Multiple Solutions for the Chern–Simons–Schrödinger Equation with Indefinite Nonlinearities in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{2}$$\end{document}

In this paper, we study the following Chern–Simons–Schrödinger equation: -Δu+u+λh2(|x|)|x|2+∫|x|+∞h(s)su2(s)dsu=f(|x|)|u|p-2u+g(|x|)|u|q-2uinR2,u∈Hr1(R2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+u+\lambda \left( \frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+\infty }\frac{h(s)}{s} u^{2}(s)\textrm{d}s\right) u =f(|x|)|u|^{p-2}u\\&+g(|x|)|u|^{q-2}u~~\text {in}~\mathbb {R}^{2},\\&u\in H_{r}^{1}(\mathbb {R}^{2}), \end{aligned} \right. \end{aligned}$$\end{document}where 1<q<2<p<4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<q<2<p<4$$\end{document}, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} is a parameter, f∈C(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C(\mathbb {R}^{2})$$\end{document} is a bounded sign-changing function, g∈Lpp-q(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in L^{\frac{p}{p-q}}(\mathbb {R}^{2})$$\end{document}, and h(s)=12∫0sru2(r)dr.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\textrm{d}r. \end{aligned}$$\end{document}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 Hr1(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{r}^{1}(\mathbb {R}^{2})$$\end{document} 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.