Ground states and high energy solutions of the planar Schrodinger-Poisson system

In this paper, we are concerned with the Schrodinger-Poisson system {-Delta u + u + empty set u = vertical bar u vertical bar(p-2)u in R-d, Delta empty set - u(2) in R-d. (0.1) 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 Schrdinger-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 Schrodinger-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.