Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent

Guo, Lun; Hu, Tingxi; Peng, Shuangjie; Shuai, Wei

doi:10.1007/s00526-019-1585-1

Cited by 50 publications

(26 citation statements)

References 35 publications

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“…In such circumstances, problems of type (1.1) have a variational structure and can be treated by using variational methods (see, for example, [8,20]). This paper is also concerned with the supercritical case p > (N + α)/(N − 2).…”

Section: Introductionmentioning

confidence: 99%

“…Here, we adapt some ideas in [3] to deal with stable solutions of (1.1) which only belong to the C 1 (R N ) class. Nevertheless, when p = (N + α)/(N − 2) and in some circumstances, solutions in H 1 (R N ) or positive solutions in D 1,2 (R N ) of (1.1) are known to be classical ones (see [2,8]). Similar Liouville-type results for stable H 1 loc (R N ) solutions of Hénon equations were obtained in [23].…”

Section: Introductionmentioning

confidence: 99%

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Stable Solutions to the Static Choquard Equation

2020

Bull. Aust. Math. Soc.

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This paper is concerned with the static Choquard equation $$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$ where $N,p>2$ and $\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$ . We prove that if $u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then $u\equiv 0$ . This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and $\unicode[STIX]{x1D6FC}\neq 2$ .

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stable Solutions to the Static Choquard Equation

2020

Bull. Aust. Math. Soc.

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“…As shown in other studies, 8,15,21 the main obstacle for such nonlocal problem (KC) is the lack of compactness. Motivated by the classical work 21 of Benci and Cerami, the main variational approach is to introduce a global compactness lemma and restore the compactness of Palais–Smale sequence when the functional energy lies in a suitable interval.…”

Section: Introductionmentioning

confidence: 87%

“…However, we remark that the researchers mentioned above have been mainly devoted to the question of existence of ground state solutions. Recently, Guo et al 8 studied the following problem:

- normalΔ u + V false(x false) u = false(I_{α} * false| u {false|}^{2_{α}^{*}} false) false| u {false|}^{2_{α}^{*} - 2} u, u \in D^{1,2} false(ℝ^{N} false),

and proved the existence of positive high‐energy solutions under the assumption of

false‖ V {false‖}_{L^{\frac{N}{2}}}

small. Moreover, if

V false(x false) = 0

and

2_{α}^{*} \geq 2

in Equation (), they also obtained the uniqueness of positive solutions.…”

Section: Introductionmentioning

confidence: 99%

Bound state solutions of Choquard equations with a nonlocal operator

Guo

2020

Math Methods in App Sciences

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In this paper, we study the following Choquard equation with Kirchhoff operator −()a+b∫ℝNfalse|∇ufalse|2normalΔu+Vfalse(xfalse)u=false(Iα*false|ufalse|2α*false)false|ufalse|2α*−2u,0.3emx∈ℝN, where a ≥ 0, b > 0, α ∈ (0, N), 2α*=N+αN−2 is the critical exponent respect to Hardy–Littlewood–Sobolev inequality, and Vfalse(xfalse)∈LN2false(ℝNfalse) is a given nonnegative function. By using the classical linking theorem and global compactness theorem, we prove that equation () has at least one bound state solution if false‖Vfalse‖LN2 is small. More intriguingly, our result covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero.

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“…Using the integral equation method, equation (2) with α = 2 and c 2 = 0 was recently investigated in [27]. Some classification results for nonnegative H α (R n ) solutions of (2) with 0 < α < n and c 2 = 0 were also established in [12,22,25,26,28,33,42] using the integral equation method.…”

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confidence: 99%

Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order

Le¹

2021

Discrete &Amp; Continuous Dynamical Systems - A

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We classify all nonnegative nontrivial classical solutions to the equation (−∆) α 2 u = c 1 1 |x| n−β * f (u) g(u) + c 2 h(u) in R n , where 0 < α, β < n, c 1 , c 2 ≥ 0, c 1 + c 2 > 0 and f, g, h ∈ C([0, +∞), [0, +∞)) are increasing functions such that f (t)/t n+β n−α , g(t)/t α+β n−α , h(t)/t n+α n−α are nonincreasing in (0, +∞). We also derive a Liouville type theorem for the equation in the case α ≥ n. The main tool we use is the method of moving spheres in integral forms.

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Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent

Cited by 50 publications

References 35 publications

Stable Solutions to the Static Choquard Equation

Stable Solutions to the Static Choquard Equation

Bound state solutions of Choquard equations with a nonlocal operator

Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order

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