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August  2017, 22(6): 2321-2338. doi: 10.3934/dcdsb.2017099

## Asymptotic behaviors of Green-Sch potentials at infinity and its applications

 School of of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China

Received  June 2016 Revised  November 2016 Published  March 2017

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant Nos. 11301140, U1304102).

The first aim in this paper is to deal with asymptotic behaviors of Green-Sch potentials in a cylinder. As an application we prove the integral representation of nonnegative weak solutions of the stationary Schrödinger equation in a cylinder. Next we give asymptotic behaviors of them outside an exceptional set. Finally we obtain a quantitative property of rarefied sets with respect to the stationary Schrödinger operator at $+\infty$ in a cylinder. Meanwhile we show that the reverse of this property is not true.

Citation: Lei Qiao. Asymptotic behaviors of Green-Sch potentials at infinity and its applications. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2321-2338. doi: 10.3934/dcdsb.2017099
##### References:
 [1] L. Ahlfors and M. Heins, Questions of regularity connected with the Phragmén-Lindelöf principle, Ann. of Math., 50 (1949), 341-346. [2] H. Aikawa, On the behavior at infinity of nonnegative superharmonic functions in a half space, Hiroshima Math. J., 11 (1981), 425-441. [3] H. Aikawa and M. Essén, Potential theory-selected topics. Lecture Notes in Mathematics, 1633, Springer-Verlag, Berlin, 1996. [4] V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an n-dimensional cone (Russian), Mat. Sb. (N.S.), 66 (1965), 248-264. [5] R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc. , New York, N. Y. , 1953. [6] M. Cranston, Conditional Brownian motion, Whitney squares and the conditional gauge theorem, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), 109-119, Progr. Probab. , 17, Birkhäuser Boston, Boston, MA, 1989. [7] M. Cranston, E. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc., 307 (1988), 171-194. [8] M. Essén and H. L. Jackson, On the covering properties of certain exceptional sets in a half-space, Hiroshima Math. J., 10 (1980), 233-262. [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [10] P. Hartman, Ordinary Differential Equations, John Wiley and Sons, Inc. , New York-LondonSydney, 1964. [11] J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159. [12] B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. A. Escassut, W. Tutschke and C. C. Yang), Science Press, 11 (2008), 323-397. [13] I. Miyamoto and H. Yoshida, Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone, J. Math. Soc. Japan., 54 (2002), 487-512. [14] I. Miyamoto, Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder, Hokkaido Math. J., 36 (2007), 507-534. [15] Y. Mizuta, Potential theory in Euclidean spaces. GAKUTO International Series. Mathematical Sciences and Applications, 6, Gakkötosho Co. , Ltd. , Tokyo, 1996. [16] L. Qiao, Weak solutions for the stationary Schrödinger equation and its application, Appl. Math. Lett., 63 (2017), 34-39. [17] L. Qiao and G. Deng, Growth properties of modified α-potentials in the upper-half space, Filomat, 27 (2013), 703-712. [18] L. Qiao and G. Deng, Minimally thin sets at infinity with respect to the Schrödinger operator, Sci. Sin. Math., 44 (2014), 1247-1256. [19] L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. [20] L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's type theorem, Bull. Sci. Math., 140 (2016), 70-85. [21] L. Qiao and Y. Ren, ntegral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. [22] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. [23] H. Yoshida and I. Miyamoto, Solutions of the Dirichlet problem on a cone with continuous data, J. Math. Soc. Japan, 50 (1998), 71-93. [24] Y. Zhang, G. Deng and K. Kou, Asymptotic behavior of fractional Laplacians in the half space, Appl. Math. Comput., 254 (2015), 125-132. [25] Y. Zhang, G. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936.

show all references

##### References:
 [1] L. Ahlfors and M. Heins, Questions of regularity connected with the Phragmén-Lindelöf principle, Ann. of Math., 50 (1949), 341-346. [2] H. Aikawa, On the behavior at infinity of nonnegative superharmonic functions in a half space, Hiroshima Math. J., 11 (1981), 425-441. [3] H. Aikawa and M. Essén, Potential theory-selected topics. Lecture Notes in Mathematics, 1633, Springer-Verlag, Berlin, 1996. [4] V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an n-dimensional cone (Russian), Mat. Sb. (N.S.), 66 (1965), 248-264. [5] R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc. , New York, N. Y. , 1953. [6] M. Cranston, Conditional Brownian motion, Whitney squares and the conditional gauge theorem, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), 109-119, Progr. Probab. , 17, Birkhäuser Boston, Boston, MA, 1989. [7] M. Cranston, E. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc., 307 (1988), 171-194. [8] M. Essén and H. L. Jackson, On the covering properties of certain exceptional sets in a half-space, Hiroshima Math. J., 10 (1980), 233-262. [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [10] P. Hartman, Ordinary Differential Equations, John Wiley and Sons, Inc. , New York-LondonSydney, 1964. [11] J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159. [12] B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. A. Escassut, W. Tutschke and C. C. Yang), Science Press, 11 (2008), 323-397. [13] I. Miyamoto and H. Yoshida, Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone, J. Math. Soc. Japan., 54 (2002), 487-512. [14] I. Miyamoto, Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder, Hokkaido Math. J., 36 (2007), 507-534. [15] Y. Mizuta, Potential theory in Euclidean spaces. GAKUTO International Series. Mathematical Sciences and Applications, 6, Gakkötosho Co. , Ltd. , Tokyo, 1996. [16] L. Qiao, Weak solutions for the stationary Schrödinger equation and its application, Appl. Math. Lett., 63 (2017), 34-39. [17] L. Qiao and G. Deng, Growth properties of modified α-potentials in the upper-half space, Filomat, 27 (2013), 703-712. [18] L. Qiao and G. Deng, Minimally thin sets at infinity with respect to the Schrödinger operator, Sci. Sin. Math., 44 (2014), 1247-1256. [19] L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. [20] L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's type theorem, Bull. Sci. Math., 140 (2016), 70-85. [21] L. Qiao and Y. Ren, ntegral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. [22] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. [23] H. Yoshida and I. Miyamoto, Solutions of the Dirichlet problem on a cone with continuous data, J. Math. Soc. Japan, 50 (1998), 71-93. [24] Y. Zhang, G. Deng and K. Kou, Asymptotic behavior of fractional Laplacians in the half space, Appl. Math. Comput., 254 (2015), 125-132. [25] Y. Zhang, G. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936.
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