Multiple Solutions for the Chern–Simons–Schrödinger Equation with Indefinite Nonlinearities in \(\mathbb {R}^{2}\)

In this paper, we study the following Chern–Simons–Schrödinger equation: \(\begin{aligned} \left\{ \begin{aligned}&-\Delta u+u+\lambda \left( \frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+\infty }\frac{h(s)}{s} u^{2}(s)\textrm{d}s\right) u =f(|x|)|u|^{p-2}u\\&+g(|x|)|u|^{q-2}u~~\text {in}~\mathbb {R}^{2},\\&u\in H_{r}^{1}(\mathbb {R}^{2}), \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& -\mathrm{\Delta }u+u+\lambda \left(\frac{h^2(|x|)}{{|x|}^2},+,{\int }_{|x|}^{+\infty },\frac{h(s)}{s},u^2,(s),\text{d},s\right)u={f(|x|)|u|}^{p-2}u\hfill \\ \hfill & +{g(|x|)|u|}^{q-2}u\text{in}{\mathbb{R}}^2,\hfill \\ \hfill & u\in H_r^1({\mathbb{R}}^2),\hfill \end{array}\right)\end{array}$$ where \(1<q<2<p<4\) $1<q<2<p<4$ , \(\lambda >0\) $\lambda >0$ is a parameter, \(f\in C(\mathbb {R}^{2})\) $f\in C({\mathbb{R}}^2)$ is a bounded sign-changing function, \(g\in L^{\frac{p}{p-q}}(\mathbb {R}^{2})\) $g\in L^{\frac{p}{p-q}}({\mathbb{R}}^2)$ , and \(\begin{aligned} h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\textrm{d}r. \end{aligned}\) $$\begin{array}{c}\hfill h(s)=\frac{1}{2}{\int }_0^{s}r{u}^2(r)\text{d}r.\end{array}$$ Such problem cannot be studied by applying variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because Palais–Smale sequences may not be bounded. In this paper, by using some inequality estimates and imposing a proper scope for the parameter, we prove that the energy functional is coercive and bounded from below on \(H_{r}^{1}(\mathbb {R}^{2})\) $H_r^1({\mathbb{R}}^2)$ and obtain two negative-energy solutions. Moreover, by developing an innovative constraint method of the Nehari manifold, three solutions are studied for the above problem.