Saddle solutions of the balanced bistable diffusion equation

Abstract

AbstractWe prove that Δu + f(u) = 0 has a unique entire solution u(x, y) on ℝ2 which has the same sign as the function xy, where f is a balanced bistable function like f(u) = u − u3. But we neither assume f is odd nor assume the monotonicity properties of f(u)/u. Our result generalizes a previous result by Dang, Fife, and Peletier [12]. Our approach combines bifurcation methods and recent results on the qualitative properties for elliptic equations in unbounded domains by Berestycki, Caffarelli and Nirenberg [5, 6]. © 2002 Wiley Periodicals, Inc.