TY - JOUR
T1 - Nonlinear Scalar Field Equations with L 2 Constraint
T2 - Mountain Pass and Symmetric Mountain Pass Approaches
AU - Hirata, Jun
AU - Tanaka, Kazunaga
N1 - Funding Information:
The second author is partially supported by JSPS Grants-in-Aid for Scientific Research (B) (Grant No. 25287025) and (B) (Grant No. 17H02855).
Funding Information:
Funding: The second author is partially supported by JSPS Grants-in-Aid for Scientific Research (B) (Grant No. 25287025) and (B) (Grant No. 17H02855).
Publisher Copyright:
© 2019 Walter de Gruyter GmbH.
PY - 2019/5/1
Y1 - 2019/5/1
N2 - 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).
AB - 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).
KW - Deformation Theory
KW - L -Constraint Problem
KW - Normalized Solutions
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U2 - 10.1515/ans-2018-2039
DO - 10.1515/ans-2018-2039
M3 - Article
AN - SCOPUS:85060714228
SN - 1536-1365
VL - 19
SP - 263
EP - 290
JO - Advanced Nonlinear Studies
JF - Advanced Nonlinear Studies
IS - 2
ER -