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).

POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR -LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS

2019 ◽  
Vol 109 (2) ◽  
pp. 193-216 ◽  
Author(s):  
J. C. DE ALBUQUERQUE ◽  
JOÃO MARCOS DO Ó ◽  
EDCARLOS D. SILVA
Keyword(s):  
Large Class ◽  
Ground State ◽  
Variational Approach ◽  
Ground States ◽  
Nehari Manifold ◽  
Coupled Systems ◽  
Coupling Term ◽  
Ground State Solutions ◽  
Image Position ◽  
The Nehari Manifold

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems $$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ where $1<p\leq q<N$. Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.

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 Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tianfang Wang ◽  
Wen Zhang
Keyword(s):  
Ground State ◽  
Nehari Manifold ◽  
Existence Results ◽  
Infinitely Many Solutions ◽  
Choquard Equation ◽  
Ground State Solutions ◽  
Existence And Multiplicity ◽  
Variational Tools ◽  
Nonlinear Choquard Equation ◽  
The Nehari Manifold

AbstractIn this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , where $N\geq 3$ N ≥ 3 , $0<\mu <N$ 0 < μ < N , $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ 2 N − μ N ≤ p < 2 N − μ N − 2 , ∗ represents the convolution between two functions. We assume that the potential function $V(x)$ V ( x ) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

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 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.

scholarly journals THE ROBIN PROBLEM FOR THE HÉNON EQUATION

2013 ◽  
Vol 88 (1) ◽  
pp. 1-11
Author(s):  
HAIYANG HE
Keyword(s):  
Symmetry Breaking ◽  
Unit Ball ◽  
Ground State ◽  
Existence Results ◽  
Robin Problem ◽  
Ground State Solutions ◽  
Hénon Equation ◽  
Image Position

AbstractIn this paper, we consider the following Robin problem:$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$where$\Omega $is the unit ball in${ \mathbb{R} }^{N} $centred at the origin, with$N\geq 3$,$p\gt 1$,$\alpha \gt 0$,$\beta \gt 0$, and$\nu $is the unit outward vector normal to$\partial \Omega $. We prove that the above problem has no solution when$\beta $is small enough. We also obtain existence results and we analyse the symmetry breaking of the ground state solutions.

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

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.

Sign in / Sign up

Export Citation Format

Share Document