Extremizers for the Strichartz Inequality for a Fourth-Order Schr\"odinger Equation

TL;DR

This paper proves the existence and analyzes the properties of extremizers for the Strichartz inequality related to a fourth-order Schrödinger equation in two spatial dimensions, revealing their exponential decay and analyticity.

Contribution

It establishes the existence of extremizers for the inequality using profile decomposition and studies their decay and regularity properties.

Findings

Extremizers exist for the Strichartz inequality in this setting.

Extremizers exhibit exponential decay.

Extremizers are analytic functions.

Abstract

In this paper, we consider the Strichartz inequality for a fourth-order Schr\"odinger equation on $\mathbb{R}^{2+1}$. We show that extremizers exist using a linear profile decomposition which follows from the endpoint version decomposition and the stationary phase method. Based on the existence of extremizers, we investigate the associated Euler-Lagrange equation to show that the extremizers have exponential decay and consequently must be analytic.