Existence and Uniqueness of Positive Solutions to a Semilinear Elliptic Problem in N

Let p 2 C loc N‘ with p > 0 and let f 2 C10;1‘; 0; 1‘‘ be such that limu&0 f u‘=u D C1, f is bounded at infinity, and the mapping u 7−! f u‘=uC ‘ is decreasing on 0;1‘, for some > 0. We prove that the problem −1u D px‘f u‘ in N, N > 2, has a unique positive C2C loc  N‘ solution that vanishes at infinity provided R1 0 r8r‘dr < 1, where 8r‘ D max ”px‘y ŽxŽ D r•. Furthermore, it is showed that this condition is nearly optimal. Our results extend previous works by Lair–Shaker and Zhang, while the proofs are based on two theorems on bounded domains, due to Brezis–Oswald and Crandall–Rabinowitz–Tartar.