Normalized solutions to the nonlinear Choquard equations with Hardy-Littlewood-Sobolev upper critical exponent

Abstract

We study the effect of lower order nonlocal perturbations in the existence of positive solutions to the following nonlinear Choquard equation−Δu=λu+μ(Iα⁎|u|p)|u|p−2u+(Iα⁎|u|2α⁎)|u|2α⁎−2ux∈RN, having prescribed L2−norm∫RN|u|2dx=a>0, where N≥3, α∈(0,N), N+αN<p<2α⁎, μ>0 and λ∈R. Under different assumptions on p, we establish several existence results and characterize the behavior of solutions as μ→0.