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 Δu+λ(u−u∣u∣p−1)=0in  B,∂u∂ν=0on  ∂B has exactly two monotone radial nodal solutions, one is decreasing and the other is increasing. Here BB is the unit ball in RnRn, p>1p>1 and λ>0λ>0.