Blow-up for biharmonic Schrodinger equation with critical nonlinearity

TL;DR

This paper investigates the existence and blow-up behavior of minimizers for a biharmonic nonlinear Schrödinger functional with critical nonlinearity, focusing on the threshold parameter where solutions become unbounded.

Contribution

It establishes the existence and blow-up behavior of minimizers at the critical parameter value for the biharmonic nonlinear Schrödinger functional with a specific critical nonlinearity.

Findings

Existence of minimizers for subcritical parameters

Blow-up of minimizers as parameter approaches critical value

Identification of the critical constant in the Gagliardo--Nirenberg inequality

Abstract

We consider the minimizers for the biharmonic nonlinear Schr\"odinger functional $$ \mathcal{E}_a(u)=\int_{\mathbb{R}^d} |\Delta u(x)|^2 d x + \int_{\mathbb{R}^d} V(x) |u(x)|^2 d x - a \int_{\mathbb{R}^d} |u(x)|^{q} d x $$ with the mass constraint $\int |u|^2=1$. We focus on the special power $q=2(1+4/d)$, which makes the nonlinear term $\int |u|^q$ scales similarly to the biharmonic term $\int |\Delta u|^2$. Our main results are the existence and blow-up behavior of the minimizers when $a$ tends to a critical value $a^*$, which is the optimal constant in a Gagliardo--Nirenberg interpolation inequality.