CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS

doi:10.1016/S0252-9602(09)60079-5

Abstract

Abstract:

We classify all positive solutions for the following integral system:

u_i(x) =∫_Rn1/ |x − y|^n−α f_i(u(y))dy, x ∈ Rⁿ, i = 1, · · · , m,

0 < α < n, and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

Here f_i(u), 1 ≤ i ≤ m, are real-valued functions of homogeneous degree n+α/ n−α and are monotone nondecreasing with respect to all the independent variables u₁, u2, · · ·, u_m. In the special case n ≥ 3 and α = 2, we show that the above system is equivalent to the following elliptic PDE system:

−△u_i(x) = f_i(u(x)), x ∈ Rⁿ, i = 1, · · · , m,

and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

This system is closely related to the stationary Schr¨odinger system with critical exponents for Bose-Einstein condensate.

Key words: integral and PDE systems, positive solutions, method of moving planes, radial symmetry, uniqueness

CLC Number:

35J45

Wenxiong Chen, Congming Li. CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS[J].Acta mathematica scientia,Series B, 2009, 29(4): 949-960.

Trendmd

References

[1] Bourgain J. Global Solutions of Nonlinear Schr¨odinger Equations. AMS Colloquium Publications, Vol 46.
Providence, Rhode Island: AMS, 1999

[2] Caffarelli L, Gidas B, Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equations
with critical Sobolev growth. Commun Pure Appl Math, 1989, 42: 271–297

[3] Chen W, Li C. Classification of solutions of some nonlinear elliptic equations. Duke Math J, 1991, 63:
615–622

[4] Chen W, Li C. Regularity of solutions for a system of integral equations. Commun Pure Appl Anal, 2005,
4: 1–8

[5] Chen W, Li C. The best constant in weighted Hardy-Littlewood-Sobolev inequality. Proc AMS, 208,
136(3): 955–962

[6] Chen W, Li C, Ou B. Classification of solutions for an integral equation. Commun Pure Appl Math, 2006,
59: 330–343

[7] Chen W, Li C, Ou B. Classification of solutions for a system of integral equations. Commun Partial Differ
Equ, 2005, 30: 59–65

[8] Chen W, Li C, Ou B. Qualitative properties of solutions for an integral equation. Disc & Cont Dynamics
Sys, 2005, 12: 347–354

[9] de Figueiredo D G, Felmer P L. On superquadratic elliptic systems. Trans Amer Math Soc, 1994, 343:
99–116

[10] de Figueiredo D G, Felmer P L. A Liouville-type theorem for elliptic systems. Ann Scuola Norm Sup Pisa
Cl Sci, 1994, 21(4): 387–397

[11] Gidas B, Ni W M, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn//in
Mathematical Analysis and Applications, Part A, Adv Math Suppl Stud, 7A. New York: Academic Press,
1981: 369–402

[12] Jin C, Li C. Symmetry of solutions to some integral equations. Proc Amer Math Soc, 2006, 134: 1661–1670

[13] Jin C, Li C. Quantitative analysis of some system of integral equations. Cal Var PDEs, 2006, 26: 447–457

[14] Kanna T, Lakshmanan M. Exact soliton solutions, shape changing collisions, and partially coherent solitons
in coupled nonlinear Schr¨odinger equations. Phys Rev Lett, 2001, 86: 5043–5046

[15] Li C. Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent Math, 1996,
123: 221–231

[16] Li C, Ma L. Uniqueness of positive bound states to Shrodinger systems with critical exponents. SIAM J
Math Analysis, 2008, 40(3): 401049–1057

[17] Lieb E. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann Math, 1983, 118:
349–374

[18] Li C, Lim J. The singularity analysis of solutions to some integral edquations. Commun Pure Applied
Analysis, 2007, 6(2): 1–12

[19] Lin T C, Wei J. Ground state of N coupled nonlinear Schr¨odinger equations in Rn, n 3. Commun Math
Phys, 2005, 255: 629–653

[20] Lin T C,Wei J. Spikes in two coupled nonlinear Schr¨odinger equations. Ann Inst H Poincar´e Anal Non-Lin,
2005, 22: 403–439

[21] Ma L, Chen D Z. A Liouville type theorem for an integral system. Commun Pure Applied Analysis, 2006,
5: 855–859

[22] Ma L, Chen D Z. Radial symmetry and monotonicity for an integral equation. J Math Anal Appl, 2008,
342: 943–949

[23] Ma L, Zhao L. Sharp thresholds of blow up and global existence for the coupled nonlinear Schrodinger
system. J Math Phys, 2008

[24] Naito M, Usami H. Existence of nonoscillatory solutions to second-order elliptic systems of Emden-Fowler
type. Indiana Univ Math J, 2006, 55: 317–339

[25] Polacik P, Quittner P, Souplet P. Singularity and decay estimates in superlinear problems via Liouville-type
theorems, I: elliptic equations and systems. Duke Math J, 2007, 139(3): 555–579

[26] Serrin J, Zou H. Non-existence of positive solutions of Lane-Emden systems. Differential Integral Equations,
1996, 9: 635–653. MR1401429 (97f:35056)

[27] Stein E M,Weiss G. Fractional integrals in n-dimensional Euclidean space. J Math Mech, 1958, 7: 503–514

[28] Wei J, Xu X. Classification of solutions of higher order conformally invariant equations. Math Ann, 1999,
313: 207–228

[29] Weinstein M I. Nonlinear Schr¨odinger equations and sharp interpolate estimates. Commun Math Phys,
1983, 87: 567–576