Normalized Solutions to a Planar Schrödinger-Poisson System with Ring-Shaped Potentials and Inhomogeneous Interactions

This paper studies the constrained minimization problem for the Schrödinger-Poisson system with a ring-shaped potential and a logarithmic convolution term in ${ℝ}^2$. The energy functional is made well-defined by introducing a weighted Sobolev space $\mathcal{X}$ and decomposing the logarithmic kernel. Define the critical constant $a^{\text{*}}={\left|Q\right|}_2^2$, where $Q$ is the unique positive radially symmetric solution of the two-dimensional scalar field equation $-\Delta u+u=u^3$. Under suitable assumptions on the interaction coefficient $K\left(x\right)$, the existence and nonexistence of minimizers for the minimization problem $e\left(\gamma ,a\right)$ are systematically studied. The main results show that minimizers exist when $a<a^{\text{*}}$ and do not exist when $a>a^{\text{*}}$ or $a=a^{\text{*}}$ (with $\gamma >0$). Furthermore, for $\gamma >0$, the asymptotic behavior of minimizers is characterized as $a\nearrow a^{\text{*}}$.