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

Lei Qiao

doi:10.3934/dcdsb.2017099

Article Contents

2017, Volume 22, Issue 6: 2321-2338. Doi: 10.3934/dcdsb.2017099

This issue Previous Article Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity Next Article On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ

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

Lei Qiao

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

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

Received: June 2016

Revised: November 2016

Published: May 2017

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary:35J05, 35J10;Secondary:35J25, 35J67.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.