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

Fengshuang Gao, Yuxia Guo
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Abstract

This paper is concerned with the existence of normalized solutions to a mass-supercritical quasilinear Schrödinger equation:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-u\Delta u^2+V(x)u+\lambda u=g(u),\hbox { in }{\mathbb {R}}^N, \\ u\ge 0, \end{array}\right. \end{aligned}$$(0.1)

satisfying the constraint \(\int _{{\mathbb {R}}^N}u^2=a\). 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 \(N=2\), 3. Moreover, when V(x) is not too small in some sense, we show the existence of a normalized solution for \(N\ge 2\) and \(g(u)={u}^{q-2}u\) with \(4+\frac{4}{N}<q<2\cdot 2^*\).

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带势能的质量超临界准薛定谔方程的归一化解的存在性
本文关注质量超临界准线性薛定谔方程的归一化解的存在: $$\begin{aligned}-Delta u-u\Delta u^2+V(x)u+\lambda u=g(u),\hbox { in }{{mathbb {R}}^N,\ u\ge 0,\end{array}\right.\end{aligned}$$(0.1)satisfying the constraint \(\int _{{mathbb {R}}^N}u^2=a\).我们将研究势和非线性如何影响归一化解的存在。因此,在V(x)较小的假设条件和g相对严格的增长条件下,我们得到了\(N=2\), 3的归一化解。 此外,当V(x)在某种意义上不算太小时,我们证明了\(N\ge 2\) and \(g(u)={u}^{q-2}u\) with \(4+\frac{4}{N}<q<2\cdot 2^*\)的归一化解的存在。
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