Multiplicity and concentration of positive solutions for critical Choquard equations with concave perturbation

Abstract

This paper focuses on three kinds of critical Choquard equations with concave perturbation λh(x)|u|q−2u, where λ>0, q∈(1,2) and h satisfies some mild conditions. For the upper critical and lower critical cases, by the Nehari manifold methods we obtain two positive solutions for λ>0 small enough, one of which is a ground state solution. For the doubly critical case, by minimization techniques and the mountain pass theorem, we derive two positive solutions for λ>0 small enough, one of which is a ground state positive solution. Moreover, the asymptotic behaviors of positive solutions as λ→0+ are verified.