Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m
Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m
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DOI：
10.4171/jems/168
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发表时间：
2009-08
影响因子：
2.6
通讯作者：
X. Cabré;Joana Terra
中科院分区：
文献类型：
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作者：
X. Cabré;Joana Terra
研究了半线性椭圆型方程1uDf.u/在全R2 m上鞍形解的存在性和不稳定性，其中f是n型的.已知在2 mD 2维上存在一个鞍形解。这是一个在R2中更改符号并仅在fjx 1 j D jx 2 jg上消失的解决方案。我们也知道，这种解决方案是不稳定的。在这篇文章中，我们证明了鞍形解的存在性，在每个偶数维，以及他们的不稳定性的情况下，维2 m D 4。更准确地说，我们的主要结果建立，如果2米D 4，每一个解决方案消失的西蒙斯锥f.x1; x2/ 2 Rm Rm：jx 1 j D jx 2 jg是不稳定的外，每一个紧集，因此，有无限的莫尔斯指数。这些结果是有关的猜想德Giorgi广泛研究，近年来，其中存在一个反例在高维仍然是一个悬而未决的问题。
We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation ���1u D f .u/ in the whole R2m, where f is of bistable type. It is known that in dimension 2m D 2 there exists a saddle-shaped solution. This is a solution which changes sign in R2 and vanishes only on fjx1j D jx2jg. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension 2m D 4. More precisely, our main result establishes that if 2m D 4, every solution vanishing on the Simons cone f.x1; x2/ 2 Rm Rm : jx1j D jx2jg is unstable outside every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.