次临界情形下海森堡群上的有界区域上带权的非线性积分方程的正解的存在性
Existence of Positive Solutions to Nonlinear Integral Equations with Weights on the Bounded Domains of the Heisenberg Group inSubcritical Case
摘要: 本文主要研究一类海森堡群Hn上的有界区域上与精确的Hardy-Littlewood-Sobolev (下面简称HLS)不等式有关的带权的非线性积分方程：，其中q > 1, 0 < α < Q，0 < β < Q − α，Q = 2n + 2是Hn的齐次维数，λ ∈ R, Ω ⊂ Hn是一个光滑的有界域且G(ξ)是Ω¯ 中的非负连续函数。这里，我们将讨论次临界情形下该方程正解的存在性结果。
Abstract: This paper is devoted to a kind of nonlinear integral equations with weights related to the sharp Hardy-Littlewood-Sobolev (hereinafter referred to as HLS) inequality on the bounded domains of the Heisenberg group Hn:, where q > 1, 0 < α < Q, 0 < β < Q − α, Q = 2n + 2 is the homogeneous dimension of Hn, λ ∈ R, Ω ⊂ Hn is a smooth bounded domain and G(ξ) is nonnegative continuous in Ω¯ . In this paper, we will study the existence results of the positive solutions for the equation in subcritical case .
文章引用：陈佳妮. 次临界情形下海森堡群上的有界区域上带权的非线性积分方程的正解的存在性[J]. 应用数学进展, 2022, 11(4): 1764-1780. https://doi.org/10.12677/AAM.2022.114193

参考文献

[1] Chen, W. and Chen, X. (2014) Regularity of Positive Solutions for an Integral System on Heisenberg Group. Journal of Mathematical Study, 47, 208-220.
https://doi.org/10.4208/jms.v47n2.14.05
[2] Han, X., Lu, G. and Zhu, J. (2012) Hardy-Littlewood-Sobolev and Stein-Weiss Inequalities and Integral Systems on the Heisenberg Group. Nonlinear Analysis, 75, 4296-4314.
https://doi.org/10.1016/j.na.2012.03.017
[3] Folland, G.B. and Stein, E.M. (1974) Estimates for the ∂¯b Complex and Analysis on the Heisenberg Group. Communications on Pure and Applied Mathematics, 27, 429-522.
https://doi.org/10.1002/cpa.3160270403
[4] Dragomir, S. and Tomassini, G. (2006) Differential Geometry and Analysis on CR Manifolds. Birkh¨auser Boston, Inc., Boston, MA.
[5] Stein, E.M. and Weiss, G. (1958) Fractional Integrals in n-Dimensional Euclidean Space. Indi- ana University Mathematics Journal, 7, 503-514.
https://doi.org/10.1512/iumj.1958.7.57030
[6] Lieb, E.H. (1983) Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities. Annals of Mathematics, 118, 349-374.
https://doi.org/10.2307/2007032
[7] Jerison, D. and Lee, J. (1988) Extremals for the Sobolev Inequality on the Heisenberg Group and the CR Yamabe Problem. Journal of the American Mathematical Society, 1, 1-13.
https://doi.org/10.1090/S0894-0347-1988-0924699-9
[8] Frank, R.L. and Lieb, E.H. (2012) Sharp Constants in Several Inequalities on the Heisenberg Group. Annals of Mathematics, 176, 349-381.
https://doi.org/10.4007/annals.2012.176.1.6
[9] Garofalo, N. and Lanconelli, E. (1992) Existence and Nonexistence Results for Semilinear Equations on the Heisenberg Group. Indiana University Mathematics Journal, 41, 71-98.
https://doi.org/10.1512/iumj.1992.41.41005
[10] Niu, P. (1999) Nonexistence for Semilinear Equations and Systems in the Heisenberg Group. Journal of Mathematical Analysis and Applications, 240, 47-59.
https://doi.org/10.1006/jmaa.1999.6574
[11] Wang, W. (2001) Positive Solution of a Subelliptic Nonlinear Equation on the Heisenberg Group. Canadian Mathematical Bulletin, 44, 346-354.
https://doi.org/10.4153/CMB-2001-035-0
[12] Brezis, H. and Nirenberg, L. (1983) Positive Solutions of Nonlinear Elliptic Equations Involving Critical Sobolev Exponents. Communications on Pure and Applied Mathematics, 36, 437-477.
https://doi.org/10.1002/cpa.3160360405
[13] Dou, J. and Zhu, M. (2019) Nonlinear Integral Equations on Bounded Domains. Journal of Functional Analysis, 277, 111-134.
https://doi.org/10.1016/j.jfa.2018.05.020
[14] Bal, K. (2016) Uniqueness of a Positive Solution for a Quasilinear Elliptic Equation in Heisen- berg Group. Electronic Journal of Differential Equations, 2016, 1-7.
[15] Huang, L., Chen, J. and Rocha, E.M. (2015) Multiple Non-Negative Solutions to a Semilinear Equation on Heisenberg Group with Indefinite Nonlinearity. Boundary Value Problems, 2015, Article No. 165.
https://doi.org/10.1186/s13661-015-0428-z
[16] Han, Y. (2020) An Integral Type Brezis-Nirenberg Problem on the Heisenberg Group. Journal of Differential Equations, 269, 4544-4565.
https://doi.org/10.1016/j.jde.2020.03.032
[17] Folland, G.B. (1975) Subelliptic Estimates and Function Spaces on Nilpotent Lie Groups. Arkiv f¨or Matematik, 13, 161-207.
https://doi.org/10.1007/BF02386204
[18] Thangavelu, S. (1998) Harmonic Analysis on the Heisenberg Group. Birkh¨auser Boston, Inc., Boston, MA.
[19] Han, Y. and Zhu, M. (2016) Hardy-Littlewood-Sobolev Inequalities on Compact Riemannian Manifolds and Applications. Journal of Differential Equations, 260, 1-25.
https://doi.org/10.1016/j.jde.2015.06.032

为你推荐