*Mathematical Methods in the Applied Sciences*Sign-changing solutions for the nonlinear
Schrödinger equation with generalized Chern-Simons
gauge theory

21 Jan 2020

25 Jan 2020

25 Jan 2020

27 Jan 2020

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
$k\equiv1$.