Existence of positive entire solutions of a semilinear elliptic problem with a gradient term

By a sub–supersolution argument and a perturbed argument, we show the existence of entire solutions to a semilinear elliptic problem −Δu+h(x)|∇u|q=b(x)g(u)−Δu+h(x)|∇u|q=b(x)g(u), u>0u>0, x∈RN,lim|x|→∞u(x)=0x∈RN,lim|x|→∞u(x)=0, where q∈(1,2]q∈(1,2], b,h∈Clocα(RN) for some α∈(0,1)α∈(0,1), h(x)≥0,b(x)>0,∀x∈RN, and 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.