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数学与物理学

一类具有-1幂指数的拟线性奇异偏微分方程

  • 张衡 ,
  • 孙义静
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  • 中国科学院大学数学科学学院, 北京 100049

收稿日期: 2022-11-11

  修回日期: 2023-02-06

  网络出版日期: 2023-02-06

基金资助

National Science Foundation of China (11971027,12171497)

A class of quasilinear equations with -1 powers

  • ZHANG Heng ,
  • SUN Yijing
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  • School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 2022-11-11

  Revised date: 2023-02-06

  Online published: 2023-02-06

Supported by

National Science Foundation of China (11971027,12171497)

摘要

研究一类具有奇异增长的拟线性椭圆方程-Δu-uΔ(u2)=a(x)u-1。对于一般的a(x)∈Lp(Ω),p>2,证明了该方程正解的存在性,其中Ω为ℝN中的有界区域且N≥1。

本文引用格式

张衡 , 孙义静 . 一类具有-1幂指数的拟线性奇异偏微分方程[J]. 中国科学院大学学报, 2025 , 42(1) : 13 -19 . DOI: 10.7523/j.ucas.2023.006

Abstract

This paper deals with quasilinear elliptic equations of singular growth like -Δu-uΔ(u2)=a(x)u-1. We establish the existence of positive solutions for general a(x)∈Lp(Ω),p>2, where Ω is a bounded domain in ℝN with N≥1.

参考文献

[1] Kurihara S. Large-amplitude quasi-solitons in superfluid films [J]. Journal of the Physical Society of Japan, 1981, 50(10):3262-3267. DOI: 10.1143/jpsj.50.3262.
[2] Hartmann B, Zakrzewski W J. Electrons on hexagonal lattices and applications to nanotubes [J]. Physical Review B, 2003, 68(18):184302. DOI: 10.1103/physrevb.68.184302.
[3] Kosevich A M, Ivanov B A, Kovalev A S. Magnetic solitons [J]. Physics Reports, 1990, 194(3/4):117-238. DOI: 10.1016/0370-1573(90)90130-T.
[4] Makhankov V G, Fedyanin V K. Non-linear effects in quasi-one-dimensional models of condensed matter theory [J]. Physics Reports, 1984, 104(1): 1-86. DOI: 10.1016/0370-1573(84)90106-6.
[5] Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations, Ⅱ [J]. Journal of Differential Equations, 2003, 187(2):473-493. DOI: 10.1016/s0022-0396(02)00064-5.
[6] Fang X D, Szulkin A. Multiple solutions for a quasilinear Schrödinger equation [J]. Physics Reports, 2013, 254(4):2015-2032. DOI: 10.1016/j.jde.2012.11.017.
[7] Liu X Q, Liu J Q, Wang Z Q. Quasilinear elliptic equations via perturbation method [J]. Proceedings of the American Mathematical Society, 2013, 141(1):253-263. DOI: 10.1090/s0002-9939-2012-11293-6.
[8] Agarwal R P, O’Regan D. Singular differential and integral equations with applications [M]. Dordretht, Netherlands: Springer Science & Business Media, 2003. DOI: 10.1007/978-94-017-3004-4.
[9] Hernández J, Mancebo F J. Singular elliptic and parabolic equations [M]//Handbook of differential equations: stationary partial differential equations, Amsterdam; Boston: Elsevier/North Holland. 2006, 3:317-400. DOI: 10.1016/s1874-5733(06)80008-2.
[10] Marcos do Ó J, Moameni A. Solutions for singular quasilinear Schrödinger equations with one parameter[J]. Communications on Pure & Applied Analysis, 2010, 9(4): 1011-1023. DOI: 10.3934/cpaa.2010.9.1011.
[11] Santos C A, Yang M B, Zhou J Z. Global multiplicity of solutions for a modified elliptic problem with singular terms[J]. Nonlinearity, 2021, 34(11):7842-7871. DOI: 10.1088/1361-6544/ac2a50.
[12] Alves R L, Reis M. About existence and regularity of positive solutions for a quasilinear Schrödinger equation with singular nonlinearity [J]. Electronic Journal of Qualitative Theory of Differential Equations, 2020, 60:1-23. DOI: 10.14232/ejqtde.2020.1.60.
[13] Sun Y J. Compatibility phenomena in singular problems [J]. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2013, 143(6):1321-1330. DOI: 10.1017/s030821051100117x.
[14] Alves R L. Existence, nonexistence and multiplicity of positive solutions for singular quasilinear problems [J]. Electronic Journal of Qualitative Theory of Differential Equations, 2022, 13:1-29. DOI: 10.14232/ejqtde.2022.1.13.
[15] Liu J Y, Liu D C, Zhao P H. Soliton solutions for a singular Schrödinger equation with any growth exponents [J]. Acta Applicandae Mathematicae, 2017, 148(1):179-199. DOI: 10.1007/s10440-016-0084-z.
[16] do Ó J M B, Miyagaki O H, Soares S H M. Soliton solutions for quasilinear Schrödinger equations: the critical exponential case [J]. Nonlinear Analysis: Theory, Methods & Applications, 2007, 67(12):3357-3372. DOI: 10.1016/j.na.2006.10.018.
[17] do Ó J M B, Miyagaki O H, Soares S H M. Soliton solutions for quasilinear Schrödinger equations with critical growth [J]. Journal of Differential Equations, 2010, 248(4):722-744. DOI: 10.1016/j.jde.2009.11.030.
[18] Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach [J]. Nonlinear Analysis: Theory, Methods & Applications, 2004, 56(2):213-226. DOI: 10.1016/j.na.2003.09.008.
[19] Lazer A C, McKenna P J. On a singular nonlinear elliptic boundary-value problem [J]. Proceedings of the American Mathematical Society, 1991, 111(3):721. DOI: 10. 1090/s0002-9939-1991-1037213-9.
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