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- Title
Ground state sign-changing solutions for the Chern-Simons-Schrödinger equation with zero mass potential.
- Authors
Zhang, Ning; Tang, Xianhua; Mi, Heilong
- Abstract
This paper is concerned with the following nonlinear Chern-Simons-Schrödinger equation with zero mass potential$ \begin{align} -\Delta u+\lambda\left(\frac{h_u^2(|x|)}{|x|^2}+\int^{\infty}_{|x|}\frac{h_u(s)}{s}u^2(s)ds\right)u = -a|u|^{p-2}u+f(u),\ \ \ x\in\mathbb{R}^2, \end{align} $where $ a, \lambda>0 $, $ p\in (2,3) $ and$ \begin{align} h_u(s) = \int^s_0\frac{\tau}{2}u^2(\tau)d\tau = \frac{1}{4\pi}\int_{B_s}u^2(x)dx \end{align} $is the so-called Chern-Simons term, $ f $ has subcritical exponential growth in the sense of Trudinger-Moser. Under some wild assumptions on $ f $, combining Trudinger-Moser inequality, Non-Nehari manifold method and constraint minimization argument, we establish the existence of a ground state sign-changing solution $ u_{\lambda} $ for the above equation, and the corresponding energy is strictly larger than twice that of the ground state solutions of Nehari-type. Moreover, we obtain the convergence property of $ u_{\lambda} $ as parameter $ \lambda \searrow 0 $. Our theorems extend the results of Xie and Chen [Appl. Anal., 99 (2020), 880-898], Kang, Li and Tang [Bull. Malays. Math. Sci. Soc., 44 (2021), 711-731] and Shen [Complex Var. Elliptic Equ., 67 (2022), 1186-1203].
- Subjects
NONLINEAR equations; EQUATIONS; MALAYS (Asian people)
- Publication
Discrete & Continuous Dynamical Systems - Series S, 2023, Vol 16, Issue 11, p1
- ISSN
1937-1632
- Publication type
Article
- DOI
10.3934/dcdss.2023137