scholarly journals Existence and decay property of ground state solutions for Hamiltonian elliptic system

2019 ◽  
Vol 18 (5) ◽  
pp. 2433-2455 ◽  
Author(s):  
Jian Zhang ◽  
◽  
Wen Zhang ◽  
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Decay Property ◽  
Ground State Solutions ◽  
Hamiltonian Elliptic System

Ground state solutions of Hamiltonian elliptic systems in dimension two

2019 ◽  
Vol 150 (4) ◽  
pp. 1737-1768 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Jianjun Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Existence Results ◽  
Ground State Solutions ◽  
Variational Framework ◽  
Hamiltonian Elliptic System ◽  
Schrödinger Systems ◽  
Image Position

AbstractThe aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

Ground State Solutions of Nehari–Pankov Type for a Superlinear Hamiltonian Elliptic System on ℝN

2015 ◽  
Vol 58 (3) ◽  
pp. 651-663 ◽  
Author(s):  
Xianhua Tang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Direct Approach ◽  
Ground State Solutions ◽  
Hamiltonian Elliptic System

AbstractThis paper is concerned with the following elliptic system of Hamiltonian typewhere the potential V is periodic and 0 lies in a gap of the spectrum of −Δ + V, W(x, u, v) is periodic in x and superlinear in u and v at infinity. We develop a direct approach to ûnding ground state solutions of Nehari–Pankov type for the above system. Our method is especially applicable to the case whenwhere with , and and hj(x, t) are nondecreasing in t ∊ ℝ+ for every x ∊ ℝN and gi(x, 0) = hj(x, 0) = 0.

scholarly journals Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

2020 ◽  
Vol 10 (1) ◽  
pp. 331-352
Author(s):  
Wen Zhang ◽  
Jian Zhang ◽  
Heilong Mi
Keyword(s):  
Variational Method ◽  
Elliptic System ◽  
Ground State ◽  
Multiple Solutions ◽  
Ground States ◽  
Existence Results ◽  
Perturbation Approach ◽  
Gradient Term ◽  
Global Information ◽  
Hamiltonian Elliptic System

Abstract This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

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 Ground state solutions for Hamiltonian elliptic systems with super or asymptotically quadratic nonlinearity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yubo He ◽  
Dongdong Qin ◽  
Dongdong Chen
Keyword(s):  
Ground State ◽  
Spectral Gap ◽  
Elliptic Systems ◽  
Technical Condition ◽  
Sigma Delta ◽  
Ground State Solutions ◽  
Least Energy Solution ◽  
Hamiltonian Elliptic System ◽  
Least Energy ◽  
Quadratic Case

Abstract This article concerns the Hamiltonian elliptic system: $$ \textstyle\begin{cases} -\Delta \varphi +V(x)\varphi =G_{\psi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ -\Delta \psi +V(x)\psi =G_{\varphi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ \varphi , \psi \in H^{1}(\mathbb {R}^{N}). \end{cases} $$ { − Δ φ + V ( x ) φ = G ψ ( x , φ , ψ ) in  R N , − Δ ψ + V ( x ) ψ = G φ ( x , φ , ψ ) in  R N , φ , ψ ∈ H 1 ( R N ) . Assuming that the potential V is periodic and 0 lies in a spectral gap of $\sigma (-\Delta +V)$ σ ( − Δ + V ) , least energy solution of the system is obtained for the super-quadratic case with a new technical condition, and the existence of ground state solutions of Nehari–Pankov type is established for the asymptotically quadratic case. The results obtained in the paper generalize and improve related ones in the literature.

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