Two solutions for an elliptic equation with fast increasing weight and concave-convex nonlinearities

We prove the existence of two nonnegative nontrivial solutions for the equation−div(K(x)∇u)=a(x)K(x)|u|q−2u+b(x)K(x)|u|p−2u,x∈RN, where N⩾3, K(x)=exp(|x|α/4), α⩾2 and the potentials a and b have indefinite sign and satisfy some mild integrability conditions. The results hold when a has small norm in a suitable weighted Lebesgue space.