This article is concerned with the quasilinear Schrödinger equation \( \Delta u-\omega u+|u|^{p-1}u+\delta \Delta (|u|^2)u=0, \) where \(\delta >0\) , \(N=2\) and \(p>1\) or \(N\geqslant 3\) and \(1<p<\frac{3N+2}{N-2}\) . After proving uniqueness and non-degeneracy of the positive solution \(u_\omega \) for all \(\omega >0\) , our main results establish the asymptotic behavior of \(u_\omega \) in the limit \(\omega \rightarrow 0^+\) . Three different regimes arise, termed ‘subcritical’, ‘critical’ and ‘supercritical’, corresponding respectively (when \(N\geqslant 3\) ) to \(1<p<\frac{N+2}{N-2}\) , \(p=\frac{N+2}{N-2}\) and \(\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}\) . In each case a limit equation is exhibited which governs, in a suitable scaling, the behavior of \(u_\omega \) in the limit \(\omega \rightarrow 0^+\) . The critical case is the most challenging, technically speaking. In this case, the limit equation is the famous Lane-Emden-Fowler equation. A substantial part of our efforts is dedicated to the study of the function \(\omega \mapsto M(\omega )=\int _{\mathbb {R}^N}u_\omega ^2\) . We find that, for small \(\omega >0\) , \(M(\omega )\) is increasing if \(1<p\leqslant 1+\frac{4}{N}\) and decreasing if \(1+\frac{4}{N}< p\leqslant \frac{N+2}{N-2}\) . In the supercritical case, the monotonicity of \(M(\omega )\) depends on the dimension, except in the regime \(p\geqslant 3+\frac{4}{N}\) , where \(M(\omega )\) is always decreasing close to \(\omega =0\) . The crucial role played by \(M(\omega )\) for the orbital stability of the standing wave \(e^{i\omega t}u_\omega \) , and for the uniqueness of normalized ground states, is discussed in the introduction.
