Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 6, 2016

Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Yutian Lei EMAIL logo
From the journal Forum Mathematicum


In this paper, we are concerned with the following quasilinear PDE with a weight:

-divA(x,u)=|x|auq(x),u>0in n,

where n1, q>p-1 with p(1,2] and a0. The positive weak solution u of the quasilinear PDE is 𝒜-superharmonic. We also consider an integral equation involving the Wolff potential

u(x)=R(x)Wβ,p(|y|auq(y))(x),u>0in n,

which the positive solution u of the quasilinear PDE satisfies. Here β>0 and pβ<n. When -a>pβ or 0<q(n+a)(p-1)n-pβ, there does not exist any positive solution to this integral equation. On the other hand, when 0-a<pβ and q>(n+a)(p-1)n-pβ, the positive solution u of the integral equation is bounded and decays with the fast rate n-pβp-1 if and only if it is integrable (i.e., it belongs to Ln(q-p+1)pβ+a(n)). However, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one pβ+aq-p+1. In addition, we also discuss the case of -a=pβ. Thus, all the properties above are still true for the quasilinear PDE.

Communicated by Christopher D. Sogge

Award Identifier / Grant number: 11471164

Funding statement: The research was supported by NSF of China (No. 11471164), and PAPD of Jiangsu Higher Education Institutions.


[1] M. Badiale and G. Tarantello, A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), 259–293. 10.1007/s002050200201Search in Google Scholar

[2] M. F. Bidaut-Veron, Local and global behavior of solutions of quasilinear equations of Emden–Fowler type, Arch. Ration. Mech. Anal. 107 (1989), 293–324. 10.1007/BF00251552Search in Google Scholar

[3] M. F. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden–Fowler equations and systems, Adv. Differential Equations 15 (2010), 1033–1082. 10.57262/ade/1355854434Search in Google Scholar

[4] M. F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math. 84 (2001), 1–49. 10.1007/BF02788105Search in Google Scholar

[5] M. F. Bidaut-Veron and L. Veron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), 489–539. 10.1007/BF01243922Search in Google Scholar

[6] J. Byeon and Z. Wang, On the Henon equation: Asymptotic profile of ground states. II, J. Differential Equations 216 (2005), 78–108. 10.1016/j.jde.2005.02.018Search in Google Scholar

[7] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271–297. 10.1002/cpa.3160420304Search in Google Scholar

[8] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259–275. Search in Google Scholar

[9] M. Calanchi and B. Ruf, Radial and non radial solutions for Hardy–Henon type elliptic systems, Calc. Var. Partial Differential Equations 38 (2010), 111–133. 10.1007/s00526-009-0280-zSearch in Google Scholar

[10] G. Caristi, L. D’Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math. 76 (2008), 27–67. 10.1007/s00032-008-0090-3Search in Google Scholar

[11] C. Cascante, J. Ortega and I. Verbitsky, Wolff’s inequality for radially nonincreasing Kernels and applications to trace inequalities, Potential Anal. 16 (2002), 347–372. 10.1023/A:1014845728367Search in Google Scholar

[12] F. Catrina and Z. Wang, On the Caffarelli–Kohn–Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229–258. 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-ISearch in Google Scholar

[13] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622. 10.1215/S0012-7094-91-06325-8Search in Google Scholar

[14] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst. 30 (2011), 1083–1093. 10.3934/dcds.2011.30.1083Search in Google Scholar

[15] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330–343. 10.1002/cpa.20116Search in Google Scholar

[16] A. Farina, On the classification of solutions of the Lane–Emden equation on unbounded domains of RN, J. Math. Pures Appl. (9) 87 (2007), 537–561. 10.1016/j.matpur.2007.03.001Search in Google Scholar

[17] M. Franca, Classification of positive solutions of p-Laplace equation with a growth term, Arch. Math. (Brno) 40 (2004), 415–434. Search in Google Scholar

[18] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161–187. 10.5802/aif.944Search in Google Scholar

[19] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations 26 (2006), 447–457. 10.1007/s00526-006-0013-5Search in Google Scholar

[20] R. Johnson, X. Pan and Y. Yi, Positive solutions of super-critical elliptic equations and asymptotics, Comm. Partial Differential Equations 18 (1993), 977–1019. 10.1080/03605309308820958Search in Google Scholar

[21] N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div(|Du|m-2Du)+K(|x|)uq=0 in n, J. Math. Soc. Japan 45 (1993), 719–742. 10.2969/jmsj/04540719Search in Google Scholar

[22] T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161. 10.1007/BF02392793Search in Google Scholar

[23] D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J. 111 (2002), 1–49. 10.1215/S0012-7094-02-11111-9Search in Google Scholar

[24] Y. Lei, Asymptotic properties of positive solutions of the Hardy–Sobolev type equations, J. Differential Equations 254 (2013), 1774–1799. 10.1016/j.jde.2012.11.008Search in Google Scholar

[25] Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a γ-Laplace system, J. Differential Equations 252 (2012), 2739–2758. 10.1016/j.jde.2011.10.009Search in Google Scholar

[26] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy–Littlewood–Sobolev system, Calc. Var. Partial Differential Equations 45 (2012), 43–61. 10.1007/s00526-011-0450-7Search in Google Scholar

[27] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math. 123 (1996), 221–231. 10.1007/s002220050023Search in Google Scholar

[28] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS) 6 (2004), 153–180. 10.4171/JEMS/6Search in Google Scholar

[29] Y. Li and W.-M. Ni, On conformal scalar curvature equations in n, Duke Math. J. 57 (1988), 895–924. 10.1215/S0012-7094-88-05740-7Search in Google Scholar

[30] E. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349–374. 10.1007/978-3-642-55925-9_43Search in Google Scholar

[31] C. Lin, Interpolation inequalities with weights, Comm. Partial Differential Equations 11 (1986), 1515–1538. 10.1080/03605308608820473Search in Google Scholar

[32] G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality, Calc. Var. Partial Differential Equations 42 (2011), 563–577. 10.1007/s00526-011-0398-7Search in Google Scholar

[33] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math. 226 (2011), 2676–2699. 10.1016/j.aim.2010.07.020Search in Google Scholar

[34] G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy–Sobolev operator, J. Differential Equations 224 (2006), 258–276. 10.1016/j.jde.2005.07.001Search in Google Scholar

[35] G. Mancini and K. Sandeep, Cylindrical symmetry of extremals of a Hardy–Sobolev inequality, Ann. Mat. Pura Appl. (4) 183 (2004), 165–172. 10.1007/s10231-003-0084-2Search in Google Scholar

[36] G. Mingione, Gradient potential estimates, J. Eur. Math. Soc. (JEMS) 13 (2011), 459–486. 10.4171/JEMS/258Search in Google Scholar

[37] E. Mitidieri and S. Pohozaev, A priori estimates and blow-up solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1–375. Search in Google Scholar

[38] E. Mitidieri and S. Pohozaev, Liouville Theorems for some classes of nonlinear non-local problems, Proc. Steklov Inst. Math. 248 (2005), 164–185. Search in Google Scholar

[39] I. Peral and J. Vazquez, On the stability or insatility of the singular solutions with exponential reaction term, Arch. Ration. Mech. Anal. 129 (1995), 201–224. 10.1007/BF00383673Search in Google Scholar

[40] Q. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy–Henon equations, J. Differential Equations 252 (2012), 2544–2562. 10.1016/j.jde.2011.09.022Search in Google Scholar

[41] N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane–Emden type, Ann. of Math. (2) 168 (2008), 859–914. 10.4007/annals.2008.168.859Search in Google Scholar

[42] P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I: Elliptic equations and systems, Duke Math. J. 139 (2007), 555–579. 10.1215/S0012-7094-07-13935-8Search in Google Scholar

[43] S. Secchi, D. Smets and M. Willem, Remarks on a Hardy–Sobolev inequality, C. R. Math. Acad. Sci. Paris 336 (2003), 811–815. 10.1016/S1631-073X(03)00202-4Search in Google Scholar

[44] J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math. 113 (1965), 219–240. 10.1007/BF02391778Search in Google Scholar

[45] J. Serrin and H. Zou, Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), 79–142. 10.1007/BF02392645Search in Google Scholar

[46] S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane–Emden type integral systems involving the Wolff potentials, J. Funct. Anal. 263 (2012), 3857–3882. 10.1016/j.jfa.2012.09.012Search in Google Scholar

Received: 2014-10-2
Revised: 2016-9-5
Published Online: 2016-11-6
Published in Print: 2017-9-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 30.1.2023 from
Scroll Up Arrow