Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

2021 ◽  
pp. 1-28
Author(s):  
B. B. V. Maia ◽  
O. H. Miyagaki
Keyword(s):  
Riesz Potential ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Exponential Critical Growth ◽  
Beta 2 ◽  
Image Position

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

scholarly journals One-dimensional periodic fractional Schrödinger equations with exponential critical growth

2022 ◽  
Author(s):  
Shuai Yuan ◽  
Xianhua Tang ◽  
Sitong Chen
Keyword(s):  
Ground State ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Energy Estimates ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Fractional Schrödinger Equations ◽  
Exponential Critical Growth ◽  
New Challenges ◽  
The One

In the present paper, we study the existence of nontrivial solutions of the following one-dimensional fractional Schr\“{o}dinger equation $$ (-\Delta)^{1/2}u+V(x)u=f(x,u), \ \ x\in \R, $$ where $(-\Delta)^{1/2}$ stands for the $1/2$-Laplacian, $V(x)\in \mathcal{C}(\R, (0,+\infty))$, and $f(x,u):\R\times\R\to \R$ is a continuous function with an exponential critical growth. Comparing with the existing works in the field of exponential-critical-growth fractional Schr\”{o}dinger equations, we encounter some new challenges due to the weaker assumptions on the reaction term $f$. By using some sharp energy estimates, we present a detailed analysis of the energy level, which allows us to establish the existence of nontrivial solutions for a wider class of nonlinear terms. Furthermore, we use the non-Nehari manifold method to establish the existence of Nehari-type ground state solutions of the one-dimensional fractional Schr\”{o}dinger equations.

scholarly journals SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 519-541
Author(s):  
HAIYANG HE
Keyword(s):  
Positive Solution ◽  
Hyperbolic Space ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Decay Estimates ◽  
Symmetry Properties ◽  
Formula Group ◽  
Solution Pair ◽  
Image Position

Abstract(0.1) \begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation} in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.

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).

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.

Existence and Concentration of Solutions for Choquard Equations with Steep Potential Well and Doubly Critical Exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yong-Yong Li ◽  
Gui-Dong Li ◽  
Chun-Lei Tang
Keyword(s):  
Critical Exponents ◽  
Ground State ◽  
Mountain Pass Theorem ◽  
Riesz Potential ◽  
Nehari Manifold ◽  
Linking Theorem ◽  
Potential Well ◽  
Nontrivial Solutions ◽  
Choquard Equation ◽  
Steep Potential Well

AbstractIn this paper, we investigate the non-autonomous Choquard equation-\Delta u+\lambda V(x)u=(I_{\alpha}\ast F(u))F^{\prime}(u)\quad\text{in}\ \mathbb{R}^{N},where N\geq 4, \lambda>0, V\in C(\mathbb{R}^{N},\mathbb{R}) is bounded from below and has a potential well, I_{\alpha} is the Riesz potential of order \alpha\in(0,N) and F(u)=\frac{1}{2_{\alpha}^{*}}\lvert u\rvert^{2_{\alpha}^{*}}+\frac{1}{2_{*}^{\alpha}}\lvert u\rvert^{2_{*}^{\alpha}}, in which 2_{\alpha}^{*}=\frac{N+\alpha}{N-2} and 2_{*}^{\alpha}=\frac{N+\alpha}{N} are upper and lower critical exponents due to the Hardy–Littlewood–Sobolev inequality, respectively. Based on the variational methods, by combining the mountain pass theorem and Nehari manifold, we obtain the existence and concentration of positive ground state solutions for 𝜆 large enough if 𝑉 is nonnegative in \mathbb{R}^{N}; further, by the linking theorem, we prove the existence of nontrivial solutions for 𝜆 large enough if 𝑉 changes sign in \mathbb{R}^{N}.

scholarly journals Ground state solutions of Nehari–Pohožaev type for a kind of nonlinear problem with general nonlinearity and nonlocal convolution term

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shuai Yuan ◽  
Fangfang Liao
Keyword(s):  
Nonlinear Problem ◽  
Ground State ◽  
Riesz Potential ◽  
Ground State Solution ◽  
State Solution ◽  
Ground State Solutions ◽  
Beta 2 ◽  
General Nonlinearity ◽  
New Inequalities

Abstract In this paper, we consider the following nonlinear problem with general nonlinearity and nonlocal convolution term: $$ \textstyle\begin{cases} -\Delta u+V(x)u+(I_{\alpha }\ast \vert u \vert ^{q}) \vert u \vert ^{q-2}u=f(u), \quad x\in {\mathbb{R}}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}), \quad \end{cases} $$ { − Δ u + V ( x ) u + ( I α ∗ | u | q ) | u | q − 2 u = f ( u ) , x ∈ R 3 , u ∈ H 1 ( R 3 ) , where $a\in (0,3)$ a ∈ ( 0 , 3 ) , $q\in [1+\frac{\alpha }{3},3+\alpha )$ q ∈ [ 1 + α 3 , 3 + α ) , $I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}$ I α : R 3 → R is the Riesz potential, $V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))$ V ∈ C ( R 3 , [ 0 , ∞ ) ) , $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ f ∈ C ( R , R ) and $F(t)=\int _{0}^{t}f(s)\,ds$ F ( t ) = ∫ 0 t f ( s ) d s satisfies $\lim_{|t|\to \infty }F(t)/|t|^{\sigma }=\infty $ lim | t | → ∞ F ( t ) / | t | σ = ∞ with $\sigma =\min \{2,\frac{2\beta +2}{\beta }\}$ σ = min { 2 , 2 β + 2 β } where $\beta =\frac{ \alpha +2}{2(q-1)}$ β = α + 2 2 ( q − 1 ) . By using new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f.

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