JournalsaihpcVol. 33, No. 1pp. 169–197

On the planar Schrödinger–Poisson system

  • Silvia Cingolani

    Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy
  • Tobias Weth

    Institut für Mathematik, Goethe-Universität Frankfurt, D-60054 Frankfurt am Main, Germany
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We develop a variational framework to detect high energy solutions of the planar Schrödinger–Poisson system

{Δu+a(x)u+γwu=0,Δw=u2inR2\left\{\begin{matrix} −\mathrm{\Delta }u + a(x)u + \gamma wu = 0, \\ \mathrm{\Delta }w = u^{2} \\ \end{matrix}\right.\:\text{in}\mathbb{R}^{2}

with a positive function aL(R2)a \in L^{\infty }(\mathbb{R}^{2}) and γ>0\gamma > 0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z2\mathbb{Z}^{2}-translations and therefore fails to satisfy a global Palais–Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential a is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions (u,w)(u,w) such that u has arbitrarily many nodal domains. Finally, in the case where a is constant, we also show that solutions of the above problem with u>0u > 0 in R2\mathbb{R}^{2} and w(x)w(x)\rightarrow −\infty as x|x|\rightarrow \infty are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in u in the first equation.

Cite this article

Silvia Cingolani, Tobias Weth, On the planar Schrödinger–Poisson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 1, pp. 169–197

DOI 10.1016/J.ANIHPC.2014.09.008