Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system

This paper is concerned with the following Schrödinger-Poisson system

$ \begin{equation*} (P_\mu): -\Delta u +u + K(x)\phi u = |u|^{p-1}u + \mu h(x)u, \ -\Delta \phi = K(x) u^2, \ x\in\mathbb{R}^3, \end{equation*} $

where $ p\in (3,5) $, $ K(x) $ and $ h(x) $ are nonnegative functions, and $ \mu $ is a positive parameter. Let $ \mu_1 > 0 $ be an isolated first eigenvalue of the eigenvalue problem $ -\Delta u + u = \mu h(x)u $, $ u\in H^1(\mathbb{R}^3) $. As $ 0<\mu\leq\mu_1 $, we prove that $ (P_{\mu}) $ has at least one nonnegative bound state with positive energy. As $ \mu > \mu_1 $, there is $ \delta > 0 $ such that for any $ \mu\in (\mu_1, \mu_1 + \delta) $, $ (P_\mu) $ has a nonnegative ground state $ u_{0,\mu} $ with negative energy, and $ u_{0,\mu^{(n)}}\to 0 $ in $ H^1(\mathbb{R}^3) $ as $ \mu^{(n)}\downarrow \mu_1 $. Besides, $ (P_\mu) $ has another nonnegative bound state $ u_{2,\mu} $ with positive energy, and $ u_{2,\mu^{(n)}}\to u_{\mu_1} $ in $ H^1(\mathbb{R}^3) $ as $ \mu^{(n)}\downarrow \mu_1 $, where $ u_{\mu_1} $ is a bound state of $ (P_{\mu_1}) $.