Existence of positive solutions for a class of the p -Laplace equations

Abstract

We are concerned with the existence of solutions of where Δ p is the p -Laplacian, p ∈ (1, ∞), and Ω is a bounded smooth domain in ℝ n . For h(x) ≡ 0 and f(x, u) satisfying proper asymptotic spectral conditions, existence of a unique positive solution is obtained by invoking the sub-supersolution technique and the spectral method. For h(x) ≢ 0 , with assumptions on asymptotic behavior of f(x, u) as u → ±∞, an existence result is also proved.