Semiclassical ground state solutions for a Choquard type equation in ℝ2 with critical exponential growth

2018 ◽  
Vol 24 (1) ◽  
pp. 177-209 ◽  
Author(s):  
Minbo Yang
Keyword(s):  
Ground State ◽  
Exponential Growth ◽  
Riesz Potential ◽  
Type Equation ◽  
Singularly Perturbed ◽  
Maximum Point ◽  
Positive Parameter ◽  
Primitive Function ◽  
Critical Exponential Growth ◽  
Point Set

In this paper we study a nonlocal singularly perturbed Choquard type equation $$-\varepsilon^2\Delta u +V(x)u =\vr^{\mu-2}\left[\frac{1}{|x|^{\mu}}\ast \big(P(x)G(u)\big)\right]P(x)g(u)$$ in ℝ2, where ε is a positive parameter, \hbox{$\frac{1}{|x|^\mu}$} with 0 < μ < 2 is the Riesz potential, ∗ is the convolution operator, V(x), P(x) are two continuous real functions and G(s) is the primitive function of g(s). Suppose that the nonlinearity g is of critical exponential growth in ℝ2 in the sense of the Trudinger-Moser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set of P(x) if V(x) is a constant.

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).

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.

Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

2021 ◽  
pp. 1-21
Author(s):  
Heng Yang
Keyword(s):  
Ground State ◽  
Exponential Growth ◽  
Riesz Potential ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Zero Mass ◽  
Subcritical Exponential Growth ◽  
New Ideas ◽  
Mass Case

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

scholarly journals Singularly perturbed quasilinear Choquard equations with nonlinearity satisfying Berestycki–Lions assumptions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Heng Yang
Keyword(s):  
Asymptotic Behavior ◽  
Exponential Decay ◽  
Ground State ◽  
Riesz Potential ◽  
Ground State Solution ◽  
State Solution ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem

AbstractIn the present paper, we consider the following singularly perturbed problem: $$ \textstyle\begin{cases} -\varepsilon ^{2}\Delta u+V(x)u-\varepsilon ^{2}\Delta (u^{2})u= \varepsilon ^{-\alpha }(I_{\alpha }*G(u))g(u), \quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases} $$ { − ε 2 Δ u + V ( x ) u − ε 2 Δ ( u 2 ) u = ε − α ( I α ∗ G ( u ) ) g ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where $\varepsilon >0$ ε > 0 is a parameter, $N\ge 3$ N ≥ 3 , $\alpha \in (0, N)$ α ∈ ( 0 , N ) , $G(t)=\int _{0}^{t}g(s)\,\mathrm{d}s$ G ( t ) = ∫ 0 t g ( s ) d s , $I_{\alpha }: \mathbb{R}^{N}\rightarrow \mathbb{R}$ I α : R N → R is the Riesz potential, and $V\in \mathcal{C}(\mathbb{R}^{N}, \mathbb{R})$ V ∈ C ( R N , R ) with $0<\min_{x\in \mathbb{R}^{N}}V(x)< \lim_{|y|\to \infty }V(y)$ 0 < min x ∈ R N V ( x ) < lim | y | → ∞ V ( y ) . Under the general Berestycki–Lions assumptions on g, we prove that there exists a constant $\varepsilon _{0}>0$ ε 0 > 0 determined by V and g such that for $\varepsilon \in (0,\varepsilon _{0}]$ ε ∈ ( 0 , ε 0 ] the above problem admits a semiclassical ground state solution $\hat{u}_{\varepsilon }$ u ˆ ε with exponential decay at infinity. We also study the asymptotic behavior of $\{\hat{u}_{\varepsilon }\}$ { u ˆ ε } as $\varepsilon \to 0$ ε → 0 .

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