Quasilinear elliptic equations on half- and quarter-spaces.

We consider quasilinear elliptic problems of the form Delta(p)u + f(u) = 0 over the half-space H = {x is an element of R-N : x(1) > 0} and over the quarter-space Q = {X is an element of R-N : x(1) > 0, x(N) > 0}. In the half-space case we assume u >= 0 on partial derivative H, and in the quarter-space case we assume that u >= 0 on {x(1) = 0} and u = 0 on {x(N) = 0). Let u not equivalent to 0 be a bounded nonnegative solution. For some general classes of nonlinearities f, we show that, in the half-space case, lim(x1 ->infinity)u(x(1), x(2), . . . , x(N)) always exists and is a positive zero of f; and in the quarter-space case, lim(x1 ->infinity)u(x(1), x(2), . . . , x(N)) = V(X-N), where V is a solution of the one-dimensional problem Delta V-p + f(V) = 0 in R+, V(0) = 0, V(t) > 0 for t > 0, V(+infinity) = z, with z a positive zero off. 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