Ground state and multiple normalized solutions of quasilinear Schrödinger equations in the $L^2$-supercritical case and the Sobolev critical case

TL;DR

This work analyzes normalized (mass-constrained) solutions to a quasilinear Schrödinger equation in dimensions $N=3,4$ with nonlinearity exponent $p$ in the $L^2$-supercritical to Sobolev-critical range. The authors develop a perturbation framework $I_\mu$ and a Pohozaev manifold to overcome nondifferentiability, establishing existence of ground-state solutions in the nonradial space and infinitely many radial normalized solutions via variational minimax methods and genus theory; they also prove nonexistence in the Sobolev critical case and characterize energy behavior as the prescribed mass $a$ grows or vanishes. The techniques include a constrained minimax on $\mathcal{S}(a)$, a detailed analysis of the scaling function $s(u)$ ensuring $P_\mu(s(u)*u)=0$, and careful compactness arguments to pass to the limit as $\mu\to0^+$. These results extend and unify several prior findings, providing a comprehensive view of ground-state and multiplicity phenomena under $L^2$ constraints.

Abstract

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schrödinger equation \begin{equation*} \begin{aligned} -Δu-uΔu^2 +λu=|u|^{p-2}u \quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N=3,4$, $λ$ appears as a Lagrange multiplier and $p \in (4+\frac{4}{N},2\cdot2^*]$. The solutions correspond to critical points of the energy functional subject to the $L^2$-norm constraint $\int_{\mathbb{R}^N}|u|^2dx=a^2>0$. In the Sobolev critical case $p=2\cdot 2^*$, the energy functional has no critical point. As for $L^2$-supercritical case $p \in (4+\frac{4}{N},2\cdot2^*)$: on the one hand, taking into account Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions for the non-radial case; on the other hand, we get the existence of infinitely many normalized solutions in $H^1_r(\mathbb{R}^N)$. Moreover, our results cover several relevant existing results. And in the end, we get the asymptotic properties of energy as $a$ tends to $+\infty$ and $a$ tends to $0^+$.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof