Doubly nonlocal system with Hardy–Littlewood–Sobolev critical nonlinearity

Giacomoni, Jacques; Mukherjee, Tuhina; Sreenadh, K.

doi:10.1016/j.jmaa.2018.07.035

Cited by 41 publications

(18 citation statements)

References 33 publications

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“…Equations of type (1.2) have been extensively studied, see e.g. [3,15,16,18,20,27,[34][35][36]43] for the study of Choquard-type equations. In the fractional Laplacian framework, we refer to the recent papers [32,40,45].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiple solutions for critical Choquard-Kirchhoff type equations

Liang

Pucci

Zhang

2020

Advances in Nonlinear Analysis

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In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiple solutions for critical Choquard-Kirchhoff type equations

Liang

Pucci

Zhang

2020

Advances in Nonlinear Analysis

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“…Let η ∈ C ∞ c (Ω) be such that η = 1 in B δ (0), η = 0 in B c 2δ (0), and 0 ≤ η ≤ 1 in Ω. Set u ǫ = ηU ǫ , then we have the following estimates Lemma 5.3 (see [19,38]) The following hold true…”

Section: Consider the Family Of Functionsmentioning

confidence: 99%

“…For δ > 0, sufficiently small such that B 2δ (0) ⋐ Ω, 2δ < δ 1 and 0 < ǫ < δ/2, we have the following estimate ([19])…”

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confidence: 99%

Unbalanced (p,2)-fractional problems with critical growth

Kumar

Sreenadh

2021

Journal of Mathematical Analysis and Applications

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We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem:where Ω ⊂ R n is a bounded domain with C 2 boundary ∂Ω, 0 < s 2 < s 1 < 1, n > 2s 1 , 1 < q < p < 2, 1 < r ≤ 2 * µ with 2 * µ = 2n−µ n−2s1 , λ, β > 0 and a ∈ L d d−q (Ω), for some q < d < 2 * s1 := 2n n−2s1 , is a sign changing function. We prove that each nonnegative weak solution of (P λ ) is bounded. Furthermore, we obtain some existence and multiplicity results using Nehari manifold method.

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“…We show that the system (1.1) has at least two positive solutions when the parameters λ, µ and weight functions f , g satisfied some certain conditions. It should be mentioned that in [8,9,10,15,22], some problems involving fractional Laplacian operator were investigated by the Nehari manifold and fibering method. We look for solutions of (1.1) in the Sobolev space From (2.2), we employ the following equivalent norm in X s 0 (Ω):…”

Section: Introductionmentioning

confidence: 99%