Nash moser methods for the solution of quasilinear schrödinger equations

Orbital stability of standing wave solution for a quasilinear Schrödinger equation

“…Please note that in the stating of this theorem, we have tacitly assumed that the potential V (x), p and the initial data z 0 belong to a class in which a unique solution of the initial-value problem for (1.1) exists for all t ∈ [0, T * ) and some T * < ∞ or T * = ∞. In view of local well-posedness results for this kind of problems studied by Lange-Poppenberg [9] and the arguments developed by Cazenave [5], we will continue to make these assumptions without any comments. Moreover, if z 0 is radially symmetrical with respect to x, then so is z(x, t).…”

“…Kurihura [8], Nakamura [14]). For more physical motivations and more references dealing with applications, we refer the interested readers to Lange et al [9], Poppenberg et al [15] and the references therein.…”

Existence of Nontrivial Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

“…Hence (V) = 0; that is, V is a weak solution of (11). In the following, we prove that V is nontrivial.…”

“…Using a standard argument, we know that (V) = 0, that is, V is a weak solution of (11). Indeed, for any ∈ ∞ 0 (R ), we have…”

“…In the case ( ) = (1 + ) 1/2 , problem (2) models the self-channeling of a high-power ultra short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity, and this leads to interesting new nonlinear wave equations; see [3][4][5][6]. For more physical motivations and more references dealing with applications, we can refer to [7][8][9][10][11][12][13][14] and references therein.…”