An existence result for ( p , q ) -Laplacian BVP with falling zeros

We show the existence of a positive solution to the ( p , q ) -Laplacian problem { − Δ p u − a Δ q u = λ f ( u ) − h ( x ) in Ω , u = 0 on ∂ Ω , for λ large, where Ω is a bounded domain in R n with smooth boundary ∂ Ω , a is a nonnegative constant, h ∈ L ∞ ( Ω ) , p > q > 1 , and f satisfies f ( 0 ) = f ( r ) = 0 with f > 0 on ( 0 , r ) for some r > 0 .