Characterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operator

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Lopez-Gomez, Julian¹  Omari, Pierpaolo² 

[1] Univ Complutense Madrid, Inst Interdisciplinar Matemat, Plaza Ciencias 3, Madrid 28040, Spain

[2] Univ Trieste, Dipartimento Matemat & Geosci, Via A Valerio 12-1, I-34127 Trieste, Italy

关键词: Mean curvature operator;    Neumann boundary condition;    Indefinite weight;    Classical solution;    Bounded variation solution;    Formation of singularities;   

DOI  :  10.1016/j.jde.2020.01.015

来源: Elsevier

【 摘 要 】

The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem -(u'/root 1 + (u')(2))' = lambda a(x) f(u) in (0, 1), u'(0) = 0, u'(1) = 0, where lambda is an element of R is a parameter, a is an element of L-infinity(0, 1) changes sign once in (0, 1) at the point z is an element of (0, 1), and f is an element of C(R) boolean AND C-1[0, +infinity) is positive and increasing in (0, +infinity) with a potential, integral(s)(0) f(t) dt, superlinear at +infinity. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as lambda -> 0(+), we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition (integral(z)(x) a(t) dt)(-1/2) is an element of L-1(0, z) and (integral(z)(x) a(t) dt)(-1/2) is an element of L-1(z, 1). No previous result of this nature is known in the context of the theory of superlinear indefinite problems. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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