Advances in Differential Equations

Global branches of sign-changing solutions to a semilinear Dirichlet problem in a disk

Yasuhito Miyamoto

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Let $D:=\{(x,y);\ x^2+y^2<1\}\subset\mathbb R^2$ and let $f\in C^2$. We study sign-changing solutions to $$ \Delta u+\lambda f(u)=0\ \ \textrm{in}\ \ D,\quad u=0\ \ \textrm{on}\ \ \partial D $$ under the conditions $f(0)=0$ and $f'(0)=1$ from the viewpoint of bifurcation theory. We show that this problem has infinitely many bifurcation points from which unbounded continua of sign-changing solutions emanate, where the eigenfunctions corresponding to each bifurcation point are nonradially symmetric. When $f(u)$ is of Allen-Cahn type (e.g., $f(u)=u-u^3$), then we show that the maximal continuum emanating from the second eigenvalue is homeomorphic to $\mathbb R\times S^1$. It is well-known that each eigenvalue is a bifurcation point. We show that the closure of the maximal continuum emanating from each bifurcation point is locally homeomorphic to a curve or disk.

Article information

Adv. Differential Equations, Volume 16, Number 7/8 (2011), 747-773.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B32: Bifurcation [See also 37Gxx, 37K50] 35B36: Pattern formation 35J60: Nonlinear elliptic equations 35J25: Boundary value problems for second-order elliptic equations


Miyamoto, Yasuhito. Global branches of sign-changing solutions to a semilinear Dirichlet problem in a disk. Adv. Differential Equations 16 (2011), no. 7/8, 747--773.

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