Blowup for biharmonic Schrödinger equation with critical nonlinearity

Abstract

We consider the minimizers for the biharmonic nonlinear Schrödinger functional

$$\begin{aligned} \mathcal {E}_a(u)=\int \limits _{\mathbb {R}^d} |\Delta u(x)|^2 \mathrm{d}x + \int \limits _{\mathbb {R}^d} V(x) |u(x)|^2 \mathrm{d}x - a \int \limits _{\mathbb {R}^d} |u(x)|^{q} \mathrm{d}x \end{aligned}$$

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 blowup behavior of the minimizers when a tends to a critical value $a^*$, which is the optimal constant in a Gagliardo–Nirenberg interpolation inequality.