Existence of Solution for Quasilinear Equations Involving Local Conditions

In this paper, we study the existence of weak solutions of the quasilinear equation (equation presented) where a: ℝ → [0, ∞) is C1 and a nonincreasing continuous function near the origin, the nonlinear term f: Ω × ℝ → ℝ is a Carathéodory function verifying certain superlinear conditions only at zero, and λ is a positive parameter. The existence of the solution relies on C1-estimates and variational arguments. Copyright © 2019 The Royal Society of Edinburgh.