# American Institute of Mathematical Sciences

May  2011, 30(2): 547-558. doi: 10.3934/dcds.2011.30.547

## Decay estimation for positive solutions of a $\gamma$-Laplace equation

 1 School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097 2 Department of Applied Mathematics, University of Colorado at Boulder 3 Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309

Received  June 2010 Published  February 2011

In this paper, we study the properties of the positive solutions of a $\gamma$-Laplace equation in $R^n$

-div$(|\nabla u|^{\gamma-2}\nabla u) =K u^p$,

Here $1<\gamma<2$, $n>\gamma$, $p=\frac{(\gamma-1)(n+\gamma)}{n-\gamma}$ and $K(x)$ is a smooth function bounded by two positive constants. First, the positive solution $u$ of the $\gamma$-Laplace equation above satisfies an integral equation involving a Wolff potential. Based on this, we estimate the decay rate of the positive solutions of the $\gamma$-Laplace equation at infinity. A new method is introduced to fully explore the integrability result established recently by Ma, Chen and Li on Wolff type integral equations to derive the decay estimate.

Citation: Yutian Lei, Congming Li, Chao Ma. Decay estimation for positive solutions of a $\gamma$-Laplace equation. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 547-558. doi: 10.3934/dcds.2011.30.547
##### References:
 [1] 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. doi: 10.1002/cpa.3160420304. [2] 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. [3] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: 10.2307/2951844. [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, Radial symmetry of solutions for some integral systems of Wolff type, preprint, 2009. [6] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [7] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [8] C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities, Potential Analysis, 16 (2002), 347-372. doi: 10.1023/A:1014845728367. [9] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 91-102. [10] S. Ding, On some imbedding theorems, Sci. Sinica, 21 (1978), 287-297. [11] L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,'' Cambridge Unversity Press, New York, 2000. doi: 10.1017/CBO9780511569203. [12] M. Franca, Classification of positive solutions of p-Laplace equation with a growth term, Archivum Mathematicum, 40 (2004), 415-434. [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in "Mathematical Analysis and Applications,'' vol. 7a, "Advances in Mathematics. Supplementary Studies,'' Academic Press, New York, 1981. [14] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187. [15] 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. [16] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. [17] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 19 (1992), 591-613. [18] 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. [19] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. [20] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [21] C. Li and L. Ma, Uniqueness of positive bound states to Schrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [22] Y. Li, Remark on some conformally invariant integral equations: the method of moving planes, Journal of European Mathematical Society, 6 (2004), 153-180. doi: 10.4171/JEMS/6. [23] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [24] 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. [25] 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. [26] 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. [27] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: 10.1007/BF00250468. [28] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645. [29] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.

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##### References:
 [1] 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. doi: 10.1002/cpa.3160420304. [2] 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. [3] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. doi: 10.2307/2951844. [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, Radial symmetry of solutions for some integral systems of Wolff type, preprint, 2009. [6] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. in Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [7] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [8] C. Cascante, J. Ortega and I. Verbitsky, Wolff's inequality for radially nonincreasing kernels and applications to trace inequalities, Potential Analysis, 16 (2002), 347-372. doi: 10.1023/A:1014845728367. [9] A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 91-102. [10] S. Ding, On some imbedding theorems, Sci. Sinica, 21 (1978), 287-297. [11] L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,'' Cambridge Unversity Press, New York, 2000. doi: 10.1017/CBO9780511569203. [12] M. Franca, Classification of positive solutions of p-Laplace equation with a growth term, Archivum Mathematicum, 40 (2004), 415-434. [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in "Mathematical Analysis and Applications,'' vol. 7a, "Advances in Mathematics. Supplementary Studies,'' Academic Press, New York, 1981. [14] L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187. [15] 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. [16] C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. [17] T. Kilpelaiinen and J. Maly, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 19 (1992), 591-613. [18] 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. [19] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. [20] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [21] C. Li and L. Ma, Uniqueness of positive bound states to Schrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [22] Y. Li, Remark on some conformally invariant integral equations: the method of moving planes, Journal of European Mathematical Society, 6 (2004), 153-180. doi: 10.4171/JEMS/6. [23] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [24] 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. [25] 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. [26] 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. [27] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: 10.1007/BF00250468. [28] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645. [29] E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
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