Normalized solutions for a Sobolev critical quasilinear Schr\"odinger equation

TL;DR

This paper investigates the existence of normalized solutions for a Sobolev critical quasilinear Schrödinger equation with various nonlinearities, providing new existence and non-existence results across different parameter regimes.

Contribution

It introduces novel energy estimates and convergence theorems to establish multiple types of solutions for the equation, extending the analysis to the entire nonlinear range.

Findings

Existence of two solutions for certain q ranges, including a local minimizer and mountain pass solution.

Existence of mountain pass solutions under specific conditions for larger q.

Non-existence results when the parameter τ is non-positive.

Abstract

In this paper, we study the existence of normalized solutions for the following quasilinear Schr\"odinger equation with Sobolev critical exponent: \begin{eqnarray*} -\Delta u-u\Delta (u^2)+\lambda u=\tau|u|^{q-2}u+|u|^{2\cdot2^*-2}u,~~~~x\in\mathbb{R}^N, \end{eqnarray*} under the mass constraint $\int_{\mathbb{R}^N}|u|^2dx=c$ for some prescribed $c>0$. Here $\tau\in \mathbb{R}$ is a parameter, $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier, $N\ge3$, $2^*:=\frac{2N}{N-2}$ and $2<q<2\cdot2^*$. By deriving precise energy level estimates and establishing new convergence theorems, we apply the perturbation method to establish several existence results for $\tau>0$ in the Sobolev critical regime: (a) For the case of $2<q<2+\frac{4}{N}$, we obtain the existence of two solutions, one of which is a local minimizer, and the other is a mountain pass type solution, under explicit conditions on $c>0$; (b) For the case of $2+\frac{4}{N}\leq q<4+\frac{4}{N}$, we obtain the existence of normalized solutions of mountain pass type under different conditions on $c>0$; (c) For the case of $4+\frac{4}{N}\leq q<2\cdot2^*$, we obtain the existence of a ground state normalized solution under different conditions on $c>0$. Moreover, when $\tau\le 0$, we derive the non-existence result for $2<q<2\cdot2^*$ and all $c>0$. Our research provides a comprehensive analysis across the entire range $q\in(2, 2 \cdot 2^*)$ and for all $N\ge3$. The methods we have developed are flexible and can be extended to a broader class of nonlinearities.