The Critical Hyperbola for a Hamiltonian Elliptic System with Weights

2007 ◽  
pp. 681-695
Author(s):  
Djairo G. de Figueiredo ◽  
Ireneo Peral ◽  
Julio D. Rossi
Keyword(s):  
Elliptic System ◽  
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).

scholarly journals Existence and concentration of semiclassical solutions for Hamiltonian elliptic system

2016 ◽  
Vol 15 (2) ◽  
pp. 599-622 ◽  
Author(s):  
Jian Zhang ◽  
Wen Zhang ◽  
Xiaoliang Xie
Keyword(s):  
Elliptic System ◽  
Hamiltonian Elliptic System

The critical hyperbola for a Hamiltonian elliptic system with weights

2007 ◽  
Vol 187 (3) ◽  
pp. 531-545 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
Ireneo Peral ◽  
Julio D. Rossi
Keyword(s):  
Elliptic System ◽  
Hamiltonian Elliptic System

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 Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

2020 ◽  
Vol 10 (1) ◽  
pp. 233-260
Author(s):  
Jian Zhang ◽  
Jianhua Chen ◽  
Quanqing Li ◽  
Wen Zhang
Keyword(s):  
Elliptic System ◽  
Local Property ◽  
Analytical Techniques ◽  
Energy Functional ◽  
Gradient Term ◽  
Penalization Method ◽  
Concentration Phenomena ◽  
Hamiltonian Elliptic System ◽  
Small Positive Parameter ◽  
Concentration Behavior

Abstract In this paper, we study the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\epsilon^{2}{\it\Delta} \psi +\epsilon \vec{b}\cdot \nabla \psi +\psi+V(x)\varphi=f(|\eta|)\varphi~~\hbox{in}~\mathbb{R}^{N},\\ -\epsilon^{2}{\it\Delta} \varphi -\epsilon \vec{b}\cdot \nabla \varphi +\varphi+V(x)\psi=f(|\eta|)\psi~~\hbox{in}~\mathbb{R}^{N},\\ \end{array} \right. \end{array}$$ where η = (ψ, φ) : ℝN → ℝ2, ϵ is a small positive parameter and b⃗ is a constant vector. We require that the potential V only satisfies certain local condition. Combining this with other suitable assumptions on f, we construct a family of semiclassical solutions. Moreover, the concentration phenomena around local minimum of V, convergence and exponential decay of semiclassical solutions are also explored. In the proofs we apply penalization method, linking argument and some analytical techniques since the local property of the potential and the strongly indefinite character of the energy functional.

scholarly journals Ground state solutions for Hamiltonian elliptic system with inverse square potential

2017 ◽  
Vol 37 (8) ◽  
pp. 4565-4583 ◽  
Author(s):  
Jian Zhang ◽  
◽  
Wen Zhang ◽  
Xianhua Tang ◽  
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Ground State Solutions ◽  
Hamiltonian Elliptic System
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