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 − 2 u + b ( x ) K ( x ) | u | p − 2 u , x ∈ R N , 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.