Semiclassical solutions for critical Schr\"odinger-Poisson systems involving multiple competing potentials

Lingzheng Kong; Haibo Chen

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arXiv · 2012.08978

Semiclassical solutions for critical Schr\"odinger-Poisson systems involving multiple competing potentials

Lingzheng Kong·Haibo Chen

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Abstract

In this paper, a class of Schr\"{o}dinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set $\{Q_i(x)|1\le i \le m\}$ 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 $v_\varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that $v_\varepsilon$ 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.

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Lingzheng Kong, Haibo Chen. 2020-12-16. Semiclassical solutions for critical Schr\"odinger-Poisson systems involving multiple competing potentials. https://arxiv.org/abs/2012.08978

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