Radial ground states and singular ground states for a spatial-dependent p-Laplace equation

We consider the following equationΔpu(x)+f(u,|x|)=0,Δpu(x)+f(u,|x|)=0, where x∈Rnx∈Rn, n>p>1n>p>1, and we assume that f   is negative for |u||u| small and limu→+∞f(u,0)u|u|q−2=a0>0 where p∗=p(n−1)n−p2q1>2. We manage to prove the existence of a radial ground state, for more generic functions f(u,|x|)f(u,|x|) and also in the case p>2p>2 and 1