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

Abstract

<p style='text-indent:20px;'>This paper is concerned with the following Schrödinger-Poisson system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> $ \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*} $ </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">$ p\in (3,5) $</tex-math></inline-formula>, <inline-formula><tex-math id="M2">$ K(x) $</tex-math></inline-formula> and <inline-formula><tex-math id="M3">$ h(x) $</tex-math></inline-formula> are nonnegative functions, and <inline-formula><tex-math id="M4">$ \mu $</tex-math></inline-formula> is a positive parameter. Let <inline-formula><tex-math id="M5">$ \mu_1 > 0 $</tex-math></inline-formula> be an isolated first eigenvalue of the eigenvalue problem <inline-formula><tex-math id="M6">$ -\Delta u + u = \mu h(x)u $</tex-math></inline-formula>, <inline-formula><tex-math id="M7">$ u\in H^1(\mathbb{R}^3) $</tex-math></inline-formula>. As <inline-formula><tex-math id="M8">$ 0<\mu\leq\mu_1 $</tex-math></inline-formula>, we prove that <inline-formula><tex-math id="M9">$ (P_{\mu}) $</tex-math></inline-formula> has at least one nonnegative bound state with positive energy. As <inline-formula><tex-math id="M10">$ \mu > \mu_1 $</tex-math></inline-formula>, there is <inline-formula><tex-math id="M11">$ \delta > 0 $</tex-math></inline-formula> such that for any <inline-formula><tex-math id="M12">$ \mu\in (\mu_1, \mu_1 + \delta) $</tex-math></inline-formula>, <inline-formula><tex-math id="M13">$ (P_\mu) $</tex-math></inline-formula> has a nonnegative ground state <inline-formula><tex-math id="M14">$ u_{0,\mu} $</tex-math></inline-formula> with negative energy, and <inline-formula><tex-math id="M15">$ u_{0,\mu^{(n)}}\to 0 $</tex-math></inline-formula> in <inline-formula><tex-math id="M16">$ H^1(\mathbb{R}^3) $</tex-math></inline-formula> as <inline-formula><tex-math id="M17">$ \mu^{(n)}\downarrow \mu_1 $</tex-math></inline-formula>. Besides, <inline-formula><tex-math id="M18">$ (P_\mu) $</tex-math></inline-formula> has another nonnegative bound state <inline-formula><tex-math id="M19">$ u_{2,\mu} $</tex-math></inline-formula> with positive energy, and <inline-formula><tex-math id="M20">$ u_{2,\mu^{(n)}}\to u_{\mu_1} $</tex-math></inline-formula> in <inline-formula><tex-math id="M21">$ H^1(\mathbb{R}^3) $</tex-math></inline-formula> as <inline-formula><tex-math id="M22">$ \mu^{(n)}\downarrow \mu_1 $</tex-math></inline-formula>, where <inline-formula><tex-math id="M23">$ u_{\mu_1} $</tex-math></inline-formula> is a bound state of <inline-formula><tex-math id="M24">$ (P_{\mu_1}) $</tex-math></inline-formula>.</p>