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<p>In this paper, we develop some new variational and analytic techniques to study the multiplicity and concentration of positive solutions for a planar Schrödinger-Poisson system involving competing weight potentials and the nonlinearity <i>K</i>(<i>x</i>)∣<i>u</i>∣<sup><i>p</i>−2</sup><i>u</i> (2 &lt; <i>p</i> &lt; 4) in ℝ<sup>2</sup>. By Nehari manifold and Ljusternik-Schnirelmann category, we relate the number of positive solutions to the category of the global minima set of a suitable ground energy function. Our results improve and extend the ones in [Du, Weth, <i>Nonlinearity</i>, <b>30</b>, 3492–3515 (2017)] and [Chen, Tang, <i>J. Differ. Equ.</i>, <b>268</b>, 945–976 (2020)]. In particular, we do not need the assumption <i>K</i>(<i>x</i>) ≡ 1 and the <i>C</i><sup>1</sup> smoothness of <i>V</i>(<i>x</i>). Furthermore, we do not use the axially symmetric condition of the potential in our second main result. Moreover, we shall show that there is a great difference in our results between <i>N</i> = 2 and <i>N</i> ≥ 3.</p>

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Multiplicity and Concentration of Positive Solutions for Planar Schrödinger-Poisson Systems with Competing Potentials

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • Haining Fan,
  • Binlin Zhang

摘要

In this paper, we develop some new variational and analytic techniques to study the multiplicity and concentration of positive solutions for a planar Schrödinger-Poisson system involving competing weight potentials and the nonlinearity K(x)∣u∣p−2u (2 < p < 4) in ℝ2. By Nehari manifold and Ljusternik-Schnirelmann category, we relate the number of positive solutions to the category of the global minima set of a suitable ground energy function. Our results improve and extend the ones in [Du, Weth, Nonlinearity, 30, 3492–3515 (2017)] and [Chen, Tang, J. Differ. Equ., 268, 945–976 (2020)]. In particular, we do not need the assumption K(x) ≡ 1 and the C1 smoothness of V(x). Furthermore, we do not use the axially symmetric condition of the potential in our second main result. Moreover, we shall show that there is a great difference in our results between N = 2 and N ≥ 3.