A new proof on quasilinear Schrödinger equations with prescribed mass and combined nonlinearities
Jianhua Chen, Jijiang Sun, Chenggui Yuan, Jian Zhang
TL;DR
This work addresses normalized standing-wave solutions for a quasilinear Schrödinger equation with combined nonlinearities under a prescribed mass. It develops a dual variational framework using a $f$-transformation to restore differentiability and employs a monotonicity trick on a family of constrained functionals $\Phi_\theta$ to obtain bounded Palais–Smale sequences. The authors prove the existence of two distinct radial normalized solutions: a local minimizer and a mountain-pass type solution, for suitable ranges of exponents and small mass, with negative Lagrange multipliers, thereby resolving an open problem in prior work. The approach is robust and extensible to other quasilinear Schrödinger equations, providing a versatile toolkit for normalized solutions via the dual method.
Abstract
In this work, we study the quasilinear Schrödinger equation \begin{equation*} \aligned -Δu-Δ(u^2)u=|u|^{p-2}u+|u|^{q-2}u+λu,\,\, x\in\R^N, \endaligned \end{equation*} under the mass constraint \begin{equation*} \int_{\R^N}|u|^2\text{d}x=a, \end{equation*} where $N\geq2$, $2<p<2+\frac{4}{N}<4+\frac{4}{N}<q<22^*$, $a>0$ is a given mass and $λ$ is a Lagrange multiplier. As a continuation of our previous work (Chen et al., 2025, arXiv:2506.07346v1), we establish some results by means of a suitable change of variables as follows: \begin{itemize} \item[{\bf(i) }] {\bf qualitative analysis of the constrained minimization}\\ For $2<p<4+\frac{4}{N}\leq q<22^*$, we provide a detailed study of the minimization problem under some appropriate conditions on $a>0$; \end{itemize} \begin{itemize} \item[{\bf(ii)}]{\bf existence of two radial distinct normalized solutions}\\ For $2<p<2+\frac{4}{N}<4+\frac{4}{N}<q<22^*$, we obtain a local minimizer under the normalized constraint;\\ For $2<p<2+\frac{4}{N}<4+\frac{4}{N}<q\leq2^*$, we obtain a mountain pass type normalized solution distinct from the local minimizer. \end{itemize} Notably, the second result {\bf (ii)} resolves the open problem {\bf(OP1)} posed by (Chen et al., 2025, arXiv:2506.07346v1). Unlike previous approaches that rely on constructing Palais-Smale-Pohozaev sequences by [Jeanjean, 1997, Nonlinear Anal. {\bf 28}, 1633-1659], we obtain the mountain pass solution employing a new method, which lean upon the monotonicity trick developed by (Chang et al., 2024, Ann. Inst. H. Poincaré C Anal. Non Linéaire, {\bf 41}, 933-959). We emphasize that the methods developed in this work can be extended to investigate the existence of mountain pass-type normalized solutions for other classes of quasilinear Schrödinger equations.