Another look at quasilinear Schrödinger equations with prescribed mass via dual method
Jianhua Chen, Vicentiu D. Radulescu, Jijiang Sun, Jian Zhang
TL;DR
This work develops a dual-method framework for proving the existence of ground state normalized solutions to a quasilinear Schrödinger equation with prescribed mass. By a nonlinear change of variables u=f(v) and a novel mass-preserving scaling v_t, the authors transform the problem into a semilinear setting and construct a constrained variational structure on 𝒮_a and a Pohozaev-type manifold 𝒩_a. They obtain broad existence results across subcritical, supercritical, and critical growth regimes in dimensions 2 and 3, including Sobolev-critical and exponential-critical scenarios, and provide asymptotic analyses of the ground-state energy. The results extend and refine prior work by Colin–Jeanjean–Squassina, Jeanjean–Luo–Wang, Li–Zou, and Zhang–Li–Wang, and illustrate the adaptability of the dual method to normalized solutions for quasilinear Schrödinger equations.
Abstract
In this paper, we aim to study the existence of ground state normalized solutions for the following quasilinear Schrödinger equation $-Δu-Δ(u^2)u=h(u)+λu,\,\, x\in\R^N$, under the mass constraint $\int_{\R^N}|u|^2\text{d}x=a,$ where $N\geq2$, $a>0$ is a given mass, $λ$ is a Lagrange multiplier and $h$ is a nonlinear reaction term with some suitable conditions. By employing a suitable transformation $u=f(v)$, we reformulate the original problem into the equivalent form $-Δv =h(f(v))f'(v)+λf(v)f'(v),\,\, x\in\R^N,$ with prescribed mass $ \int_{\R^N}|f(v)|^2\text{d}x=a. $ To address the challenge posed by the $L^2$-norm $\|f(v)\|^2_2$ not necessarily equaling $a$, we introduce a novel stretching mapping: $ v_t(x):=f^{-1}(t^{N/2}f(v(tx))). $ This construction, combined with a dual method and detailed analytical techniques, enables us to establish the following existence results: (1)Existence of solutions via constrained minimization using dual methods; (2) Existence of ground state normalized solutions under general $L^2$-supercritical growth conditions, along with nonexistence results, analyzed via dual methods; (3)Existence of normalized solutions under critical growth conditions, treated via dual methods. Additionally, we analyze the asymptotic behavior of the ground state energy obtained in {\bf(P2)}. Our results extend and refine those of Colin-Jeanjean-Squassina [Nonlinearity 20: 1353-1385, 2010], of Jeanjean-Luo-Wang [J. Differ. Equ. 259: 3894-3928, 2015], of Li-Zou [Pacific J. Math. 322: 99-138, 2023], of Zhang-Li-Wang [Topol. Math. Nonl. Anal. 61: 465-489, 2023] and so on. We believe that the methodology developed here can be adapted to study related problems concerning the existence of normalized solutions for quasilinear Schrödinger equations via the dual method.