Improved results on normalized solutions for planar Schrödinger–Poisson equations with [formula omitted]-supercritical case

Abstract

We prove the existence of normalized solutions for the following planar Schrödinger–Poisson equation −Δu+λu+12πln|⋅|∗|u|2u=|u|q−2u,x∈R2,∫R2u2dx=c, where c>0, q>4 and λ∈R arises as a Lagrange multiplier and is not a priori given. In the axially symmetric setting, new variational techniques and inequalities related with logarithmic convolution potential are developed to detect the geometry structure of the constrained functional, we believe, which can be applicable for more planar L2-constrained problems. Our study improves the one of Cingolani and Jeanjean (2019) from some small values of c to arbitrary c>0, which seems to be new in the context of normalized solutions.