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- 10.1007/s10440-023-00579-1
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Abstract
This paper is concerned with the number of multi-peak solutions for the Schrödinger-Poisson system {−ε2Δu+V(x)u+Φ(x)u=|u|p−1u, in R3,−ΔΦ=u2, in R3,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \left \{ \textstyle\begin{array}{lll} -\varepsilon ^{2} \Delta u + V(x) u + \Phi (x) u = |u|^{p-1} u, && \text{ in } \mathbb{R}^{3}, \\ - \Delta \Phi = u^{2}, && \text{ in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $$\end{document} where ε\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\varepsilon $\end{document} is a parameter, p∈(1,5)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$p \in (1, 5)$\end{document} and V(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$V (x)$\end{document} is the potential function. We obtain the number of peak solutions for the system when the solutions concentrate at the critical points of V(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$V(x)$\end{document} through Pohozaev identities and the blow-up analysis.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Jun 1, 2023
Keywords: Schrödinger-Poisson systems; Peak solutions; Local uniqueness; 35J20; 35J60