Journal Article•
Nonlinear scalar field equations
01 Jan 1992-Differential and Integral Equations (Khayyam Publishing, Inc.)-Vol. 5, Iss: 4, pp 777-792
About: This article is published in Differential and Integral Equations.The article was published on 1992-01-01 and is currently open access. It has received 1096 citations till now. The article focuses on the topics: Scalar field & Scalar theories of gravitation.
Citations
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TL;DR: In this paper, the authors derive a generic theorem for a wide class of functionals, having a mountain pass geometry, and show how to obtain, for a given functional, a special Palais-Smale sequence possessing extra properties that help to ensure its convergence.
Abstract: Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the formWe assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.
815 citations
TL;DR: In this paper, the existence of radially symmetric solitary waves for nonlinear Klein-Gordon equations and nonlinear Schrodinger equations coupled with Maxwell equations was studied using a variational approach and the solutions were obtained as mountain-pass critical points for the associated energy functional.
Abstract: In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrodinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.
454 citations
TL;DR: In this article, the authors study the initial value problem associated to some canonical dispersive equations and establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem.
Abstract: We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.
447 citations
TL;DR: For a class of quasilinear Schrodinger equations, the authors established the existence of ground states of soliton-type solutions by a variational method for soliton type solutions.
439 citations
437 citations
References
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TL;DR: In this article, a constrained minimization method was proposed for the case of dimension N = 1 (Necessary and sufficient conditions) for the zero mass case, where N is the number of dimensions in the dimension N.
Abstract: 1. The Main Result; Examples . . . . . . . . . . . . . . . . . . . . . . . 316 2. Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 319 3. The Constrained Minimization Method . . . . . . . . . . . . . . . . . . 323 4. Further Properties of the Solution . . . . . . . . . . . . . . . . . . . . 328 5. The \"Zero Mass\" Case . . . . . . . . . . . . . . . . . . . . . . . . . 332 6. The Case of Dimension N = 1 (Necessary and Sufficient Conditions) . . . . . 335 Appendix. Technical Results . . . . . . . . . . . . . . . . . . . . . . . . 338
2,385 citations
TL;DR: In this paper, it was shown that Δu=F(u) possesses non-trivial solutions in R n which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.
Abstract: The elliptic equation Δu=F(u) possesses non-trivial solutions inR n which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.
1,812 citations
TL;DR: In this article, divers resultats de compacite for les sousespaces d'espaces de Sobolev tres generaux constitues par les fonctions ayant un certain nombre de symetries: symetrie spherique, symetria cylindrique, etc.
374 citations