Existence of steady states for the Maxwell-Schrödinger-Poisson system: exploring the applicability of the concentration-compactness principle. (English) Zbl 1283.35117
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 \(\alpha\in [0\,2]\).
Reviewer: Eugene Postnikov (Kursk)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 82D10 | Statistical mechanics of plasmas |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 82D37 | Statistical mechanics of semiconductors |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
Keywords:
steady states; variational methods; subadditivity inequality; constrained minimization; sharp nonexistence. \(L^{2}\)-norm constraint; Schrödinger-Poisson; Maxwell-Schrödinger; Schrödinger-Poisson-\(X^{\alpha}\); concentration-compactness; standing waves; semiconductors; plasma physicsReferences:
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