Multiple existence of solutions for a nonhomogeneous elliptic problem with critical exponent on RNRN

Let N⩾3N⩾3, 2*=2N/(N−2)2*=2N/(N−2). Our purpose in this paper is to consider the existence and multiplicity of solutions of problem{−Δu+αu=|u|2*−2u+fon RN,u∈H1(RN) where α>0α>0, f∈L2*/(2*−1)(RN)∩L∞(RN)f∈L2*/(2*−1)(RN)∩L∞(RN), f⩾0f⩾0 and f≢0f≢0.