Blowup for biharmonic Schrödinger equation with critical nonlinearity

Abstract

We consider the minimizers for the biharmonic nonlinear Schrodinger 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.