Existence and uniqueness of monotone nodal solutions of a semilinear Neumann problemNonlinear Analysis: Theory, Methods & Applications (2016)
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)=0 in B,∂u∂ν=0on ∂B
has exactly two monotone radial nodal solutions, one is decreasing and the other is increasing. Here B is the unit ball in ℝn, p>1 and λ>0.
- Nodal solution
Publication DateMarch, 2016
Citation InformationRuofei Yao, Yi Li and Hongbin Chen. "Existence and uniqueness of monotone nodal solutions of a semilinear Neumann problem" Nonlinear Analysis: Theory, Methods & Applications (2016)
Available at: http://works.bepress.com/yi_li/102/