Existence and uniqueness of monotone nodal solutions of a semilinear Neumann problem

In this paper, we study monotone radially symmetric solutions of semilinear equations with Allen-Cahn type nonlinearities by the bifurcation method. Under suitable conditions imposed on the nonlinearities, we show that the structure of the monotone nodal solutions consists of a continuous U-shaped curve bifurcating from the trivial solution at the third eigenvalue of the Laplacian. The upper branch consists of a decreasing solution and the lower branch consists of an increasing solution. In particular, we show the following equation Delta u + lambda(u - u vertical bar u vertical bar(p-1)) = 0 in B, partial derivative u/partial derivative v = 0 on partial derivative B has exactly two monotone radial nodal solutions, one is decreasing and the other is increasing. Here B is the unit ball in R-n, p > 1 and lambda > 0. (C) 2016 Elsevier Ltd. All rights reserved.