## Hokkaido Mathematical Journal

### Existence and decay of solutions to a semilinear Schrödinger equation with magnetic field

Shin-ichi SHIRAI

#### Abstract

In this paper we study the decay property of solutions of a semilinear Schrödinger equation, $-(∇ - iA)^{2}u+(V-E)u=Q|u|^{p-2}u$, on $\mathbb R^{n}$, where $n \geq 2$ and $2<p <2^{*}$. We give a lower bound estimate of nontrivial solutions at infinity. In two-dimensional case, we give super-exponential decay estimates of solutions at infinity. Moreover, we show the existence of a nontrivial solution under additional assumptions on potentials.

#### Article information

Source
Hokkaido Math. J., Volume 37, Number 2 (2008), 241-273.

Dates
First available in Project Euclid: 21 September 2009

https://projecteuclid.org/euclid.hokmj/1253539554

Digital Object Identifier
doi:10.14492/hokmj/1253539554

Mathematical Reviews number (MathSciNet)
MR2415900

Zentralblatt MATH identifier
1144.35048

#### Citation

SHIRAI, Shin-ichi. Existence and decay of solutions to a semilinear Schrödinger equation with magnetic field. Hokkaido Math. J. 37 (2008), no. 2, 241--273. doi:10.14492/hokmj/1253539554. https://projecteuclid.org/euclid.hokmj/1253539554