On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Li, Gongbao^1,2  Ye, Hongyu^1,2 

[1] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China

[2] Wuhan Univ Sci & Technol, Coll Sci, Wuhan 430065, Hubei, Peoples R China

关键词: Kirchhoff equation;    Mass concentration;    Constrained minimization;    Normalized solutions;    Sharp existence;   

DOI  :  10.1016/j.jde.2018.11.024

来源: Elsevier

【 摘 要 】

In this paper, we study the existence and concentration behavior of minimizers for i(V)(c) = inf (u is an element of Sc) I-V(u), here S-c ={u is an element of H-1(R-N)vertical bar integral(RN) V(x)vertical bar u vertical bar(2) < +infinity, vertical bar u vertical bar(2) = c > 0} and IV(u) =1/2 integral(RN) (a vertical bar del u vertical bar(2)+ V(x)vertical bar u vertical bar(2)) + b/4 (integral(RN)vertical bar del u vertical bar(2))(2) - 1/p integral(RN)vertical bar u vertical bar(p), where N = 1, 2, 3 and a, b > 0 are constants. By the Gagliardo-Nirenberg inequality, we get the sharp existence of global constraint minimizers of i(V) (c) for 2 < p < 2* when V(x) >= 0, V(x) is an element of L-loc(infinity)(R-N) and lim(vertical bar x vertical bar ->+infinity) V(x) = +infinity. For the case p is an element of(2, 2N+8/N)\{4}, we prove that the global constraint minimizers u(c) of iV(c) behave like u(c)(x) approximate to c/vertical bar Q(p)vertical bar(2) (m(c)/c)(n/2) Q(p) (mc/c x - z(c)) , for some z(c) is an element of R-N when c is large, where Q(p) is, up to translations, thw unique positive solution of -N(p-2)/4 Delta Q(p) + 2N-p(N-2)/4 Q(p) = vertical bar Q(p)vertical bar(p-2) Q(p) in R-N and m(c) = (root a(2)D(1)(2)-4bD(2)i(0)(c)+aD(1)/2bD(2))1/2, D-1 = Np-2N-4/2N(p-2) and D-2 = 2N+8-Np/4N(p-2) . (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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