Nonlinear Scalar Field Equations with L 2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches

Jun Hirata, Kazunaga Tanaka

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ℝ N (N ≥ 2): (Equation Presented) where g(ξ) ∈ C(ℝ, ℝ), m > 0 is a given constant and μ ∈ ℝ is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem (∗)m. We develop a new deformation argument under a new version of the Palais-Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313.345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347.375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in ℝ N : Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253.276], it enables us to apply minimax argument for L 2 constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem (Equation Presented).

Original languageEnglish
Pages (from-to)263-290
Number of pages28
JournalAdvanced Nonlinear Studies
Volume19
Issue number2
DOIs
Publication statusPublished - 2019 May 1

Keywords

  • Deformation Theory
  • L -Constraint Problem
  • Normalized Solutions

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematics(all)

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