Another look at quasilinear Schrödinger equations with prescribed mass via dual method

Abstract: In this paper, we aim to study the existence of ground state normalized solutions for the following quasilinear Schr\"{o}dinger equation $-\Delta u-\Delta(u^2)u=h(u)+\lambda u,\,\, x\in\R^N$, under the mass constraint $\int_{\R^N}|u|^2\text{d}x=a,$ where $N\geq2$, $a>0$ is a given mass, $\lambda$ 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 $-\Delta v =h(f(v))f'(v)+\lambda f(v)f'(v),\,\, x\in\R^N,$ 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 Schr\"{o}dinger equations via the dual method.