Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:395

Article

Candito, P.¹  Carl, S.²  Livrea, R.¹ 

[1] Univ Mediterranea Reggio Calabria, MECMAT Dipartimento Meccan & Mat, I-89100 Reggio Di Calabria, Italy

[2] Univ Halle Wittenberg, Inst Math, D-06099 Halle, Germany

关键词: Critical points;    p-Laplacian;    Extremal constant-sign solutions;    Sign-changing solutions;   

DOI  :  10.1016/j.jmaa.2012.05.003

来源: Elsevier

【 摘 要 】

We study a quasilinear elliptic problem depending on a parameter lambda of the form -Delta(p)u = lambda f(u) in Omega, u = 0 on partial derivative Omega. We present a novel variational approach that allows us to obtain multiplicity, regularity and a priori estimate of solutions by assuming certain growth and sign conditions on f prescribed only near zero. More precisely, we describe an interval of parameters lambda for which the problem under consideration admits at least three nontrivial solutions: two extremal constant-sign solutions and one sign-changing solution. Our approach is based on an abstract localization principle of critical points of functionals of the form E = phi - lambda psi on open sublevels phi(-1) (]- infinity, r[), combined with comparison principles and the sub-supersolution method. Moreover, variational and topological arguments, such as the mountain pass theorem, in conjunction with truncation techniques are the main tools for the proof of sign-changing solutions. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free