some x ∈ R N and asymptotically quadratic at the other x ∈ R N . Under a local super-quadratic condition of F, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.

scholarly journals Synchronized and ground-state solutions to a coupled Schrödinger system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohammad Ali Husaini ◽  
Chuangye Liu
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Coupled Schrödinger System ◽  
Nonlinear Schrödinger System ◽  
Schrödinger System

<p style='text-indent:20px;'>In this paper, we study the following coupled nonlinear Schrödinger system of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{array}{l} -\Delta u_i-\kappa_iu_i = g_i(u_i)+\lambda\partial_iF(\vec{u}), \\ \vec{u} = (u_1,u_2,\cdots,u_m), u_i\in D_0^{1,2}(\Omega), \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for <inline-formula><tex-math id="M1">\begin{document}$ m = 2,3 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is a bounded domain or <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ F(t_1,t_2\cdots,t_m)\in C^1(\mathbb{R}^m,\mathbb{R}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \kappa_i\in\mathbb{R} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ g_i\in C(\mathbb{R}) \ (i = 1,2,\cdots,m) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is large enough. In this work we mainly focus on the existence of fully nontrivial ground-state solutions and synchronized ground-state solutions under certain conditions.</p>

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