Articles

LIOUVILLE THEOREM FOR CHOQUARD EQUATION WITH FINITE MORSE INDICES

  • Xiaojun ZHAO
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  • School of Economics, Peking University, Beijing 100871, China

Received date: 2016-10-24

  Revised date: 2017-09-01

  Online published: 2018-04-25

Abstract

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation
-△u=∫RN (|u(y)|p)/(|x-y|α)dy|u(x)|p-2u(x) in RN,
where N ≥ 3, 0< α < min{4, N}. Suppose that 2< p < (2N-α)/(N-2), we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2N-α)/(N-2), if i(u) < ∞, then we have ∫RNRN (|u(x)|p|u(y)|p)/(|x-y|α) dxdy < ∞ and ∫RN|▽u|2 dx=∫RNRN(|u(x)|p|u(y)|p)/(|x-y|α dxdy.

Cite this article

Xiaojun ZHAO . LIOUVILLE THEOREM FOR CHOQUARD EQUATION WITH FINITE MORSE INDICES[J]. Acta mathematica scientia, Series B, 2018 , 38(2) : 673 -680 . DOI: 10.1016/S0252-9602(18)30773-2

References

[1] Bahri A, Lions P L. Solutions of superlinear elliptic equations and their Morse indices. Commun Pure App Math, 1992, 45:1205-1215
[2] Chen W, Fang Y, Li C. Super poly-harmonic property of solutions for Navier boundary problems on a half space. J Funct Anal, 2013, 265:1522-1555
[3] Chen W, Fang Y, Yang R. Liouville theorems involving the fractional Laplacian on a half space. Adv Math, 2015, 274:167-198
[4] Chen W, Li C. Classification of solutions of some nonlinear elliptic equations. Duke Math J, 1991, 63:615-622
[5] Chen W, Li C, Li Y. A direct method of moving planes for the fractional Laplacian. Adv Math, 2017, 308:404-437
[6] Chen W, Li C, Ou B. Classification of solutions for an integral equation. Comm Pure Appl Math, 2006,59:330-343
[7] Chen W, Li Y, Zhang R. A direct method of moving spheres on fractional order equations. J Funct Anal, 2017, 272:4131-4157
[8] Damascelli L, Gladiali F. Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Rev Mat Iberoamericana, 2004, 20:67-86
[9] Fang Y, Chen W. A Liouville type theorem for poly-harmonic Dirichlet problems in a half space. Adv Math, 2012, 229:2835-2867
[10] Farina A. On the classification of solutions of the Lane-Emden equation on unbounded domains of RN. J Math Pures Appl, 2007, 87:537-561
[11] de Figueiredo D G, Felmer P L. A Liouville type theorem for Elliptic systems. Ann Scuola Norm Sup Pisa Cl Sci, 1994, 21:387-397
[12] de Figueiredo D G, Yang J. On a semilinear elliptic problem without (PS) condition. J Differential Equations, 2003, 187:412-428
[13] Gidas B, Spruck J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm Pure Appl Math, 1981, 34:525-598
[14] Gidas B, Spruck J. A priori bounds of positive solutions of nonlinear elliptic equations. Comm Part Differ Eq, 1981, 6:883-901
[15] Harrabi A, Ahmedou M, Rebhi S, Selmi A. A priori estimates for superlinear and subcritical elliptic equations:the Neumann boundary condition case. Manuscripta Mathematica, 2012, 137:525-544
[16] Harrabi A, Rahal B. On the sixth-order Joseph-Lundgren exponent. Ann Henri Poincaré, 2017, 18:1055-1094
[17] Harrabi A, Rebhi S, Selmi S. Solutions of superlinear equations and their Morse indices I. Duke Math J, 1998, 94:141-157
[18] Harrabi A, Rebhi S, Selmi S. Solutions of superlinear equations and their Morse indices Ⅱ. Duke Math J, 1998, 94:159-179
[19] Li Y. Remark on some conformally invariant integral equations:the method of moving spheres. J Eur Math Soc, 2004, 6:153-180
[20] Li Y, Zhang L. Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations. J Anal Math, 2003, 90:27-87
[21] Lieb E H. Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl Math, 1976/77, 57:93-105
[22] Moroz V, Van J. Schaftingen Groundstates of nonlinear Choquard equations:existence, qualitative properties and decay asymptotics. J Funct Anal, 2013, 265:153-184
[23] Yu X. Liouville type theorems for integral equations and integral systems. Calc Var PDE, 2013, 46:75-95
[24] Yu X. Solutions of mixed boundary problems and their Morse indices. Nonlinear Analysis TMA, 2014, 96:146-153
[25] Yu X. Liouville theorem for elliptic equations with nonlinear boundary value conditions and finite Morse indices. J Math Anal Appl, 2015, 421:436-443
[26] Yu X. Solutions of fractional Laplacian equations and their Morse indices. J Differential Equations, 2016, 260:860-871
[27] Zhao, X, Wang X. Liouville theorem for Robin boundary value problems and finite Morse indices. J Math Anal Appl, 2014, 419:796-803
[28] Zhao X, Yu X. Liouville type theorem for some Choquard type equation (in Chinese). Sci Sin Math, 2017, 47:713-722
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