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On a planar Schrödinger–Poisson system involving a non-symmetric potential
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 4 / November 2022
- Published online by Cambridge University Press:
- 05 December 2022, pp. 1133-1146
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- Article
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We prove the existence of a ground state positive solution of Schrödinger–Poisson systems in the plane of the form
\[ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \quad\text{in}\ \mathbb{R}^2, \]where $p\ge 4$
, $\gamma,b>0$
and the potential $V$
is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition.
