An existence and stability result for standing waves of nonlinear Schrödinger equations

By Louis Jeanjean, Stefan Le Coz. Advances in Differential Equations, 2006

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We consider a nonlinear Schrodinger equation with a nonlinearity of the form V (x)g(u). Assuming that V (x) behaves like |x|−b at infinity and g(s) like |s|p around 0, we prove the existence and orbital stability of travelling waves if 1 < p < 1 + (4− 2b)/N .

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8084889f2cccbe93141e4810c00e0c699b233a82

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Citations in the corpus

Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent (2014, methodology): “Inspired by [6,20], we introduce a truncation method, already used in earlier researches [17,18].”
Existence of a positive solution to Kirchhoff type problems without compactness conditions (2012, unknown): “To overcome the difficulty of finding bounded Palais–Smale sequences for the associated functional Jλ , following [7,9], we use a cut-off function ψ ∈ C∞(R+,R) satisfying ⎪⎪⎪⎨ ⎪⎪⎪⎩ ψ(t) = 1, t ∈ [0,1], 0 ψ(t) 1, t ∈ (1,2), ψ(t) = 0, t ∈ [2,∞), ∥∥ψ ′∥∞ 2, and study the following modified functional J”
On the Schrödinger-Maxwell equations under the effect of a general nonlinear term (2009, methodology): “To overcome these difficulties, we will use a combined technique consisting in a truncation argument (see [17,21]) and a monotonicity trick à la Jeanjean [15] (see also Struwe [26]).”
Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves (2008, background): “Furthermore, v(k) ∈ C(RN) ∩ L∞(RN) for all k ∈ [0, k0) and |v(k)|L∞(RN ) remains bounded as k → 0.”
Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent (2014, unknown): “To overcome the difficulty of finding bounded PS sequences for the associated functional Jq, following [16, 18], we use a cut-off function ψ ∈ C∞(R+, [0, 1]) satisfying   ψ(t) = 1, t ∈ [0, 1], ψ(t) = 0, t ∈ [2,∞), |ψ′|∞ ≤ 2 and study the following modified functional J q : Hb → R defined by J ”
A multiplicity result for a fractional Kirchhoff equation in ℝN with a general nonlinearity (2016, methodology): “More precisely, we follow the ideas in [8], and by combining the Mountain Pass approach introduced in [19] with the truncation argument of [20], we prove the multiplicity result above stated.”