Generalized Minimizer Solutions for Equations with the p-Laplacian and a Potential Term

Let Ω be a domain in R (possibly unbounded), N ≥ 2, 1 < p <∞, and let V ∈ Lloc(Ω). Consider the energy functional QV on C∞ c (Ω) and its Gâteaux derivative QV , respectively, given by QV (u) def = 1 p ∫ Ω (|∇u| + V |u|) dx, QV (u) = div(|∇u|p−2∇u) + V |u|p−2u, for u ∈ C∞ c (Ω). Assume that QV > 0 on C∞ c (Ω) \ {0} and QV does not have a ground state (in the sense of Pinchover and Tintarev [13, Def. 1.4]). Finally, let f ∈ D′(Ω) be such that the functional u 7→ QV (u) − 〈u, f〉 : C∞ c (Ω) → R is bounded from below. Then the equation QV (u) = f has a solution u0 ∈ W 1,p loc (Ω) in the sense of distributions. This solution also minimizes the functional u 7→ Q∗∗ V (u) − 〈u, f〉 : C∞ c (Ω) → R where Q∗∗ V denotes the bipolar (Γ-regularization) of QV ; Q∗∗ V is the largest convex, weakly lower semicontinuous functional on C∞ c (Ω) that satisfies Q∗∗ V ≤ QV . (The original energy functional QV is not necessarily convex.) 2000 AMS subject classifications: Primary 35J20, 49J45; Secondary 35J70, 46E35