scholarly journals Entire Solutions to Nonlinear Scalar Field Equations with Indefinite Linear Part

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Gilles Evéquoz ◽  
Tobias Weth
Keyword(s):  
Scalar Field ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Linear Part ◽  
Continuous Functions ◽  
Entire Solutions ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Scalar Field Equations ◽  
Semilinear Schrödinger Equation

AbstractWe consider the stationary semilinear Schrödinger equation−Δu + a(x)u = f (x, u), u ∈ Hwhere a and f are continuous functions converging to some limits a

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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