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Positive solutions to involving Wolff potentials
1. | School of Mathematical Sciences, Shandong Normal University, Jinan 250014 |
2. | School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116 |
References:
[1] |
C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalityes, Potential Anal., 16 (2002), 347-372.
doi: 10.1023/A:1014845728367. |
[2] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093.
doi: 10.3934/dcds.2011.30.1083. |
[3] |
W. Chen and C. Li, Regularity of solutions for a system of intgral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. |
[4] |
W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
[5] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[6] |
L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. |
[7] |
X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potentials on bounded domains, Nonlinear Anal., 75 (2012), 5601-5611.
doi: 10.1016/j.na.2012.05.007. |
[8] |
X. Huang, D. Li and L. Wang, Symmetry and monotonicity of integral equation systems, Nonlinear Anal., 12 (2011), 3515-3530.
doi: 10.1016/j.nonrwa.2011.06.012. |
[9] |
X. Huang, D. Li and L. Wang, Radial symmetry results for systems of integral equations on $\Omega\in R^n$, Manuscripta Math., 137 (2012), 317-330.
doi: 10.1007/s00229-011-0465-6. |
[10] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: 10.1090/S0002-9939-05-08411-X. |
[11] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[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. |
[13] |
T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (1992), 591-613. |
[14] |
D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.
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), 387-402.
doi: 10.1007/s11118-010-9218-5. |
[16] |
Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation, Disc. Cont. Dyn. Sys., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[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. |
[18] |
Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Commun. Pure Appl. Anal., 10 (2011) 193-207.
doi: 10.3934/cpaa.2011.10.193. |
[19] |
C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[20] |
S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal.: Theory, Methods Applications, 71 (2009),1796-1806.
doi: 10.1016/j.na.2009.01.014. |
[21] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[22] |
J. Maly, Wolff potential estimates of superminnimizers of Orilicz type Dirichlet integrals, Manuscripta Math., 110 (2003), 513-525.
doi: 10.1007/s00229-003-0358-4. |
[23] |
N. Pfuc 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. |
[24] |
E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[25] |
Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999.
doi: 10.1016/j.na.2011.09.051. |
show all references
References:
[1] |
C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalityes, Potential Anal., 16 (2002), 347-372.
doi: 10.1023/A:1014845728367. |
[2] |
W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093.
doi: 10.3934/dcds.2011.30.1083. |
[3] |
W. Chen and C. Li, Regularity of solutions for a system of intgral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. |
[4] |
W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: 10.1090/S0002-9939-07-09232-5. |
[5] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[6] |
L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. |
[7] |
X. Huang, G. Hong and D. Li, Some symmetry results for integral equations involving Wolff potentials on bounded domains, Nonlinear Anal., 75 (2012), 5601-5611.
doi: 10.1016/j.na.2012.05.007. |
[8] |
X. Huang, D. Li and L. Wang, Symmetry and monotonicity of integral equation systems, Nonlinear Anal., 12 (2011), 3515-3530.
doi: 10.1016/j.nonrwa.2011.06.012. |
[9] |
X. Huang, D. Li and L. Wang, Radial symmetry results for systems of integral equations on $\Omega\in R^n$, Manuscripta Math., 137 (2012), 317-330.
doi: 10.1007/s00229-011-0465-6. |
[10] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: 10.1090/S0002-9939-05-08411-X. |
[11] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.
doi: 10.1007/s00526-006-0013-5. |
[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. |
[13] |
T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (1992), 591-613. |
[14] |
D. Labutin, Potential eatimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49.
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), 387-402.
doi: 10.1007/s11118-010-9218-5. |
[16] |
Y. Lei, C. Li and C. Ma, Decay estimation for positve solutions of a $\gamma$-Laplace equation, Disc. Cont. Dyn. Sys., 30 (2011), 547-558.
doi: 10.3934/dcds.2011.30.547. |
[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. |
[18] |
Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Commun. Pure Appl. Anal., 10 (2011) 193-207.
doi: 10.3934/cpaa.2011.10.193. |
[19] |
C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[20] |
S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal.: Theory, Methods Applications, 71 (2009),1796-1806.
doi: 10.1016/j.na.2009.01.014. |
[21] |
C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[22] |
J. Maly, Wolff potential estimates of superminnimizers of Orilicz type Dirichlet integrals, Manuscripta Math., 110 (2003), 513-525.
doi: 10.1007/s00229-003-0358-4. |
[23] |
N. Pfuc 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. |
[24] |
E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[25] |
Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999.
doi: 10.1016/j.na.2011.09.051. |
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