Infinitely many turning points for some supercritical problems
Infinitely many turning points for some supercritical problems
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DOI：
10.1007/bf02505896
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发表时间：
2000-12
影响因子：
1
通讯作者：
E. N. Dancer
中科院分区：
文献类型：
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作者：
E. N. Dancer
-Au=~ f (u) in t9u = 0 on~ t9u > in~ 0，对于一些非线性在无穷远处渐近似超临界幂函数且~ 9是相当对称的域。我们证明了正解的分支在变成无界（可能只有方向的改变）时必须经历无限次分叉。这表明我们的问题在许多域上的行为与球的行为非常相似。我们的结果似乎是对非球域的第一个这样的结果。事实上，即使在球上，我们的结果也比之前的结果适用于更大的非线性类。早期的结果倾向于特定的非线性，如b[22]。事实上，我们把这个问题在更一般的域和更一般的非线性上简化为一个关于rn的问题。我们相信在这个问题上可以做更多的事情。
-Au=~ f (u) in t9 u= 0 on~ t9 u> 0 in~, for some nonlinearities which are asymptotically like a supercritical power at infinity and~ 9 is a rather symmetric domain. We prove that the branch of positive solutions must undergo infinitely many bifurcations as it becomes unbounded (possibly only changes of direction). This shows that the behaviour of our problem on many domains is very similar to that for a ball. Our result seems to be the first such result for domains which are not balls. Indeed even on balls, our results hold for a much larger class of nonlinearities than the earlier results. The earlier results tended to be for particular nonlinearities as in [22]. Indeed, we reduce the question on more general domains and for more general nonlinearities to a question on R n. We believe much more can be done on this problem.