We consider the radially symmetric positive solutions to quasilinear problem

\begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*}

having prescribed mass $\int_{\mathbb{R}^{N}}|u|^2 =a^2,$ where a > 0 is a constant, λ appears as a Lagrange multiplier. We focus on the pure L2-supercritical case and combination case of L2-subcritical and L2-supercritical nonlinearities

\begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*}

where $\bar{p}:=4+\frac{4}{N}$ is the L2-critical exponent. Our work extends and develops some recent results in the literature.

In this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$

This paper is concerned with the stability and instability of standing waves for the quasilinear Schrödinger equation of the form which has been derived in many models from mathematical physics. We find the exact threshold depending upon the interplay of quasilinear and nonlinear terms that separates stability and instability. More precisely, we prove that for α ∈ and odd p ∈ , when , the standing wave is stable, and when (where for N ≥ 3 and 2 α ċ 2* = +∞ for N = 2), the standing wave is strongly unstable. Our results show that the quasilinear term 2 α(△|φ|2α)|φ|2α−2φ makes the standing waves more stable, which is consistent with the physical phenomena.