Existence theorems for a semilinear elliptic boundary value problem

Abstract

Let Ω be a bounded domain in R, n ≥ 3, with a smooth boundary ∂Ω; let L be a linear, second order, elliptic operator; let f and g be two real-valued functions defined on Ω × R such that f(x, z) ≤ g(x, z) for almost every x ∈ Ω and every z ∈ R. In this paper we prove that, under suitable assumptions, the problem { f(x, u) ≤ Lu ≤ g(x, u) in Ω, u = 0 on ∂Ω, has at least one strong solution u ∈W 2,p(Ω)∩W 1,p 0 (Ω). Next, we present some remarkable special cases. Introduction. Let Ω be a bounded domain in R, n≥3, with a smooth boundary ∂Ω; let L be a linear, second order, elliptic differential operator; let f and g be two real-valued functions defined on Ω×R such that f(x, z) ≤ g(x, z) for almost every x ∈ Ω and every z ∈ R. Consider the problem (P) { f(x, u) ≤ Lu ≤ g(x, u) in Ω, u = 0 on ∂Ω. A function u : Ω → R is said to be a strong solution of (P) if u ∈W (Ω)∩ W 1,p 0 (Ω), p ∈ ]n/2,∞[, and, for almost every x ∈ Ω, one has f(x, u(x)) ≤ Lu(x) ≤ g(x, u(x)). Remarkable special cases of problem (P) are those where f(x, z) = g(x, z), (x, z) ∈ Ω × R, or, roughly speaking, f(x, z) = lim infw→z φ(x,w) and g(x, z) = lim supw→z φ(x,w), (x, z) ∈ Ω ×R, with φ a locally bounded real-valued function defined on Ω ×R. Both have been extensively studied, mainly by variational methods [6], [11], [15], or topological methods [5], [17], or suband super-solution arguments [9], [14]. 1991 Mathematics Subject Classification: 35R70, 35J60.