Existence of steady states for the Maxwell-Schrödinger-Poisson system: exploring the applicability of the concentration-compactness principle (Q2868592)

The work considers the existence of solutions in the form \(\psi(x,t)=\exp(i\ell_{M}t)\varphi(x)\) for the non-linear non-local Schödinger-type equation \(i\partial_t\psi=-\Delta_x\psi+\epsilon(|\psi|^2\star|x|^{-1})\psi-C|\psi|^{2\alpha}\psi\), which is the reduction of the Maxwell-Schrödinger-Poisson system. Here \(\star\) means a convolution. The problem of existence is considered from the point of view of the concentration-compactness method and the table of the existence results is provided for various intervals of \(\alpha\in [0\,2]\).