Positive solutions of nonlinear scalar field equations involing critical Sobolev exponent

1987 ◽  
Vol 3 (1) ◽  
pp. 27-37 ◽  
Author(s):  
Zhang Dong
Keyword(s):  
Scalar Field ◽  
Positive Solutions ◽  
Critical Sobolev Exponent ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Sobolev Exponent ◽  
Scalar Field Equations

Quasilinear Scalar Field Equations Involving Critical Sobolev Exponents and Potentials Vanishing at Infinity

2018 ◽  
Vol 61 (3) ◽  
pp. 705-733 ◽  
Author(s):  
Athanasios N. Lyberopoulos
Keyword(s):  
Scalar Field ◽  
Weak Solutions ◽  
Bound States ◽  
Critical Sobolev Exponent ◽  
Field Equations ◽  
Critical Sobolev Exponents ◽  
Sobolev Exponent ◽  
Image Position ◽  
Scalar Field Equations

AbstractWe are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions inW1,p(ℝN)), for quasilinear scalar field equations of the form$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$where Δpu: =div(|∇u|p−2∇u), 1 <p<N,p*: =Np/(N−p) is the critical Sobolev exponent,q∈ (p, p*), whileV(·) andK(·) are non-negative continuous potentials that may decay to zero as |x| → ∞ but are free from any integrability or symmetry assumptions.

scholarly journals Nonlinear scalar field equations with general nonlinearity

2020 ◽  
Vol 190 ◽  
pp. 111604 ◽  
Author(s):  
Louis Jeanjean ◽  
Sheng-Sen Lu
Keyword(s):  
Scalar Field ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Scalar Field Equations ◽  
General Nonlinearity

A perturbation of nonlinear scalar field equations

2019 ◽  
Vol 45 ◽  
pp. 531-541 ◽  
Author(s):  
Jiu Liu ◽  
Tao Liu ◽  
Jia-Feng Liao
Keyword(s):  
Scalar Field ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Scalar Field Equations

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.

scholarly journals Nonradial solutions of nonlinear scalar field equations

Nonlinearity ◽  
2020 ◽  
Vol 33 (12) ◽  
pp. 6349-6380
Author(s):  
Jarosław Mederski
Keyword(s):  
Scalar Field ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Scalar Field Equations
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