The existence of normalized solutions for $L^2$-critical
quasilinear Schrödinger equations
Abstract
In this paper, we study the existence of critical points for the
following functional
$$I(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla
u|^2+\ds\int_{\R^N}|u|^2|\nabla
u|^2-\frac{N}{4(N+1)}\ds\int_{\R^N}|u|^{\frac{4(N+1)}{N}},$$
constrained on $S_c=\{u\in
H^1(\R^N)|~\int_{\R^N}|u|^2|\nabla
u|^2<+\infty,~|u|_2=c,c>0\}$,
where $N\geq1$. The constraint problem is
$L^2$-critical. We prove that the minimization problem
$i_c=\inf\limits_{u\in
S_c}I(u)$ has no minimizer for all $c>0$. We also
obtain a threshold value of $c$ separating the existence and
nonexistence of critical points for $I(u)$ restricted to $S_c$.