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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Journal of Differential Equations
( Vol. 80 NO.2 / (1989))
On Non-Radially Symmetric Bifurcation in the Annulus
Song-Sun Lin,
Pages. 251-279
Abstract We discuss the radially symmetric solutions and the non-radially symmetric bifurcation of the semilinear elliptic equation $Delta u+2 delta e^u=0$ in $Omega$ and $u=0$ on $partial Omega$, where $Omega={x in mathbb R^2:a^2<|x|<1}$. We prove that, for each $a in (0,1)$, there exists a decreasing sequence ${delta^*(k,a)}^infty_{k=0}$ with $delta^*(k,a) o 0$ as $k o infty$ such that the equation has exactly two radial solutions for $delta in (0,delta^*(0,a))$, exactly one for $delta=delta^*(0,a)$, and none for $delta>delta^*(0,a)$. The upper branch of radial solutions has a non-radially symmetric bifurcation (symmetry breaking) at each $delta^*(k,a),; k geqslant 1$. As $a o 0$, the radial solutions will tend to the radial solutions on the disk and $delta^*(0,a) o delta^*=1$, the critical number on the disk.
Contents 1. Introduction
2. Radially Symmetric Solutions
3. Linearized Eigenvalue Problems
4. Symmetry Breaking
5. Annuli and Disk
Key words
Mathmatical Subject Classification