Standing waves for nonlinear Schrödinger equations involving critical growth

Standing waves with a critical frequency for nonlinear Schrödinger equations involving critical growth

This paper is concerned with the existence and qualitative property of standing wave solutions ψ(t, x) = e−iEt/h̄v(x) for the nonlinear Schrödinger equation h̄ ∂ψ ∂t + h̄2 2 ψ − V (x)ψ + |ψ |p−1ψ = 0 with E being a critical frequency in the sense that minRN V (x) = E. We show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude g...

Quasilinear Schrödinger equations involving critical exponents in $mathbb{textbf{R}}^2$

‎We study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{R}}^2$ with critical exponential growth‎. ‎This model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.

Standing waves in nonlinear Schrödinger equations

In the theory of nonlinear Schrödinger equations, it is expected that the solutions will either spread out because of the dispersive effect of the linear part of the equation or concentrate at one or several points because of nonlinear effects. In some remarkable cases, these behaviors counterbalance and special solutions that neither disperse nor focus appear, the so-called standing waves. For...

Standing Waves for a Class of Schrödinger–poisson Equations in R Involving Critical Sobolev Exponents

−ε2∆u + V (x)u + ψu = f(u) in R, −ε2∆ψ = u in R, u > 0, u ∈ H(R), has been studied extensively, where the assumption for f(u) is that f(u) ∼ |u|p−2u with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz condition which forces the boundedness of any Palais– Smale sequence of the corresponding energy functional of the equation. The more difficult critical case is studied in this paper. As g(u) :...

Stability and instability of standing waves for nonlinear Schrödinger equations