Multiple solutions with compact support for a semilinear elliptic problem with critical growth

For a sign-changing function a(x)a(x), we consider solutions of the following semilinear elliptic problem in RNRN with N⩾3N⩾3:−Δu=(γa+−a−)uq+up,u⩾0 and u∈D(RN), where γ>0γ>0, 00. When Ω+={x∈RN|a(x)>0}Ω+={x∈RN|a(x)>0} has several connected components, we prove that there exists an interval on γ  , in which two solutions exist and are positive in Ω+Ω+, moreover one solution blows up as γ→0γ→0. A uniqueness result for solution with small L∞L∞-norm is also given.