CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

doi:10.1016/S0252-9602(17)30082-6

Abstract

Abstract:

In this paper,we are concerned with the following Hardy-Sobolev type system

where 0 < α < n,0 < t₁,t₂ < min{α,k},and 1 < p ≤ τ₁:=(n+α-2t₁)/n-α,1 < q ≤ τ₂:=(n+α-α2t₂)/n-α. We first establish the equivalence of classical and weak solutions between PDE system (0.1) and the following integral equations (IE) system

where G_α(x,ξ)=(c_n,α)/|x-ξ|^n-α is the Green's function of (-△)^（α）/2 in Rⁿ.Then,by the method of moving planes in the integral forms,in the critical case p=τ₁ and q=τ₂,we prove that each pair of nonnegative solutions (u,v) of (0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z₀ in R^n-k.In the subcritical case （n-t₁）/p+1 + （n-t₂）/q+1 > n-α, 1 < p ≤ τ₁ and 1 < q ≤ τ₂,we derive the nonexistence of nontrivial nonnegative solutions for (0.1)

Key words: Hardy-Sobolev type systems, systems of fractional Laplacian, systems of integral equations, method of moving planes in integral forms, radial symmetry, nonexistence

Wei DAI, Zhao LIU. CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS[J].Acta mathematica scientia,Series B, 2017, 37(5): 1415-1436.

Trendmd

References

[1] Caffarelli L, Gidas B, Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth. Comm Pure Appl Math, 1989, 42:271-297
[2] Caffarelli L, Kohn R, Nirenberg L. First order interpolation inequalities with weights. Compos Math, 1984, 53:259-275
[3] Cao D, Li Y. Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator. Methods Appl Anal, 2008, 15(1):81-96
[4] Catrina F, Wang Z. On the Caffarelli-Kohn-Nirenberg inequalities:sharp constant, existence (and nonexistence) and symmetry of extremal functions. Comm Pure Appl Math, 2001, 54(2):229-258
[5] Chen W, Fang Y. Higher and fractional order Hardy-Sobolev type equations. Bull Institute Math, Academia Sinica New Series, 2014, 9(3):317-349
[6] Chen W, Li C. Methods on Nonlinear Elliptic Equations. AIMS Book Series on Diff Equa and Dyn Sys, Vol 4. Amer Inst Math Sci, 2010
[7] Chen W, Li C. Classification of solutions of some nonlinear elliptic equations. Duke Math J, 1991, 63:615-622
[8] Chen W, Li C. Classification of positive solutions for nonlinear differential and integral systems with critical exponents. Acta Math Sci, 2009, 29B(4):949-960
[9] Chen W, Li C. The best constant in a weighted Hardy-Littlewood-Sobolev inequality. Proc Amer Math Soc, 2008, 136:955-962
[10] Chen W, Li C. Super polyharmonic property of solutions for PDE systems and its applications. Comm Pure Appl Anal, 2013, 12(6):2497-2514
[11] Chen W, Li C, Ou B. Classification of solutions for an integral equation. Comm Pure Appl Math, 2006, 59:330-343
[12] Chen W, Li C, Ou B. Classification of solutions for a system of integral equations. Comm PDE, 2005, 30:59-65
[13] Chou K, Chu C. On the best constant for a weighted Sobolev-Hardy inequality. J Lond Math Soc, 1993, 2:137-151
[14] D'Ambrosio L, Lucente S. Nonlinear Liouville theorems for Grushin and Tricomi operators. J Differ Equ, 2003, 193(2):511-541
[15] D'Ambrosio L, Mitidieri E, Pohozaev S I. Representation formulae and inequalities for solutions of a class of second order partial differential equations. Trans Amer Math Soc, 2005, 358:893-910
[16] Dai W, Liu Z, Lu G. Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space. Potential Anal, 2017, 46:569-588
[17] Gidas B, Ni W, Nirenberg L. Symmetry and related properties via the maximum principle. Comm Math Phys, 1979, 68(3):209-243
[18] Gidas B, Ni W, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rⁿ//Mathematical analysis and applications, Vol 7a of the book series Advances in Mathematics. New York:Academic Press, 1981
[19] Huang X, Li D, Wang L. Symmetry and monotonicity of integral equation systems. Nonlinear Anal:RWA, 2011, 12(6):3515-3530
[20] Liu Z, Dai W. A Liouville type theorem for poly-harmonic system with dirichlet boundary conditions in a half space. Adv Nonlinear Stud, 2015, 15:117-134
[21] Lei Y. Asymptotic properties of positive solutions of the Hardy-Sobolev type equations. J Diff Equ, 2013, 254:1774-1799
[22] Li C. Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent Math, 1996, 123:221-231
[23] Lin C S. A classification of solutions of a conformally invariant fourth order equation in Rⁿ. Comment Math Helv, 1998, 73:206-231
[24] Lieb E, Loss M. Analysis. 2nd edition, Graduate Studies in Mathematics, Vol. 14. Providence, RI:Amer Math Soc, 2001
[25] Lu G, Wei J, Xu X. On conformally invariant equation (-△)^pu-K(x)u ^(N+2p)/N-2p=0 and its generalizations. Ann Mat Pur Appl, 2001, 179(1):309-329
[26] Lu G, Zhu J. Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality. Calc Var PDEs, 2011, 42:563-577
[27] Mancini G, Fabbri I, Sandeep K. Classification of solutions of a critical Hardy-Sobolev operator. J Diff Equ, 2006, 224:258-276
[28] Wei J, Xu X. Classification of solutions of higher order conformally invariant equations. Math Ann, 1999, 313:207-228
[29] Zhao Y. Regularity and symmetry for solutions to a system of weighted integral equations. J Math Anal Appl, 2012, 391:209-222
[30] Zhang Z. Nonexistence of positive solutions of an integral system with weights. Adv Diff Equ, 2011