In this paper, we are concerned with the Schrödinger–Poisson system 0.1 { − Δ u + u + ϕ u = | u | p − 2 u i n R d , Δ ϕ = u 2 i n R d . Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case d ⩾ 3 . In contrast, much less information is available in the planar case d = 2 which is the focus of the present paper. It has been observed by Cingolani S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) that the variational structure of (0.1) differs substantially in the case d = 2 and leads to a richer structure of the set of solutions. However, the variational approach of Cingolani S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) is restricted to the case p ⩾ 4 which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case 2 < p < 4 with a different variational approach.
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Du et al. (2017) studied this question.