Existence and Uniqueness of Solution for P-Laplacian Dirichlet Problem

whereΔp is the p-Laplacian, Ω ∈ C0,1 be a bounded domain inRN . Let p ≥ 2, λ > 0 and f : Ω×R −→ R be a caratheodory function which is decreasing with respect to the second variable, i.e., f(x, s1) ≥ f(x, s2) for a.a. x ∈ Ω ands1, s2 ∈ R, s1 ≤ s2 (2) Assume, moreover, that there exists f0 ∈ Lp(Ω), p′ = p p−1 and c > 0 such that ∣f(x, s)∣ ≤ f0(x) + c∣s∣p−1 (3) We considered such problems with numerical methods in [2-5] Definition 1 We say that u ∈ W 1,p 0 (Ω) is a weak solution to (1) if ∫ ∣∇u∣p−2∇u∇v + λ ∫ ∣u(x)∣p−2u(x)v(x) = ∫ f(x, u(x)) v(x)dx for all v ∈ W 1,p 0 (Ω). Definition 2 Let H be a real Hilbert space. An operator T : H → H satisfying (T (u)− T (v), u− v) ≥ 0 (4) for any u, v ∈ H is called a monotone operator. An operator T is called strictly monotone if for u ∕= v the strict inequality holds in (4). An operator T is called strongly monotone if there exists c > 0 such that (T (u)− T (v), u− v) ≥ c ∥ u− v ∥2 for any u, v ∈ H .