Pairs of positive solutions of quasilinear elliptic equations in exterior domains

Takaŝi Kusano, Charles Andrew Swanson and Hiroyuki Usami

Vol. 120 (1985), No. 2, 385–399

DOI: 10.2140/pjm.1985.120.385

Abstract

Our main objective is to prove the existence of infinitely many pairs (u₁,u₂) of positive solutions of quasilinear elliptic differential equations

throughout exterior domains Ω_α ⊂ R^N, N ≥ 2, of the type

where x = (x₁,…,x_N), ∇u = (∂u∕∂x₁,…,∂u∕∂x_N), and Δ = ∇⋅∇. Each pair has the property that u₁(x)∕u₂(x) has uniform limit zero in Ω_α as |x|→∞. In particular, if q(t) ≡ 0 and N ≥ 3, u₁(x) has limit 0 as |x|→∞, and u₂(x) is bounded above and below by positive constants in Ω_α.

Mathematical Subject Classification 2000

Primary: 35J65

Milestones

Received: 24 February 1984

Published: 1 December 1985