Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth. (English) Zbl 1448.35160
Summary: This paper is concerned with the following planar Schrödinger-Poisson system
\[
\begin{cases}
- \Delta u + V (x) u + \phi u = f (x, u), & x \in \mathbb{R}^2, \\
\Delta \phi = u^2, & x \in \mathbb{R}^2,
\end{cases}
\]
where \(V \in \mathcal{C}(\mathbb{R}^2, [0, \infty))\) is axially symmetric and \(f \in \mathcal{C}(\mathbb{R}^2 \times \mathbb{R}, \mathbb{R})\) is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak assumptions on \(V\) and \(f\). Our theorems extend the results of
S. Cingolani and T. Weth [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33, No. 1, 169–197 (2016; Zbl 1331.35126)]
and of M. Du and T. Weth [Nonlinearity 30, No. 9, 3492–3515 (2017; Zbl 1384.35010)]
and the authors [J. Differ. Equations 268, No. 3, 945–976 (2020; Zbl 1431.35030)], where \(f(x, u)\) has polynomial growth on \(u\). In particular, some new tricks and approaches are introduced to overcome the double difficulties resulting from the appearance of both the convolution \(\phi_{2, u}(x)\) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity \(f(x, u)\).
MSC:
| 35J47 | Second-order elliptic systems |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J62 | Quasilinear elliptic equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
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