On an indefinite, non-Lipschitz semilinear elliptic problem: Compactly supported solutions, multiplicity and uniqueness

For a sign-changing function a(x)a(x), we consider solutions of the following semilinear elliptic problem in RNRN with N⩾3N⩾3:−△u=(γa+−a−)uq+up,u⩾0 and u∈D1,2(RN), where γ>0γ>0 and 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+. We also give a uniqueness result for solution with small L∞L∞-norm. In the end if a(x)=a(|x|)a(x)=a(|x|) and a(x)a(x) is strictly decreasing in |x||x|, we show that all solutions are radially symmetric and are decreasing in |x||x|.