scholarly journals On the Ground State to Hamiltonian Elliptic System with Choquard’s Nonlinear Term

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Wenbo Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Ground State ◽  
Nonlinear Term ◽  
Smooth Boundary ◽  
Ground State Solution ◽  
State Solution ◽  
Hamiltonian Elliptic System

In the present paper, we consider the following Hamiltonian elliptic system with Choquard’s nonlinear term −Δu+Vxu=∫ΩGvy/x−yβdygv in Ω,−Δv+Vxv=∫ΩFuy/x−yαdyfu in Ω,u=0,v=0 on ∂Ω,where Ω⊂ℝN is a bounded domain with a smooth boundary, 0<α<N, 0<β<N, and F is the primitive of f, similarly for G. By establishing a strongly indefinite variational setting, we prove that the above problem has a ground state solution.

scholarly journals On a class of critical elliptic systems in ℝ4

2020 ◽  
Vol 10 (1) ◽  
pp. 548-568
Author(s):  
Xin Zhao ◽  
Wenming Zou
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
State Solution ◽  
Smooth Bounded Domain ◽  
Nonexistence Result ◽  
Coprime Integers ◽  
Two Parameters

Abstract In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2 v^3+\beta u^2v+2 y u^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ4 is a smooth bounded domain with smooth boundary ∂Ω; p, q are positive coprime integers with 1 < $\begin{array}{} \displaystyle \frac{2q}{p} \end{array}$ < 2; μi > 0 and λi ∈ ℝ are fixed constants, i = 1, 2; β > 0, y > 0 are two parameters. We prove a nonexistence result and the existence of the ground state solution to the above system under proper assumptions on the parameters. It seems that this system has not been explored directly before.

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.

Pseudo-Arclength Continuation Algorithms for Binary Rydberg-Dressed Bose-Einstein Condensates

2016 ◽  
Vol 19 (4) ◽  
pp. 1067-1093 ◽  
Author(s):  
Sirilak Sriburadet ◽  
Y.-S. Wang ◽  
C.-S. Chien ◽  
Y. Shih
Keyword(s):  
Ground State ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
State Solution ◽  
Divide And Conquer ◽  
Continuation Methods ◽  
Solution Curve ◽  
Bose Einstein ◽  
Curve Tracing ◽  
Nonlocal Nonlinear

AbstractWe study pseudo-arclength continuation methods for both Rydberg-dressed Bose-Einstein condensates (BEC), and binary Rydberg-dressed BEC which are governed by the Gross-Pitaevskii equations (GPEs). A divide-and-conquer technique is proposed for rescaling the range/ranges of nonlocal nonlinear term/terms, which gives enough information for choosing a proper stepsize. This guarantees that the solution curve we wish to trace can be precisely approximated. In addition, the ground state solution would successfully evolve from one peak to vortices when the affect of the rotating term is imposed. Moreover, parameter variables with different number of components are exploited in curve-tracing. The proposed methods have the advantage of tracing the ground state solution curve once to compute the contours for various values of the coefficients of the nonlocal nonlinear term/terms. Our numerical results are consistent with those published in the literatures.

Ground State for a Coupled Elliptic System with Critical Growth

2018 ◽  
Vol 18 (1) ◽  
pp. 1-15
Author(s):  
Huiling Wu ◽  
Yongqing Li
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Ground State Solution ◽  
Critical Growth ◽  
State Solution ◽  
Content Type ◽  
Critical Nonlinearities ◽  
Differentiable Functions ◽  
Graphic Content

Abstract We study the following coupled elliptic system with critical nonlinearities: \left\{\begin{aligned} &\displaystyle-\triangle{u}+u=f(u)+\beta h(u)K(v),&&% \displaystyle x\in{\mathbb{R}}^{N},\\ &\displaystyle-\triangle{v}+v=g(v)+\beta H(u)k(v),&&\displaystyle x\in{\mathbb% {R}}^{N},\\ &\displaystyle u,v\in H^{1}({\mathbb{R}}^{N}),\end{aligned}\right. where {\beta>0} ; f, g are differentiable functions with critical growth; and {H,K} are primitive functions of h and k, respectively. Under some assumptions on f, g, h and k, we obtain the existence of a positive ground state solution of this system for {N\geq 2} .

Exact Ground State Solution for Anyons with spin-$\frac {1}{2}$

1992 ◽  
Vol 17 (1) ◽  
pp. 107-110
Author(s):  
Shun-Qing Shen ◽  
J. Cai ◽  
Rui-Bao Tao
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Exact Ground State
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