Least energy solution for a scalar field equation with a singular nonlinearity

We are concerned with a nonnegative solution to the scalar field equation Δu + f(u) = 0 in ℝN, lim|x|→∞u(x) = 0. A classical existence result by Berestycki and Lions [3] considers only the case when f is continuous. In this paper, we are interested in the existence of a solution when f is singular. For a singular nonlinearity f, Gazzola, Serrin and Tang [8] proved an existence result when f ϵ L1loc(ℝ) ∩ Liploc(0,∞) with u0 f(s) ds < 0 for small u > 0. Since they use a shooting argument for their proof, they require the property that f ϵ Liploc(0,∞). In this paper, using a purely variational method, we extend the previous existence results for f ϵ L1loc(ℝ) ∩ C(0,∞). We show that a solution obtained through minimization has the least energy among all radially symmetric weak solutions. Moreover, we describe a general condition under which a least energy solution has compact support.