Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials

We study the equation Δu+u|u|p−1+V(x)u+f(x)=0 in Rn, where n⩾3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|−2 where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.