September/October 2022 On Chern-Simons-Schrödinger systems involving steep potential well and concave-convex nonlinearities
Yingying Xiao, Chuanxi Zhu
Adv. Differential Equations 27(9/10): 543-570 (September/October 2022). DOI: 10.57262/ade027-0910-543

Abstract

In this paper, we study the following gauged nonlinear Schrödinger equation \begin{align*} -\Delta u + \lambda V(|x|)u & + \mu\Big(\int_{|x|}^{ + \infty} \frac{h(s)}{s} u^2(s)\text{d}s + \frac{h^2(|x|)}{|x|^2}\Big)u \\ & = a(|x|)|u|^{q-2}u + b(|x|)|u|^{p-2}u~~~~~~~ \text{in}~~\mathbb R^2, \end{align*} where $\lambda > 0$, $\mu > 0$, $h(s) = \frac{1}{2}\int_0^s{u^2(l)l}\text{d}l$ and $a\in L^\infty(\mathbb R^2)$, $b\in L^{\frac{q}{q-p}}(\mathbb R^2, \mathbb R^ + )$, $1 < p < 2$ $ < q < 6$ and the potential well $V \in \mathcal{C}(\mathbb R^2,\mathbb R)$, $V^{-1}(0)$ has nonempty interior. First, the existence of the positive energy solutions is obtained by using the truncation technique. Secondly, we also explore the asymptotic behavior of the positive energy solutions as $\lambda\rightarrow + \infty$ and $\mu\rightarrow 0$. Finally, we give the existence of a negative energy solution via Ekeland variational principle. Our result extend and supplement some recent results in related literatures.

Citation

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Yingying Xiao. Chuanxi Zhu. "On Chern-Simons-Schrödinger systems involving steep potential well and concave-convex nonlinearities." Adv. Differential Equations 27 (9/10) 543 - 570, September/October 2022. https://doi.org/10.57262/ade027-0910-543

Information

Published: September/October 2022
First available in Project Euclid: 2 June 2022

Digital Object Identifier: 10.57262/ade027-0910-543

Subjects:
Primary: 35J20 , 35J60

Rights: Copyright © 2022 Khayyam Publishing, Inc.

Vol.27 • No. 9/10 • September/October 2022
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