Existence of Normalized Solutions for Mass Super-Critical Quasilinear Schrödinger Equation with Potentials

This paper is concerned with the existence of normalized solutions to a mass-supercritical quasilinear Schrödinger equation: 0.1 \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-u\Delta u^2+V(x)u+\lambda u=g(u),\hbox { in }{\mathbb {R}}^N, \\ u\ge 0, \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u-u\mathrm{\Delta }{u}^2+V(x)u+\lambda u=g(u),\text{in}{\mathbb{R}}^N,\hfill \\ u\ge 0,\hfill \end{array}\right)\end{array}$$ satisfying the constraint \(\int _{{\mathbb {R}}^N}u^2=a\) ${\int }_{{\mathbb{R}}^N}{u}^2=a$ . We will investigate how the potential and the nonlinearity effect the existence of the normalized solution. As a consequence, under a smallness assumption on V(x) and a relatively strict growth condition on g, we obtain a normalized solution for \(N=2\) $N=2$ , 3. Moreover, when V(x) is not too small in some sense, we show the existence of a normalized solution for \(N\ge 2\) $N\ge 2$ and \(g(u)={u}^{q-2}u\) $g(u)=u^{q-2}u$ with \(4+\frac{4}{N}<q<2\cdot 2^*\) $4+\frac{4}{N}<q<2·2^{\ast }$ .