Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Yutian Lei

doi:10.1515/forum-2014-0173

Published by De Gruyter November 6, 2016

Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Yutian Lei

From the journal Forum Mathematicum

https://doi.org/10.1515/forum-2014-0173

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Abstract

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

-div⁡A⁢(x,∇⁡u)=|x|a⁢uq⁢(x),u>0 in ⁢ℝn,

where n≥1, q>p-1 with p∈(1,2] and a≤0. 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|a⁢uq⁢(y))⁢(x),u>0 in ⁢ℝn,

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

Keywords: Wolff potential; decay rate; quasilinear equation; Hardy–Sobolev inequality

MSC 2010: 35B40; 35J62; 45G05; 45M05

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11471164

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

References

[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⁢(|D⁢u|m-2⁢D⁢u)+K⁢(|x|)⁢uq=0 in ℝn, 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 ℝn, 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

Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Abstract

References

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