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<p>In this paper, we investigate the two dimensional Schrödinger-Poisson equation</p><p><Equation ID="Equ1"> <EquationNumber>(0.1)</EquationNumber> <EquationSource Format="TEX">\(-\Delta u+|x|^2u+\left(\int_{\mathbb{R}^{2}}\ln |x-y|u^2(y)\rm{d}y\right)u=\lambda u+a|u|^{p-2}u\ \ {\rm in} \ \mathbb{R}^{2}\)</EquationSource> </Equation></p><p>in the mass-supercritical case. We show that problem (0.1) admits a mountain pass solution <i>u</i><sub><i>p</i></sub>, which blows up as <i>p</i> → 4<sup>+</sup>. Then we consider the concentration and local uniqueness of <i>u</i><sub><i>p</i></sub> as <i>p</i> → 4<sup>+</sup>.</p>

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Concentration and local uniqueness of normalized solutions to mass-supercritical 2D Schrödinger-Possion equations

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Shuyao Lu,
  • Dehong Yang,
  • Jinge Yang,
  • Xiongjun Zheng

摘要

In this paper, we investigate the two dimensional Schrödinger-Poisson equation

(0.1) \(-\Delta u+|x|^2u+\left(\int_{\mathbb{R}^{2}}\ln |x-y|u^2(y)\rm{d}y\right)u=\lambda u+a|u|^{p-2}u\ \ {\rm in} \ \mathbb{R}^{2}\)

in the mass-supercritical case. We show that problem (0.1) admits a mountain pass solution up, which blows up as p → 4+. Then we consider the concentration and local uniqueness of up as p → 4+.