On the existence of stationary solutions of nonlinear Schrödinger equations

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.