scholarly journals Groundstates for Choquard type equations with Hardy–Littlewood–Sobolev lower critical exponent

2019 ◽  
Vol 150 (3) ◽  
pp. 1377-1400 ◽  
Author(s):  
Daniele Cassani ◽  
Jean Van Schaftingen ◽  
Jianjun Zhang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Riesz Potential ◽  
Type Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Choquard Equation ◽  
Resonant Case ◽  
Image Position ◽  
Nonlocal Nonlinear

AbstractFor the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, $$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$where $N\ges 3$, Vμ,ν :ℝN → ℝ is an external potential defined for μ, ν > 0 and x ∈ ℝN by Vμ,ν(x) = 1 − μ/(ν2 + |x|2) and $I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0, N), we exhibit two thresholds μν, μν > 0 such that the equation admits a positive ground state solution if and only if μν < μ < μν and no ground state solution exists for μ < μν. Moreover, if μ > max{μν, N2(N − 2)/4(N + 1)}, then equation still admits a sign changing ground state solution provided $N \ges 4$ or in dimension N = 3 if in addition 3/2 < α < 3 and $\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $, namely in the non-resonant case.

scholarly journals Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang ◽  
Qiongfen Zhang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Sobolev Inequality ◽  
Existence Of Solutions ◽  
Riesz Potential ◽  
Ground State Solution ◽  
Pohozaev Identity ◽  
Choquard Equation ◽  
Minimax Methods ◽  
Variable Potential

AbstractIn this paper, we focus on the existence of solutions for the Choquard equation $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ { − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + λ | u | p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , where $\lambda >0$ λ > 0 is a parameter, $\alpha \in (0,N)$ α ∈ ( 0 , N ) , $N\ge 3$ N ≥ 3 , $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ I α : R N → R is the Riesz potential. As usual, $\alpha /N+1$ α / N + 1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if $\lambda >\lambda _{*}$ λ > λ ∗ for some given number $\lambda _{*}$ λ ∗ in three cases: (i) $2< p<\frac{4}{N}+2$ 2 < p < 4 N + 2 , (ii) $p=\frac{4}{N}+2$ p = 4 N + 2 , and (iii) $\frac{4}{N}+2< p<2^{*}$ 4 N + 2 < p < 2 ∗ . Our result improves the previous related ones in the literature.

The existence of a ground-state solution for a class of Kirchhoff-type equations in ℝN

2016 ◽  
Vol 146 (2) ◽  
pp. 371-391 ◽  
Author(s):  
Jiu Liu ◽  
Jia-Feng Liao ◽  
Chun-Lei Tang
Keyword(s):  
Ground State ◽  
Type Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Image Position

In this paper, we study the following Kirchhoff-type equation: where a, b are positive constants and N = 1, 2, 3. Under appropriate assumptions on V, K and g, we obtain a ground-state solution by using the approach developed by Szulkin and Weth in 2010.

scholarly journals Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

2020 ◽  
Vol 10 (1) ◽  
pp. 152-171
Author(s):  
Sitong Chen ◽  
Xianhua Tang ◽  
Jiuyang Wei
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Ground State ◽  
Exponential Growth ◽  
Sobolev Inequality ◽  
Critical Case ◽  
Riesz Potential ◽  
Ground State Solution ◽  
Choquard Equation ◽  
Critical Exponential Growth

Abstract This paper deals with the following Choquard equation with a local nonlinear perturbation: $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), & x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$ where α ∈ (0, 2), Iα : ℝ2 → ℝ is the Riesz potential and f ∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponent $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ and the critical exponential growth of f(u).

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 Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

2019 ◽  
Vol 9 (1) ◽  
pp. 413-437 ◽  
Author(s):  
Xianhua Tang ◽  
Sitong Chen
Keyword(s):  
Ground State ◽  
Riesz Potential ◽  
Ground State Solution ◽  
State Solution ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
New Approach ◽  
Related Literature

Abstract In the present paper, we consider the following singularly perturbed problem: $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}*F(u))f(u), & x\in \mathbb{R}^N; \\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{array}$$ where ε > 0 is a parameter, N ≥ 3, α ∈ (0, N), F(t) = $\begin{array}{} \int_{0}^{t} \end{array}$f(s)ds and Iα : ℝN → ℝ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution (ε ∈ (0, ε0)) and a ground state solution (ε = 1) under the general “Berestycki-Lions assumptions” on the nonlinearity f which are almost necessary, as well as some weak assumptions on the potential V. When ε = 1, our results generalize and improve the ones in [V. Moroz, J. Van Schaftingen, T. Am. Math. Soc. 367 (2015) 6557-6579] and [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, we propose a new approach to recover the compactness for a (PS)-sequence, and our approach is useful for many similar problems.

scholarly journals SYMMETRY BREAKING FOR GROUND-STATE SOLUTIONS OF HÉNON SYSTEMS IN A BALL

2010 ◽  
Vol 53 (2) ◽  
pp. 245-255 ◽  
Author(s):  
HAIYANG HE
Keyword(s):  
Symmetry Breaking ◽  
Unit Ball ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Asymptotic Estimates ◽  
Ground State Solutions ◽  
Image Position

AbstractWe consider in this paper the problem (1) where Ω is the unit ball in ℝN centred at the origin, 0 ≤ α < pN,β > 0, N ≥ 3. Suppose qϵ → q as ϵ → 0+ and qϵ, q satisfy, respectively, we investigate the asymptotic estimates of the ground-state solutions (uϵ, vϵ) of (1) as β → + ∞ with p, qϵ fixed. We also show the symmetry-breaking phenomenon with α, β fixed and qϵ → q as ϵ → 0+. In addition, the ground-state solution is non-radial provided that ϵ > 0 is small or β is large enough.

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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