Existence and Exact Multiplicity for Quasilinear Elliptic Equations in Quarter-Spaces

We consider positive solutions of quasilinear elliptic problems of the form Δp^u + ƒ(u) = 0 over the quarter-space Q = {x ∈ ℝ^N : x₁ > 0, x₂ > 0}, with u = 0 on ∂Q. For a general class of nonlinearities ƒ ≥ 0 with finitely many positive zeros, we show that, for each z > 0 such that ƒ(z) = 0, there is a bounded positive solution satisfying

lim u(x1, x2, ..., x_N) = V (x2), <br/> x1→∞ <br/>lim u(x1, x2, ..., x_N) = V (x1),<br/>x2→∞

where V is the unique solution of the one-dimensional problem

ΔpV + ƒ(V) = 0 in [0,∞), V (0) = 0, V (t) > 0 for t > 0, V (∞) = z.

When p = 2, we show further that such a solution is unique, and there are no other types of bounded positive solutions to the quarter-space problem. Thus in this case the number of bounded positive solutions to the quarter-space problem is exactly the number of positive zeros of ƒ.