Quasilinear Elliptic Equations on Half- and Quarter-spaces

Abstract

Abstract We consider quasilinear elliptic problems of the form Δpu + f(u) = 0 over the half-space H = {x ∈ ℝN : x1 > 0} and over the quarter-space Q = {x ∈ ℝN : x1 > 0, xN > 0}. In the half-space case we assume u ≥ 0 on ∂H, and in the quarter-space case we assume that u ≥ 0 on {x1 = 0} and u = 0 on {xN = 0}. Let u ≢ 0 be a bounded nonnegative solution. For some general classes of nonlinearities f , we show that, in the half-space case, limx1→∞ u(x1, x2, ..., xN) always exists and is a positive zero of f ; and in the quarter-space case, where V is a solution of the one-dimensional problem ΔpV + f(V) = 0 in ℝ+, V(0) = 0, V(t) > 0 for t > 0, V(+∞) = z, with z a positive zero of f . Our results extend most of those in the recent paper of Efendiev and Hamel [6] for the special case p = 2 to the general case p > 1. Moreover, by making use of a sharper Liouville type theorem, some of the results in [6] are improved. To overcome the difficulty of the lack of a strong comparison principle for p-Laplacian problems, we employ a weak sweeping principle.