The second bifurcation branch for radial solutions of the Brezis-Nirenberg problem in dimension four
The second bifurcation branch for radial solutions of the Brezis-Nirenberg problem in dimension four
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DOI：
10.1007/s00030-007-6034-8
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发表时间：
2008-01
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影响因子：
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通讯作者：
G. Arioli;F. Gazzola;H. Grunau;Edoardo Sassone
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文献类型：
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作者：
G. Arioli;F. Gazzola;H. Grunau;Edoardo Sassone
半线性Brezis-Nirenberg特征值问题的存在性结果表明，相应的作用泛函的紧性问题是更严重的小尺寸。在n = 3维空间中，甚至可以证明在特征参数一定范围内不存在正解.本文研究了在n = 4维情况下，这种紧性问题的不存在性。我们考虑Dirichlet边界条件下单位球上的方程-Δu= λu+ u3。研究了由-Δ的第二径向特征值引起的分支分支。已知当解的L ∞范数趋于爆破时，它渐近地趋于第一特征值.相反，在空间维数n = 5，我们表明，它不交叉的第一个特征值。特别地，当λ与第一特征值重合时，所提到的Dirichlet问题inn= 4不允许非平凡径向解。
Existence results available for the semilinear Brezis-Nirenberg eigenvalue problem suggest that the compactness problems for the corresponding action functionals are more serious in small dimensions. In space dimensionn= 3, one can even prove nonexistence of positive solutions in a certain range of the eigenvalue parameter. In the present paper we study a nonexistence phenomenon manifesting such compactness problems also in dimensionn= 4.We consider the equation –Δu= λu+u3in the unit ball ofunder Dirichlet boundary conditions. We study the bifurcation branch arising from the second radial eigenvalue of –Δ. It is known that it tends asymptotically to the first eigenvalue as theL∞-norm of the solution tends to blow up. Contrary to what happens in space dimensionn= 5, we show that it does not cross the first eigenvalue. In particular, the mentioned Dirichlet problem inn= 4 does not admit a nontrivial radial solution when λ coincides with the first eigenvalue.