Sign-changing Solutions for the Chern-Simons-Schrödinger Equation with Concave-convex Nonlinearities

Abstract

In this paper, we study the existence of sign-changing solutions for the Chern-Simons-Schrödinger equation with concave-convex nonlinearities

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+\omega u+\left( \frac{h^{2}(\vert x\vert )}{\vert x\vert ^{2}}+\int _{\vert x\vert }^{\infty }\frac{h(s)}{s}u^{2}(s)ds\right) u=\lambda K(|x|)|u|^{p-2}u+ |u|^{q-2}u \text { in}\,\,\mathbb {R}^2,\,\,\,\\&u\in H_r^1(\mathbb {R}^2), \end{aligned} \right. \end{aligned}$$

(0.1)

where $\omega , \lambda >0$ and $K\in L^{\frac{p}{2-p}}(\mathbb {R}^2, \mathbb {R}_{+}),\,\mathbb {R}_{+}:=(0,\infty ), \,1<p<2, \,q>6$. Using constrained minimization arguments and the quantitative deformation lemma, we prove that Eq. (0.1) has a sign-changing solution $u_{\lambda }$ with positive energy when there exists a constant $\lambda ^{*}>0$ such that for any $\lambda <\lambda ^{*}$.