Blow-up for biharmonic Schrodinger equation with critical nonlinearity

Abstract

We consider the minimizers for the biharmonic nonlinear Schrödinger functional $${\mathcal{E}}_a(u)={\int }_{{\mathbb{R}}^d}|\Delta u(x){|}^{2}dx+{\int }_{{\mathbb{R}}^d}V(x)|u(x){|}^{2}dx-a{\int }_{{\mathbb{R}}^d}|u(x){|}^{q}dx$$ 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 $a^{\ast }$, which is the optimal constant in a Gagliardo--Nirenberg interpolation inequality.

13 pages, To appear Zeitschrift Angewandte Mathematik und Physik