On singular equations with critical and supercritical components

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Bhakta, Mousomi¹  Santra, Sanjiban² 

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

[2] Ctr Invest Math, Dept Basic Math, Guanajuato, Mexico

关键词: Super-critical exponent;    Local estimates;    Asymptotic behavior;    Entire solution;    Large solutions;   

DOI  :  10.1016/j.jde.2017.04.018

来源: Elsevier

【 摘 要 】

We study the problem (I-epsilon) {-Delta u - mu u/vertical bar x vertical bar(2) = u(p) - epsilon u(q) in Omega, u > 0 in Omega, u is an element of H-0(1)(Omega) boolean AND Lq+1 (Omega) where q > p >= 2* - 1, epsilon > 0, Omega subset of R-N is a bounded domain with smooth boundary, 0 is an element of Omega, N >= 3 and 0 < mu < (mu) over bar:= (N-2/2)(2). We completely classify the singularity of solution at 0 in the supercritical case. Using the transformation v = vertical bar x vertical bar(v) u, we reduce the problem (l(epsilon)) to (J(epsilon)) (J(epsilon)) {-div(vertical bar x vertical bar(-2v)del nu) = vertical bar x vertical bar(-(p+1)nu)vp - epsilon vertical bar x vertical bar(-(q+1)nu)vq in Omega, u > 0 in Omega, v is an element of H-0(1)(Omega, vertical bar x vertical bar(-2v)) boolean AND Lq+1(Omega, vertical bar x vertical bar(-(q+1)v)), and then formulating a variational problem for (J(epsilon)), we establish the existence of a variational solution v epsilon and characterize the asymptotic behavior of v(epsilon) as epsilon -> 0 by variational arguments when p = 2* - 1. (C) 2017 Elsevier Inc. All rights reserved.

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