Existence of Ground State Solutions for Hamiltonian Elliptic Systems with Gradient Terms

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Yunjuan Jin ◽  
Minbo Yang
Keyword(s):  
Real Function ◽  
Ground State ◽  
Elliptic Systems ◽  
Continuous Real Function ◽  
Ground State Solutions

AbstractIn this paper we consider the following Hamiltonian elliptic terns in R[XXX]Where V(x) > 0 is a periodic continuous real Function, b̅(x) = (b

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).

On semiclassical ground state solutions for Hamiltonian elliptic systems

2014 ◽  
Vol 94 (7) ◽  
pp. 1380-1396 ◽  
Author(s):  
Jian Zhang ◽  
Xianhua Tang ◽  
Wen Zhang
Keyword(s):  
Ground State ◽  
Elliptic Systems ◽  
Ground State Solutions

Existence of semiclassical ground-state solutions for semilinear elliptic systems

2012 ◽  
Vol 142 (4) ◽  
pp. 867-895 ◽  
Author(s):  
Jun Wang ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Ground State ◽  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Limit Problem ◽  
State Solution ◽  
Ground State Solutions ◽  
Semilinear Elliptic ◽  
Small Positive Parameter ◽  
Image Position

This paper is concerned with the following semilinear elliptic equations of the formwhere ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε > 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

scholarly journals Infinitely Many Nontrivial Solutions of Resonant Cooperative Elliptic Systems with Superlinear Terms

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Guanwei Chen ◽  
Shiwang Ma
Keyword(s):  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Ground State Solutions

We study a class of resonant cooperative elliptic systems and replace the Ambrosetti-Rabinowitz superlinear condition with general superlinear conditions. We obtain ground state solutions and infinitely many nontrivial solutions of this system by a generalized Nehari manifold method developed recently by Szulkin and Weth.

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