Nonlinear scalar field equations with L2 constraint: Mountain pass and symmetric mountain pass approaches

Jun Hirata, Kazunaga Tanaka

Research output: Contribution to journalArticlepeer-review

Abstract

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in RN (N ≥ 2): (∗)m{−∆u = g(u) − µu in RN , kukL2(RN ) = m, u ∈ H1(RN ), 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 the 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 [BL1, BL2, HIT], it enables us to apply minimax argument for L2 constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem: inf{ ZRN 21 |∇u|2 − G(u) dx; kuk2L2(RN ) = m}, G(ξ) = Z0ξ g(τ) dτ.

35J60, 58E05

Original languageEnglish
JournalUnknown Journal
Publication statusPublished - 2018 Mar 14

ASJC Scopus subject areas

  • General

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