标题： 扩展区域中Schrödinger-Poisson系统的极小归一化解：多重性和渐近行为 显示英文标题

标题： Small normalised solutions for a Schrödinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour

摘要： 给定一个光滑有界区域$\Omega\subset \mathbb R^3$，我们考虑以下非线性薛定谔-泊松型系统\begin{equation*} \left\{ \begin{array}{ll} -\Delta u+ \phi u -\abs{u}^{p-2}u = \omega u & \quad \text{in } \lambda\Omega, -\Delta\phi =u^{2}& \quad \text{in }\lambda\Omega, u>0 &\quad \text{in }\lambda\Omega, u =\phi=0 &\quad \text{on }\partial (\lambda\Omega), \int_{\lambda\Omega}u^{2} \,\text{d} x=\rho^2 \end{array} \right. \end{equation*}在扩展区域$\lambda\Omega\subset \mathbb R^{3}, \lambda>1$和$p\in (2,3)$中，未知数为$(u,\phi,\omega)$。 我们证明，对于膨胀参数$\lambda$的任意大值和质量$\rho>0$的任意小值，解的数量至少是$\lambda\Omega$的Ljusternick-Schnirelmann范畴。 此外，我们证明当$\lambda\to+\infty$时，所找到的解收敛到整个空间$\mathbb R^{3}$中问题的基态。 显示英文摘要

摘要： Given a smooth bounded domain $\Omega\subset \mathbb R^3$, we consider the following nonlinear Schr\"odinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+ \phi u -\abs{u}^{p-2}u = \omega u & \quad \text{in } \lambda\Omega, -\Delta\phi =u^{2}& \quad \text{in }\lambda\Omega, u>0 &\quad \text{in }\lambda\Omega, u =\phi=0 &\quad \text{on }\partial (\lambda\Omega), \int_{\lambda\Omega}u^{2} \,\text{d} x=\rho^2 \end{array} \right. \end{equation*} in the expanding domain $\lambda\Omega\subset \mathbb R^{3}, \lambda>1$ and $p\in (2,3)$, in the unknowns $(u,\phi,\omega)$. We show that, for arbitrary large values of the expanding parameter $\lambda$ and arbitrary small values of the mass $\rho>0$, the number of solutions is at least the Ljusternick-Schnirelmann category of $\lambda\Omega$. Moreover we show that as $\lambda\to+\infty$ the solutions found converge to a ground state of the problem in the whole space $\mathbb R^{3}$.