scholarly journals On semiclassical ground states for Hamiltonian elliptic system with critical growth

2016 ◽  
Vol 48 (1) ◽  
pp. 1
Author(s):  
Jian Zhang ◽  
Xianhua Tang ◽  
Wen Zhang
Keyword(s):  
Elliptic System ◽  
Ground States ◽  
Critical Growth ◽  
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.

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

scholarly journals Existence of bound and ground states for an elliptic system with double criticality

2022 ◽  
Vol 216 ◽  
pp. 112730
Author(s):  
Eduardo Colorado ◽  
Rafael López-Soriano ◽  
Alejandro Ortega
Keyword(s):  
Elliptic System ◽  
Ground States

scholarly journals Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth

2018 ◽  
Vol 8 (1) ◽  
pp. 1184-1212 ◽  
Author(s):  
Daniele Cassani ◽  
Jianjun Zhang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Sobolev Inequality ◽  
Ground States ◽  
Critical Growth ◽  
Qualitative Behavior ◽  
Schrödinger Equations ◽  
Qualitative Properties ◽  
Concentration Phenomena ◽  
Qualitative Properties Of Solutions

Abstract We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.

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 Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth

2020 ◽  
Vol 19 (1) ◽  
pp. 123-144
Author(s):  
Hua Jin ◽  
◽  
Wenbin Liu ◽  
Huixing Zhang ◽  
Jianjun Zhang ◽  
...  
Keyword(s):  
Ground States ◽  
Critical Growth
Sign in / Sign up

Export Citation Format

Share Document