On Chern-Simons-Schrödinger systems involving steep potential well and concave-convex nonlinearities

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.