# Existence to Fractional Critical Equation with Hardy-Littlewood-Sobolev Nonlinearities

Authors
• 1 Razi University, Kermanshah, Iran , Kermanshah (Iran)
• 2 Imam Khomeini International University, Qazvin, 34149-16818, Iran , Qazvin (Iran)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 01, 2021
Volume
41
Issue
4
Pages
1321–1332
Identifiers
DOI: 10.1007/s10473-021-0418-4
Source
Springer Nature
Keywords
In this paper, we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{c}} {{{\left( {a + b\iint_{{\mathbb{R}^N}} {\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N + ps}}}}\text{d}x\text{d}y}} \right)}^{p - 1}}}&{( - \Delta )_p^su + \lambda V(x){{\left| u \right|}^{p - 2}}u} \\ { = \left( {\int\limits_{{\mathbb{R}^N}} {\frac{{{{\left| u \right|}^{p_{\mu ,s}^*}}}}{{{{\left| {x - y} \right|}^\mu }}}\text{d}y} } \right){{\left| u \right|}^{p_{\mu ,{s^{ - 2}}}^*}}u,\;x \in {\mathbb{R}^N}}&\; \end{array}\;$$\end{document} where (−Δ)sp is the fractional p-Laplacian with 0 < s < 1 < p, 0 < μ < N, N > ps, a, b > 0, λ > 0 is a parameter, V: ℝN → ℝ+ is a potential function, θ ∈ [1, 2μ,s*) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{p_{\mu,s}^ * = {{pN - p{\mu \over 2}} \over {N - ps}}}$$\end{document} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.