On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints
On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints
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非齐次约束下 Sobolev 型嵌入的最大化问题
DOI：
10.1080/00036811.2018.1491971
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Hidemitsu Wadade
中科院分区：
文献类型：
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作者：
Michinori Ishiwata;Hidemitsu Wadade
本文研究了一类Sobolev嵌入极大化问题的可达性，其中D：=supu∈W1，p（RN），n =1，n = 1与指数和b有关的Dare最大化器的存在性和不存在性。事实上，对于α的某些范围，尽管问题的亚临界设置和径向性质，我们还是得到了不存在的结果。我们证明了值α=α：=infu∈W1，p（RN），u pa + u pb=1（1− u pp）/u qq是D可达性的阈值.我们还考虑了D的可达性，并计算了指数在一定范围内的精确值。
We consider the attainability of a maximizing problem associated with the Sobolev embeddingof the form D:=supu∈W1,p(RN),∥∇u∥pa+∥u∥pb=1∥u∥pp+α∥u∥qq, where,, p<q<(Np/(N−p)) ifp<N,=+∞ ifp=N and. The existence and non-existence of a maximizer forDare related to the exponentsandb. Indeed, for some range ofα, we have a non-existence result in spite of the sub-critical setting and the radial nature of the problem. We show that the value α=α∗:=infu∈W1,p(RN),∥∇u∥pa+∥u∥pb=1(1−∥u∥pp)/∥u∥qq is the threshold in terms of the attainability ofD. We also consider the attainability ofDwithand compute the exact valuefor some range of the exponents.