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Wolff type potential estimates and application to nonlinear equations with negative exponents
1. | Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023 |
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.
doi: 10.1023/A:1014845728367. |
[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.
doi: 10.3934/dcds.2014.34.1879. |
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615.
doi: 10.1215/S0012-7094-91-06325-8. |
[4] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.
doi: 10.3934/dcds.2011.30.1083. |
[5] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.
doi: 10.1002/cpa.20116. |
[6] |
Y. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents,, J. Differential Equations, 246 (2009), 216.
doi: 10.1016/j.jde.2008.06.027. |
[7] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.
doi: 10.1002/cpa.3160340406. |
[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.
doi: 10.3934/dcds.2014.34.2561. |
[9] |
L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenobel), 33 (1983), 161.
doi: 10.5802/aif.944. |
[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.
doi: 10.1016/j.anihpc.2011.03.004. |
[11] |
T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, Ann. Seuola Norm. Sup. Pisa, 19 (1992), 591.
|
[12] |
T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.
doi: 10.1007/BF02392793. |
[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.
doi: 10.2969/jmsj/04540719. |
[14] |
D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.
doi: 10.1215/S0012-7094-02-11111-9. |
[15] |
Y. Lei, Decay rates for solutions of an integral system of Wolff type,, Potential Anal., 35 (2011), 387.
doi: 10.1007/s11118-010-9218-5. |
[16] |
Y. Lei, On the integral systems with negative exponents,, Discrete Contin. Dyn. Syst., 35 (2015), 1039.
doi: 10.3934/dcds.2015.35.1039. |
[17] |
Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system,, J. Differential Equations, 252 (2012), 2739.
doi: 10.1016/j.jde.2011.10.009. |
[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.
doi: 10.1007/s00526-011-0450-7. |
[19] |
Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.
doi: 10.4171/JEMS/6. |
[20] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.
doi: 10.2307/2007032. |
[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.
doi: 10.1515/forum.2010.057. |
[22] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676.
doi: 10.1016/j.aim.2010.07.020. |
[23] |
J. Maly, Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals,, Manuscripta Math., 110 (2003), 513.
doi: 10.1007/s00229-003-0358-4. |
[24] |
G. Mingione, Gradient potential estimates,, J. Eur. Math. Soc., 13 (2011), 459.
doi: 10.4171/JEMS/258. |
[25] |
N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.
doi: 10.4007/annals.2008.168.859. |
[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.
doi: 10.1016/j.jfa.2012.09.012. |
[27] |
X. Xu, Exact solution of nonlinear conformally invarient integral equations in $R^3$,, Adv. Math., 194 (2005), 485.
doi: 10.1016/j.aim.2004.07.004. |
[28] |
X. Xu, Uniqueness theorem for integral equations and its application,, J. Funct. Anal., 247 (2007), 95.
doi: 10.1016/j.jfa.2007.03.005. |
[29] |
X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var. Partial Differential Equations, 46 (2013), 75.
doi: 10.1007/s00526-011-0474-z. |
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.
doi: 10.1023/A:1014845728367. |
[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.
doi: 10.3934/dcds.2014.34.1879. |
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615.
doi: 10.1215/S0012-7094-91-06325-8. |
[4] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.
doi: 10.3934/dcds.2011.30.1083. |
[5] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.
doi: 10.1002/cpa.20116. |
[6] |
Y. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents,, J. Differential Equations, 246 (2009), 216.
doi: 10.1016/j.jde.2008.06.027. |
[7] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.
doi: 10.1002/cpa.3160340406. |
[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.
doi: 10.3934/dcds.2014.34.2561. |
[9] |
L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory,, Ann. Inst. Fourier (Grenobel), 33 (1983), 161.
doi: 10.5802/aif.944. |
[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.
doi: 10.1016/j.anihpc.2011.03.004. |
[11] |
T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials,, Ann. Seuola Norm. Sup. Pisa, 19 (1992), 591.
|
[12] |
T. Kilpelaiinen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.
doi: 10.1007/BF02392793. |
[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.
doi: 10.2969/jmsj/04540719. |
[14] |
D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.
doi: 10.1215/S0012-7094-02-11111-9. |
[15] |
Y. Lei, Decay rates for solutions of an integral system of Wolff type,, Potential Anal., 35 (2011), 387.
doi: 10.1007/s11118-010-9218-5. |
[16] |
Y. Lei, On the integral systems with negative exponents,, Discrete Contin. Dyn. Syst., 35 (2015), 1039.
doi: 10.3934/dcds.2015.35.1039. |
[17] |
Y. Lei and C. Li, Integrability and asymptotics of positive solutions of a $\gamma$-Laplace system,, J. Differential Equations, 252 (2012), 2739.
doi: 10.1016/j.jde.2011.10.009. |
[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.
doi: 10.1007/s00526-011-0450-7. |
[19] |
Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.
doi: 10.4171/JEMS/6. |
[20] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.
doi: 10.2307/2007032. |
[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.
doi: 10.1515/forum.2010.057. |
[22] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676.
doi: 10.1016/j.aim.2010.07.020. |
[23] |
J. Maly, Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals,, Manuscripta Math., 110 (2003), 513.
doi: 10.1007/s00229-003-0358-4. |
[24] |
G. Mingione, Gradient potential estimates,, J. Eur. Math. Soc., 13 (2011), 459.
doi: 10.4171/JEMS/258. |
[25] |
N. Phuc and I. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.
doi: 10.4007/annals.2008.168.859. |
[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.
doi: 10.1016/j.jfa.2012.09.012. |
[27] |
X. Xu, Exact solution of nonlinear conformally invarient integral equations in $R^3$,, Adv. Math., 194 (2005), 485.
doi: 10.1016/j.aim.2004.07.004. |
[28] |
X. Xu, Uniqueness theorem for integral equations and its application,, J. Funct. Anal., 247 (2007), 95.
doi: 10.1016/j.jfa.2007.03.005. |
[29] |
X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var. Partial Differential Equations, 46 (2013), 75.
doi: 10.1007/s00526-011-0474-z. |
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