On the existence of soliton solutions to quasilinear Schrödinger equations

Abstract

Variational techniques are applied to prove the existence of standing wave solutions for quasilinear Schrödinger equations containing strongly singular nonlinearities which include derivatives of the second order. Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Direct methods of the calculus of variations and minimax methods like the Mountain Pass Theorem are used. The difficulties introduced by the nonconvex functional Φ(u) = |∇u| 2 u 2 are substantially different from the semilinear case. Mathematics Subject Classification (1991):35J20, (35Q55, 35J60)

Soliton solutions for quasilinear Schrödinger equations, I

“…( (2) in [20]. We emphasize here that for α > 1 2 , 2α2 * > 2 * and the condition 4α ≤ p + 1 < 2α2 * can be satisfied for all dimensions N .…”

“…We emphasize here that for α > 1 2 , 2α2 * > 2 * and the condition 4α ≤ p + 1 < 2α2 * can be satisfied for all dimensions N . The result in Theorem 1.1 was proved in [20] for the case: N = 1, α = 1, 2 < p + 1; and the result in Theorem 1.2 was proved in [20] under the assumptions: N = 1, α = 1, 4 ≤ p + 1. Remark 1.4.…”

“…In the following we always assume V ∈ C(R N , R) and inf R N V (x) > 0. We 442 JIAQUAN LIU AND ZHI-QIANG WANG also assume N ≥ 2 since results for N = 1 have been given in [20]. Let 2 * = 2N N −2 for N ≥ 3 and 2 * = ∞ for N = 2.…”

MULTIPLE SOLUTIONS FOR A CLASS OF QUASILINEAR SCHRÖDINGER SYSTEM IN ℝN

“…(1.2) models the self-channeling of a high-power ultra short laser in matter (see [10]). For more mathematical models in physics described by (1.2), see [14] and the references therein.…”

A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations

“…The implementation of a Nash-Moser iterative scheme is however very technical, and is only used as a last recourse to solve nonlinear evolution equations, though some recent works show that it is a useful tool (e.g. [16,17,14,13,10,8]). We develop here a Nash-Moser theorem specific to the general class of IVP (1), which allows us to greatly simplify the general theory (at the cost, sometimes, of optimality -see also [18] for a simplified general Nash-Moser implicit function theorem).…”