Existence and uniqueness of monotone nodal solutions of a semilinear Neumann problem. (English) Zbl 1408.34028
Summary: 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 that the following equation
\[
\Delta u + \lambda(u - u | u |^{p - 1}) = 0 \text{ in } B,\, \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial B
\]
has exactly two monotone radial nodal solutions, one is decreasing and the other is increasing. Here \(B\) is the unit ball in \(\mathbb{R}^n\), \(p > 1\) and \(\lambda > 0\).
MSC:
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 35J15 | Second-order elliptic equations |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
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