Positive solutions to a class of Choquard type equations with a competing perturbation

Abstract

In this paper, we study a class of Choquard type equations with a competing perturbation{−Δu+λV(x)u=(Iα⁎K|u|p)K|u|p−2u−|u|q−2uinRN,u∈H1(RN), where N≥3,λ>0 a parameter, V∈C(RN,R+), Iα is the Riesz potential, K(x)≥0 in RN, 1+αN<p<N+αN−2 and 2<q<2⁎=2NN−2. Unlike the existing literature, we are interested in the existence and multiplicity of positive solutions under the different relationship between p and q when the competing effect of the nonlocal term with the perturbation happens. In particular, when 2p<q, variational methods can not be applied in a standard way, even restricting the energy functional on the Nehari manifold, because Palais-Smale sequences may not be bounded. A new constraint approach developed by us recently is employed in this case.