On singular problems involving the (1,q)-Laplacian

In this paper we study the asymptotic behavior of the singular ( p , q ) -equation − Δ p u − Δ q u = u − γ + λ u r − 1 in Ω , u = 0 on ∂ Ω , as p → 1 + , where Ω ⊆ R N , N ≥ 2 , is a bounded domain with Lipschitz boundary ∂Ω, λ > 0 , 0 < γ < 1 < p < q < N and q < r < q ∗ . Based on a purely singular auxiliary problem along with the Nehari manifold and the fibering map we prove the existence of at least two weak solutions belonging to the usual Sobolev space W 0 1 , q ( Ω ) of the ( 1 , q ) -Laplace problem − div ( ∇ u | ∇ u | ) − Δ q u = u − γ + λ u r − 1 in Ω , u = 0 on ∂ Ω , provided the parameter λ is sufficiently small. As far as we know this is the first result for singular ( 1 , q ) -problems. Moreover, in contrast to other works dealing with the 1-Laplacian, we get much more regularity of the solutions in the sense that the solutions of the problem above belong to the Sobolev space W 0 1 , q ( Ω ) instead of the space BV ( Ω ) of functions of bounded variation.