Nonlinear scalar field equations, II existence of infinitely many solutions

1983 ◽  
Vol 82 (4) ◽  
pp. 347-375 ◽  
Author(s):  
H. Berestycki ◽  
P. -L. Lions
Keyword(s):  
Scalar Field ◽  
Infinitely Many Solutions ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Scalar Field Equations

scholarly journals Nonlinear Scalar Field Equations with L2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches

2019 ◽  
Vol 19 (2) ◽  
pp. 263-290 ◽  
Author(s):  
Jun Hirata ◽  
Kazunaga Tanaka
Keyword(s):  
Scalar Field ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Field Equations ◽  
Lagrange Formulation ◽  
Nonlinear Anal ◽  
Nonlinear Scalar Field ◽  
Palais Smale Condition ◽  
Formula Group ◽  
Scalar Field Equations

AbstractWe study the existence of radially symmetric solutions of the following nonlinear scalar field equations in {\mathbb{R}^{N}} ({N\geq 2}):${(*)_{m}}$\displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},% \cr\lVert u\rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N})% ,\end{cases}where {g(\xi)\in C(\mathbb{R},\mathbb{R})}, {m>0} is a given constant and {\mu\in\mathbb{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 {\mathbb{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\inf\Bigg{\{}\int_{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u% \rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int_{0}^{\xi}g(% \tau)\,d\tau.

scholarly journals Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter

2019 ◽  
Vol 150 (5) ◽  
pp. 2417-2441 ◽  
Author(s):  
Takafumi Akahori ◽  
Slim Ibrahim ◽  
Hiroaki Kikuchi
Keyword(s):  
Scalar Field ◽  
Ground State ◽  
Linear Instability ◽  
Frequency Parameter ◽  
Field Equations ◽  
Power Type ◽  
Nonlinear Scalar Field ◽  
Scalar Field Equations ◽  
Sobolev Critical Exponent ◽  
Large Frequency

AbstractWe consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In [3], it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schrödinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.

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