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)
Full Text:

References:

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