题目：Normalized ground states for a quasi-linear Schr\"odinger equations: Mass super-critical case
摘要：In this paper, we show the existence of normalized ground states of the following quasilinear elliptic equation:
$$-\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u~\quad \hbox{in}~\R^N, N\geq 1,$$
with prescribed mass $\displaystyle\int_{\R^N}|u|^2 \mathrm{d}x=a$. We are interest in the mass super-critical case $4+\frac{4}{N}<p<2\cdot 2^*$, where $2^*:=\frac{2N}{N-2}$ for $N\geq 3$, while $2^*:=+\infty$ for $N=1,2$.
Our existence results are optimal. Precisely, for $1\leq N\leq 4,4+\frac{4}{N}<p<2\cdot 2^*$, we can prove the existence of normalized ground state for all mass $a>0$. For $N\geq 5, 4+\frac{4}{N}<p<2\cdot 2^*$, we can find a precise number $a_0$ such that the existence of normalized ground state is true if and only if $a\in (0, a_0]$.
This is the first result in the super $2^*$ case (i.e. $p>2^*$) for $N\geq 3$. In previous literature, scholars have utilized the key condition $p\leq 2^*$ to obtain the $L^2$-compactness. And thus this is also the first result for the mass super-critical $p>4+\frac{4}{N}$ when $N\geq 4$.
We also study the asymptotic behavior of the normalized ground states as the mass $a\rightarrow \infty$ as well as $a\rightarrow 0$. For the completeness of this project, we also study the uniqueness of positive radial solutions to a class of overdetermined problem, which extend the results of Serrin and Tang[Indiana Univ. Math. J., 2000, 897--923] for the semilinear case to the low dimension case. This is a joint work with Prof. Louis Jeanjean and Prof. Jianjun Zhang.
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