Multiplicity results for the scalar curvature equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Flores, Isabel¹  Franca, Matteo² 

[1] Univ Tecn Federico Santa Maria, Dept Matemat, Valparaiso, Chile

[2] Univ Politecn Marche, Dipartimento Sci Matemat, Via Brecce Bianche 1, I-60131 Ancona, Italy

关键词: Critical exponent;    Ground state;    Fowler transformation;    Singular perturbations;    Melnikov theory;   

DOI  :  10.1016/j.jde.2015.05.020

来源: Elsevier

【 摘 要 】

This paper is devoted to the study of positive radial solutions of the scalar curvature equation, i.e. Delta u(x) + K(vertical bar x vertical bar)u(sigma-1) (x) = 0 where sigma = 2n/n-2 and we assume that K (vertical bar x vertical bar) = k(vertical bar x vertical bar(epsilon)) and k(r) is an element of C-1 is bounded and epsilon > 0 is small. It is known that we have at least a ground state with fast decay for each positive critical point of k for epsilon small enough. In fact if the critical point k(r(0)) is unique and it is a maximum we also have uniqueness; surprisingly we show that if k(r(0)) is a minimum we have an arbitrarily large number of ground states with fast decay. The results are obtained using Fowler transformation and developing a dynamical approach inspired by Melnikov theory. We emphasize that the presence of subharmonic solutions arising from zeroes of Melnikov functions has not appeared previously, as far as we are aware. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free