Nonlocal scalar field equations: Qualitative properties, asymptotic profiles and local uniqueness of solutions

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Bhakta, Mousomi¹  Mukherjee, Debangana¹ 

[1] Indian Inst Sci Educ & Res, Dept Math, Dr Homi Bhaba Rd, Pune 411008, Maharashtra, India

关键词: Nonlocal;    Symmetry;    Blow-up and decay estimates;    Uniqueness;    Subcritical/critical;    Supercritical nonlinearities;   

DOI  :  10.1016/j.jde.2018.11.023

来源: Elsevier

【 摘 要 】

We study the nonlocal scalar field equation with a vanishing parameter: {(-Delta)(s)u + epsilon u = vertical bar u vertical bar(p-2)u - vertical bar u vertical bar(q-2)u in R-N u is an element of H-S (R-N), (P-epsilon) where s is an element of( 0, 1), N > 2s, q > p > 2 are fixed parameters and epsilon > 0 is a vanishing parameter. For epsilon small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2* = 2N/N-2s. For p < 2*, the ground state solution asymptotically coincides with unique positive ground state solution of (-Delta)(S)u + u = u(p), whereas for p = 2* the asymptotic behavior of the solutions is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. For p > 2*, the solution asymptotically coincides with a ground-state solution of (-Delta)(S)u = u(p)- u(q). Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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