PDE Seminar Abstracts

# On the existence of stationary solutions of nonlinear Schrödinger equations

Huan-Song Zhou
Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan
26 Oct 2009 3-4pm, Carslaw Room 454

## Abstract

In this talk, we will talk about the following type of nonlinear stationary Schrödinger equation: $-\Delta u+\left(1+\lambda g\left(x\right)\right)u=f\left(u\right)$ and $u>0$ in ${ℝ}^{N}$, $u\in {H}^{1}\left({ℝ}^{N}\right)$ $N\ge 3$, where $\lambda \ge 0$, $g\left(x\right)\in {L}^{\infty }\left({ℝ}^{N}\right)$ which vanishes on a bounded domain in ${ℝ}^{N}$, and $f\left(s\right)~\left(\alpha +1\right)s$ at $+\infty$. We discuss how the existence and behavior of the solutions of the equation depend on the parameters $\lambda$ and $\alpha$. This is joint work with C.A.Stuart and Z.P. Wang.