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Sign-changing solutions for the nonlinear Schrödinger equation with generalized Chern-Simons gauge theory
  • Liejun Shen
Liejun Shen
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University
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Peer review status:UNDER REVIEW

21 Jan 2020Submitted to Mathematical Methods in the Applied Sciences
25 Jan 2020Assigned to Editor
25 Jan 2020Submission Checks Completed
27 Jan 2020Reviewer(s) Assigned


We study the existence and asymptotic behavior of least energy sign-changing solutions for the nonlinear Schr\“{o}dinger equation coupled with the Chern-Simons gauge theory \[ \left\{ \begin{gathered} -\Delta u+ \omega u+\lambda \sum_{j=1}^k\bigg( \frac{h^2(|x|)}{|x|^2}u^{2(j-1)} +\frac{1}{j}\int_{|x|}^\infty \frac{h(s)}{s}u^{2j}(s) ds \bigg)u= f(u) \ \ \text{in}\ \ \mathbb{R}^2 , \hfill \\ {\text{ }}u \in {H^1_r}({\mathbb{R}^2}), \hfill \\ \end{gathered} \right. \] where $\omega, ~\lambda >0$ are constants, $k\in \mathbb{N}^+$ and \[ h(s)=\int_0^s\frac{r}{2}u^2(r)dr. \] Under some suitable assumptions on $f\in C(\R)$, with the help of the Gagliardo-Nirenberg inequality, we apply the constraint minimization argument to obtain a least energy sign-changing solution $u_\lambda$ with precisely two nodal domains. Furthermore, we prove that the energy of $u_\lambda$ is strictly larger than two times of the ground state energy and analyze the asymptotic behavior of $u_\lambda$ as $\lambda\searrow0^+$. Our results cover and improve the existing ones for the gauged nonlinear Schr\”{o}dinger equation when $k\equiv1$.