scholarly journals Asymptotic behavior of ground states for a fractional Choquard equation with critical growth

2021 ◽  
Vol 6 (4) ◽  
pp. 3838-3856
Author(s):  
Xianyong Yang ◽  
◽  
Qing Miao ◽  
Keyword(s):  
Asymptotic Behavior ◽  
Ground States ◽  
Critical Growth ◽  
Choquard Equation ◽  
Fractional Choquard Equation

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

2021 ◽  
Vol 19 (1) ◽  
pp. 259-267
Author(s):  
Liuyang Shao ◽  
Yingmin Wang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Asymptotic Behavior Of Solutions ◽  
Suitable Assumption ◽  
Choquard Equation ◽  
Existence Of Positive Solutions

Abstract In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

scholarly journals Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation

2019 ◽  
pp. 1-27
Author(s):  
P. Belchior ◽  
H. Bueno ◽  
O.H. Miyagaki ◽  
G.A. Pereira
Keyword(s):  
Asymptotic Behavior ◽  
Ground States ◽  
Hartree Equation

scholarly journals Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth

2018 ◽  
Vol 8 (1) ◽  
pp. 1184-1212 ◽  
Author(s):  
Daniele Cassani ◽  
Jianjun Zhang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Sobolev Inequality ◽  
Ground States ◽  
Critical Growth ◽  
Qualitative Behavior ◽  
Schrödinger Equations ◽  
Qualitative Properties ◽  
Concentration Phenomena ◽  
Qualitative Properties Of Solutions

Abstract We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.

scholarly journals Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth

2020 ◽  
Vol 19 (1) ◽  
pp. 123-144
Author(s):  
Hua Jin ◽  
◽  
Wenbin Liu ◽  
Huixing Zhang ◽  
Jianjun Zhang ◽  
...  
Keyword(s):  
Ground States ◽  
Critical Growth
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