Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations (Q525547)

The author studies the asymptotic behavior of standing waves for the following nonlinear Schrödinger equation \[ i u_t=-\Delta u + V(x) u-a_q|u|^q u, \] where \(a_q>0\), \(0<q\leq 2\), and \(V(x)\) is a trapping potential. Let \(a^*= \|Q\|_2^2\), where \(Q\) is the unique (up to translations) positive radial solution of \(\Delta u-u+u^3=0\) in \(\mathbb R^2\). It is proved that if \(\lim_{q\nearrow 2}a_q =a<a^*\), then \(\lim_{q\nearrow 2}d_{a_q}(q) =d_a(2)\), where \(d_{a_q}\) is the minimizer of a suitable functional. Moreover, for any sequence \(\{q_k\}\) with \(\lim_{k\to \infty}q_k=2\), there exists a subsequence such that \(\lim_{k\to\infty}u_{q_k}(x)=u_0 \) with \(u_0\) being a minimizer of \(d_a(2)\). If \(\lim_{q\nearrow 2}a_q =a\geq a^*\), by directly using asymptotic analysis, the author proves that all minimizers blow up and the detailed asymptotic behavior of the minimizers is given.