Superlinear and Sublinear Dirichlet problem with the (p(y), q(y))-Laplacian operator

Abstract

We study the following nonlinear elliptic problem involving the (p(y), q(y))- Laplacian operator: −div(a(y)|∇v| p(y)−2∇v) + b(y)|v| p(y)−2 v − div(|∇v| q(y)−2∇v) = g(y, v) y ∈ Ω, v = 0 on ∂Ω, where Ω ⊂ R n is a smooth bounded domain, 1 < q(y) < p(y) < n. The functions a, b ∈ L∞(Ω) and a(y) ≥ a0 > 0, b(y) ≥ b0 > 0 for all y ∈ Ω. We prove the existence of weak solutions in W1,p(y) 0 (Ω) for the superlinear case g(y, v) = h(y)v β(y) , p(y) − 1 < β(y) < p∗ (y) − 1, and sublinear case g(y, v) = f(y)v α(y) , 0 ≤ α(y) < q(y) < p(y) − 1, by using the Mountain Pass Theorem.