Nonlinear critical problems for the biharmonic operator with Hardy potential
Nonlinear critical problems for the biharmonic operator with Hardy potential
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DOI：
10.1007/s00526-014-0789-7
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发表时间：
2015-09
影响因子：
2.1
通讯作者：
L. D’Ambrosio;E. Jannelli
中科院分区：
文献类型：
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作者：
L. D’Ambrosio;E. Jannelli
本文研究L μ u：= Δ 2 u-μ u问题|X| 4= λ u+| u| 2 <$-2 u in Ω u=<$u <$n= 0 on <$Ω其中Ω <$R^ n Ω <$Rn是包含原点的有界开集，n ≥ 5 n≥ 5且2^*= 2n/（n-4）2 <$= 2 n/（n-4）.我们发现这个问题是临界的（在Pucci-Serrin和Grunau意义下），取决于μ ∈ 0，μ）μ∈ 0，μ <$）的值，μ <$是Rellich不等式中的最佳常数.为了获得我们的存在性结果，研究极限问题L μ u= u^2 ^*-1 L μ u = u2 <$-1的径向解（其解析表达式未知）在整个空间R^nRn中的行为是至关重要的.另一方面，我们的不存在性结果依赖于一个合适的Pohozaev型恒等式，这反过来又依赖于一些加权Hardy-Rellich不等式。
In this paper we study the problem L μ u:= Δ 2 u-μ u| x| 4= λ u+| u| 2∗-2 u in Ω u=∂ u∂ n= 0 on∂ Ω where Ω ⊂ R^ n Ω⊂ R n is a bounded open set containing the origin, n ≥ 5 n≥ 5 and 2^*= 2n/(n-4) 2∗= 2 n/(n-4). We find that this problem is critical (in the sense of Pucci–Serrin and Grunau) depending on the value of μ ∈ 0, μ) μ∈ 0, μ¯), μ μ¯ being the best constant in Rellich inequality. To achieve our existence results it is crucial to study the behavior of the radial solutions (whose analytic expression is not known) of the limit problem\mathcal L _ μ u= u^ 2^*-1 L μ u= u 2∗-1 in the whole space R^ n R n. On the other hand, our non–existence results depend on a suitable Pohozaev-type identity, which in turn relies on some weighted Hardy–Rellich inequalities.