Semiclassical solutions for critical Schrödinger–Poisson systems involving multiple competing potentials

Lingzheng Kong and Haibo Chen

Mathematische Nachrichten, 2025, vol. 298, issue 6, 1808-1838

Abstract: In this paper, a class of Schrödinger–Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied using the same argument for the nonlinear term with only a positive potential, because the weight potentials set {Qi(x)|1≤i≤m}$\lbrace Q_i(x)|1\le i \le m\rbrace$ contains nonpositive, sign changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution vε$v_\varepsilon$ in the semiclassical limit via the Nehari manifold and concentration–compactness principle. Then we show that vε$v_\varepsilon$ converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.

Date: 2025
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