Multiple Positive Solutions to Nonlinear Schrödinger Equations with Competing Potential Functions

Analysis for Nonlinear Schrödinger Equations with Potential

We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besi...

Multiple Positive Solutions for Nonlinear Semipositone Fractional Differential Equations

Nonlinear Schrödinger Equations with Repulsive Harmonic Potential and Applications

We study the Cauchy problem for Schrödinger equations with repulsive quadratic potential and power-like nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the no...

Asymptotic Stability of Nonlinear Schrödinger Equations with Potential Zhou Gang

We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.

2 Nonlinear Schrödinger Equations with Repulsive Harmonic Potential and Applications

We study the Cauchy problem for Schrödinger equations with repulsive quadratic potential and power-like nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the no...