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On the planar Schrödinger-Poisson system with the axially symmetric potential. (English) Zbl 1431.35030

Summary: In this paper, we develop some new variational and analytic techniques to prove that the following planar Schrödinger-Poisson system
\[\begin{cases} -\Delta u + V(x) u + \phi u = f(u), & x \in \mathbb{R}^2, \\ \Delta \phi = u^2, & x \in \mathbb{R}^2, \end{cases}\]
admits a nontrivial solution and a ground state solution possessing the least energy in the axially symmetric functions space, where \(V(x)\) is axially symmetric. Our results improve and extend the ones in the case \(V = 1\) and \(f(u) = |u|^{p-2} u\) with \(2 < p < 6\). In particular, we use the assumption that \(2V(x) + \nabla V(x) \cdot x\) is bounded from below instead of the usually one that \(\lim_{|x| \to \infty} V(x) = 1\). Moreover, \(V(x)\) is even admitted to be unbounded.

MSC:

35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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