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May  2015, 35(5): 2067-2078. doi: 10.3934/dcds.2015.35.2067

Wolff type potential estimates and application to nonlinear equations with negative exponents

Yutian Lei 1,

1. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023

Received  December 2013 Revised  September 2014 Published  December 2014

In this paper, we are concerned with the positive continuous entire solutions of the Wolff type integral equation $$ u(x)=c(x)W_{\beta,\gamma}(u^{-p})(x), \quad u>0 ~in~ R^n, $$ where $n \geq 1$, $p>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma \neq n$. In addition, $c(x)$ is a double bounded function. Such an integral equation is related to the study of the conformal geometry and nonlinear PDEs, such as $\gamma$-Laplace equations and $k$-Hessian equations with negative exponents. By some Wolff type potential integral estimates, we obtain the asymptotic rates and the integrability of positive solutions, and discuss the existence and nonexistence results of the radial solutions.
Keywords: asymptotic behavior., Wolff potential, $\gamma$-Laplace equation, $k$-Hessian equation, conformal geometry.
Mathematics Subject Classification: 35J60, 35J92, 45G05, 45M0.
Citation: Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067
References:
[1]

C. Caseante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities, Potential Anal., 16 (2002), 347-372. doi: 10.1023/A:1014845728367.  Google Scholar

[2]

H. Chen and Z. Lü, The properties of positive solutions to an integral system involving Wolff potential, Discrete Contin. Dyn. Syst., 34 (2014), 1879-1904. doi: 10.3934/dcds.2014.34.1879.  Google Scholar

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

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W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

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Y. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234. doi: 10.1016/j.jde.2008.06.027.  Google Scholar

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B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[8]

Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580. doi: 10.3934/dcds.2014.34.2561.  Google Scholar

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L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187. doi: 10.5802/aif.944.  Google Scholar

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T. Kilpelaiinen, T. Kuusi and A. Tuhola-Kujanpaa, Superharmonic functions are locally renormalized solutions, Ann. Inst. H. Poincare Analyse Non Lineaire, 28 (2011), 775-795. doi: 10.1016/j.anihpc.2011.03.004.  Google Scholar

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T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Seuola Norm. Sup. Pisa, Cl. Sci., 19 (1992), 591-613.  Google Scholar

[12]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

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N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(|Du|^{m-2} Du)+K(|x|)u^q=0$ in $R^n$, J. Math. Soc. Japan, 45 (1993), 719-742. doi: 10.2969/jmsj/04540719.  Google Scholar

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

[15]

Y. Lei, Decay rates for solutions of an integral system of Wolff type, Potential Anal., 35 (2011), 387-402. doi: 10.1007/s11118-010-9218-5.  Google Scholar

[16]

Y. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst., 35 (2015), 1039-1057. doi: 10.3934/dcds.2015.35.1039.  Google Scholar

[17]

Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758. doi: 10.1016/j.jde.2011.10.009.  Google Scholar

[18]

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. doi: 10.1007/s00526-011-0450-7.  Google Scholar

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

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E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[21]

T. Lukkari, F.-Y. Maeda and N. Marola, Wolff potential estimates for elliptic equations with nonstandard growth and applications, Forum. Math., 22 (2010), 1061-1087. doi: 10.1515/forum.2010.057.  Google Scholar

[22]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[23]

J. Maly, Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals, Manuscripta Math., 110 (2003), 513-525. doi: 10.1007/s00229-003-0358-4.  Google Scholar

[24]

G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486. doi: 10.4171/JEMS/258.  Google Scholar

[25]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.  Google Scholar

[26]

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. doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[27]

X. Xu, Exact solution of nonlinear conformally invarient integral equations in $R^3$, Adv. Math., 194 (2005), 485-503. doi: 10.1016/j.aim.2004.07.004.  Google Scholar

[28]

X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005.  Google Scholar

[29]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

show all references

References:
[1]

C. Caseante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities, Potential Anal., 16 (2002), 347-372. doi: 10.1023/A:1014845728367.  Google Scholar

[2]

H. Chen and Z. Lü, The properties of positive solutions to an integral system involving Wolff potential, Discrete Contin. Dyn. Syst., 34 (2014), 1879-1904. doi: 10.3934/dcds.2014.34.1879.  Google Scholar

[3]

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

[4]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[6]

Y. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234. doi: 10.1016/j.jde.2008.06.027.  Google Scholar

[7]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[8]

Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580. doi: 10.3934/dcds.2014.34.2561.  Google Scholar

[9]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187. doi: 10.5802/aif.944.  Google Scholar

[10]

T. Kilpelaiinen, T. Kuusi and A. Tuhola-Kujanpaa, Superharmonic functions are locally renormalized solutions, Ann. Inst. H. Poincare Analyse Non Lineaire, 28 (2011), 775-795. doi: 10.1016/j.anihpc.2011.03.004.  Google Scholar

[11]

T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Seuola Norm. Sup. Pisa, Cl. Sci., 19 (1992), 591-613.  Google Scholar

[12]

T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[13]

N. Kawano, E. Yanagida and S. Yotsutani, Structure theorems for positive radial solutions to div$(|Du|^{m-2} Du)+K(|x|)u^q=0$ in $R^n$, J. Math. Soc. Japan, 45 (1993), 719-742. doi: 10.2969/jmsj/04540719.  Google Scholar

[14]

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

[15]

Y. Lei, Decay rates for solutions of an integral system of Wolff type, Potential Anal., 35 (2011), 387-402. doi: 10.1007/s11118-010-9218-5.  Google Scholar

[16]

Y. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst., 35 (2015), 1039-1057. doi: 10.3934/dcds.2015.35.1039.  Google Scholar

[17]

Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system, J. Differential Equations, 252 (2012), 2739-2758. doi: 10.1016/j.jde.2011.10.009.  Google Scholar

[18]

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. doi: 10.1007/s00526-011-0450-7.  Google Scholar

[19]

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

[20]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[21]

T. Lukkari, F.-Y. Maeda and N. Marola, Wolff potential estimates for elliptic equations with nonstandard growth and applications, Forum. Math., 22 (2010), 1061-1087. doi: 10.1515/forum.2010.057.  Google Scholar

[22]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[23]

J. Maly, Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals, Manuscripta Math., 110 (2003), 513-525. doi: 10.1007/s00229-003-0358-4.  Google Scholar

[24]

G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486. doi: 10.4171/JEMS/258.  Google Scholar

[25]

N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.  Google Scholar

[26]

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. doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[27]

X. Xu, Exact solution of nonlinear conformally invarient integral equations in $R^3$, Adv. Math., 194 (2005), 485-503. doi: 10.1016/j.aim.2004.07.004.  Google Scholar

[28]

X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005.  Google Scholar

[29]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

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