Planar Schrödinger-Choquard equations with potentials vanishing at infinity: The critical case

Abstract

We study the following class of stationary Schrödinger equations of Choquard type−Δu+V(x)u=[|x|−μ⁎(Q(x)F(u))]Q(x)f(u),x∈R2, where the potential V and the weight Q decay to zero at infinity like (1+|x|γ)−1 and (1+|x|β)−1 for some (γ,β) in variously different ranges, ⁎ denotes the convolution operator with μ∈(0,2), and F is the primitive of f that fulfills a critical exponential growth in the Trudinger-Moser sense. By establishing a version of the weighted Trudinger-Moser inequality, we investigate the existence of nontrivial solutions of mountain-pass type for the given problem. Furthermore, we shall establish that the nontrivial solution is a bound state, namely a solution belonging to H1(R2), for some particular (γ,β).