A remark on the existence of positive entire solutions of a sublinear elliptic problem

By means of a sub–supersolutions argument and a perturbed argument, we show the existence of entire solutions to a semilinear elliptic problem −△u=b(x)g(u)−△u=b(x)g(u), u>0u>0, x∈RNx∈RN, lim|x|→∞u(x)=0lim|x|→∞u(x)=0, where b∈Clocα(RN) for some α∈(0,1)α∈(0,1) and b(x)>0,∀x∈RN, g∈C1((0,∞),(0,∞))g∈C1((0,∞),(0,∞)) which may be singular at 0. No monotonicity condition is imposed on the functions g(s)g(s) and g(s)/sg(s)/s.