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Boundedness in a class of duffing equations with oscillating potentials via the twist theorem
Asymptotic behavior for solutions of some integral equations
1. | School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, China |
2. | Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309, United States |
$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,
$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy. $
We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.
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: doi:10.1002/cpa.3160420304. |
[2] |
W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Disc. & Cont. Dynamics Sys. S, (2005), 164-173. |
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: doi:10.1215/S0012-7094-91-06325-8. |
[4] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
doi: doi:10.2307/2951844. |
[5] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8. |
[6] |
W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: doi:10.1090/S0002-9939-07-09232-5. |
[7] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. & Cont. Dynamics Sys., 24 (2009), 1167-1184. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343.
doi: doi:10.1002/cpa.20116. |
[9] |
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: doi:10.1081/PDE-200044445. |
[10] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354. |
[11] |
A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12. |
[12] |
L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge Unversity Press, New York, 2000. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, collected in the book "Mathematical Analysis and Applications," which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. |
[14] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: doi:10.1090/S0002-9939-05-08411-X. |
[15] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. |
[16] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. |
[17] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.
doi: doi:10.3934/cpaa.2007.6.453. |
[18] |
C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: doi:10.1137/080712301. |
[19] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: doi:10.2307/2007032. |
[20] |
E. Lieb and M. Loss, "Analysis," 2nd edition, American Mathematical Society, Rhode Island, 2001. |
[21] |
C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 8 (2009), 1925-1932.
doi: doi:10.3934/cpaa.2009.8.1925. |
[22] |
L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.
doi: doi:10.3934/cpaa.2006.5.855. |
[23] |
B. Ou, A Remark on a singular integral equation, Houston J. of Math., 25 (1999), 181-184. |
[24] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
doi: doi:10.1007/BF00250468. |
[25] |
E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton University Press, Princeton, 1971. |
[26] |
E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[27] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: doi:10.1007/s002080050258. |
show all references
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: doi:10.1002/cpa.3160420304. |
[2] |
W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Disc. & Cont. Dynamics Sys. S, (2005), 164-173. |
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: doi:10.1215/S0012-7094-91-06325-8. |
[4] |
W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
doi: doi:10.2307/2951844. |
[5] |
W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure and Appl. Anal., 4 (2005), 1-8. |
[6] |
W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.
doi: doi:10.1090/S0002-9939-07-09232-5. |
[7] |
W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. & Cont. Dynamics Sys., 24 (2009), 1167-1184. |
[8] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure and Appl. Math., 59 (2006), 330-343.
doi: doi:10.1002/cpa.20116. |
[9] |
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: doi:10.1081/PDE-200044445. |
[10] |
W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. & Cont. Dynamics Sys., 12 (2005), 347-354. |
[11] |
A. Chang and P. Yang, On uniqueness of an n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12. |
[12] |
L. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge Unversity Press, New York, 2000. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, collected in the book "Mathematical Analysis and Applications," which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. |
[14] |
C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
doi: doi:10.1090/S0002-9939-05-08411-X. |
[15] |
C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. PDEs, 26 (2006), 447-457. |
[16] |
C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. |
[17] |
C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.
doi: doi:10.3934/cpaa.2007.6.453. |
[18] |
C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.
doi: doi:10.1137/080712301. |
[19] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: doi:10.2307/2007032. |
[20] |
E. Lieb and M. Loss, "Analysis," 2nd edition, American Mathematical Society, Rhode Island, 2001. |
[21] |
C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 8 (2009), 1925-1932.
doi: doi:10.3934/cpaa.2009.8.1925. |
[22] |
L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.
doi: doi:10.3934/cpaa.2006.5.855. |
[23] |
B. Ou, A Remark on a singular integral equation, Houston J. of Math., 25 (1999), 181-184. |
[24] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
doi: doi:10.1007/BF00250468. |
[25] |
E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton University Press, Princeton, 1971. |
[26] |
E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514. |
[27] |
J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: doi:10.1007/s002080050258. |
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