Jin-Cai Kang | Searcharxiv

arXiv2023

Prescribed mass standing waves for Schrödinger-Maxwell equations with combined nonlinearities

In the present paper, we study the following Schrödinger-Maxwell equation with combined nonlinearities \begin{align*} \displaystyle - Δu+λu+ \left(|x|^{-1}\ast |u|^2\right)u =|u|^{p-2}u +μ|u|^{q-2}u\quad \text{in} \ \mathbb{ R}^3 \quad \quad \text{and}\quad \quad \int_{\mathbb{R}^3}|u|^2dx=a^2, \end{align*} where $a>0$, $μ\in \mathbb{R}$, $2<q\leq \frac{10}{3}\leq p<6$ with $q\neq p$, $\ast$ denotes the convolution and $λ\in \mathbb{R}$ appears as a Lagrange multiplier. Under some mild assumptions on $a$ and $μ$, we prove some existence, nonexistence and multiplicity of normalized solution to the above equation. Moreover, the asymptotic behavior of normalized solutions is verified as $μ\rightarrow 0$ and $q\rightarrow \frac{10}{3}$, and the stability/instability of the corresponding standing waves to the related time-dependent problem is also discussed.

math.AP↗

arXiv2022

Normalized solutions for the nonlinear Schrodinger equation with potential and combined nonlinearities

In present paper, we study the following nonlinear Schrödinger equation with combined power nonlinearities \begin{align*} - Δu+V(x)u+λu=|u|^{2^*-2}u+μ|u|^{q-2}u \quad \quad \text{in} \ \mathbb{ R}^N, \ N\geq 3 \end{align*} having prescribed mass \begin{align*} \int_{ \mathbb{ R}^N}u^2dx=a^2, \end{align*} where $μ, a>0$, $q\in(2, 2^*)$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent, $V$ is an external potential vanishing at infinity, and the parameter $λ\in \mathbb{R}$ appears as a Lagrange multiplier. Under some mild assumptions on $V$, for the $L^2$-subcritical perturbation $q\in(2, 2+\frac{4}{N})$, we prove that there exists $a_0>0$ such that the normalized solution with negative energy to the above problem with $μ>0$ can be obtained for $a\in (0, a_0)$; for the $L^2$-critical perturbation $q=2+\frac{4}{N}$, by limiting the range of $μ$, the positive ground state normalized solution to the above problem for any $a>0$ is also found with the aid of the Pohožaev constraint; moreover, for the $L^2$-supercritical perturbation $q\in( 2+\frac{4}{N}, 2^*)$, we get a positive ground state normalized solution for the above problem with $a>0$ and $μ>0$ by using the Pohožaev constraint. At the same time, the exponential decay property of the positive normalized solution is established, which is important for the instability analysis of the standing waves. Furthermore, we give a description of the ground state set and obtain the strong instability of the standing waves for $q\in[2+\frac{4}{N}, 2^*)$. This paper can be regarded as a generalization of Soave [J. Funct. Anal. (2020)] in a sense.

math.AP↗