An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well

In this paper, we study an eigenvalue problem for Schrödinger-Poisson system with indefinite nonlinearity and potential well as follows:

$ \begin{equation*} \begin{cases} -\Delta u+\mu V(x)u+K(x)\phi u = \lambda f(x)u+g(x)|u|^{p-2}u\ \ &\mbox{in}\ \ \mathbb{R}^3, \\ -\Delta \phi = K(x)u^2\ \ &\mbox{in}\ \ \mathbb{R}^3, \end{cases} \end{equation*} $

where $ 4\leq p<6 $, the parameters $ \mu, \lambda>0 $, $ V\in C(\mathbb{R}^3) $ is a potential well with the bottom $ \overline\Omega: = \{x\in\mathbb{R}^3 : V(x) = 0\} $, and the functions $ f $ and $ g $ are allowed to be sign-changing. By establishing an approximate estimate between the positive principal eigenvalue of $ -\Delta u+\mu V(x)u = \lambda f(x)u $ in $ \mathbb{R}^3 $ and the positive principal eigenvalue $ \lambda_1(f_{\Omega}) $ of $ -\Delta u = \lambda f_{\Omega}(x)u $ in $ \Omega $, we prove that at least a positive solution exists in $ 0<\lambda\leq\lambda_1(f_{\Omega}) $ while at least two positive solutions exist in $ \lambda>\lambda_1(f_{\Omega}) $ and near $ \lambda_1(f_{\Omega}) $, where $ f_{\Omega}: = f|_{\Omega} $. The results are obtained via variational method and steep potential. Furthermore, we also study the concentration of solutions as $ \mu\to\infty $ and the decay rate of solutions at infinity.