A uniqueness result for a semilinear elliptic problem: A computer-assisted proof
A uniqueness result for a semilinear elliptic problem: A computer-assisted proof
复制标题
半线性椭圆问题的唯一性结果：计算机辅助证明
DOI：
10.1016/j.jde.2009.06.023
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发表时间：
2009
影响因子：
2.4
通讯作者：
D. Roth
中科院分区：
文献类型：
--
作者：
P. J. McKenna;F. Pacella;M. Plum;D. Roth
从著名的文章[A。Gidas，W.M.尼湖Nirenberg，Symmetry and related properties via the maximum principle，Comm. Math. Phys. 68（1979）209-243]，许多论文致力于研究−Δu=λu+upin Ω，u=0在Ω上的正解的唯一性问题，其中p>1且λ的范围在0和−Δ的第一Dirichlet特征值λ1（Ω）之间。对于Ω是球的情形，主要利用常微分方程技巧，可以证明其唯一性。但是当Ω不是球时，我们知道的很少，而且只有当λ=0时。本文证明了在Ω=（0，1）2，p=2的情形下，对所有λ∈[0，λ1（Ω）），存在唯一性.这构成了在不同于球的域中唯一性问题的第一个肯定答案。我们的证明大量使用计算机辅助：我们计算一个分支的近似解，并证明存在一个真正的解决方案分支接近它，使用不动点技术。通过特征值包络方法，并对λ接近λ1（Ω）的情况作了额外的分析论证，我们推导出了沿该分支的所有解沿着的非退化性，从而由问题的已知分支结构得出了唯一性.
Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209–243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+upin Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ1(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ1(Ω)), in the case Ω=(0,1)2and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ1(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.