Lane Emden problems with large exponents and singular Liouville equations
Lane Emden problems with large exponents and singular Liouville equations
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具有大指数和奇异刘维尔方程的 Lane Emden 问题
DOI：
10.1016/j.matpur.2013.06.011
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
F. Pacella
中科院分区：
文献类型：
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作者：
M. Grossi;C. Grumiau;F. Pacella
考虑Lane-Emden Dirichlet问题{−Δ u=| u| p−1 u，在∂B中，u= 0，其中p> 1和B表示r2中的单位球。研究了最小能量节点径向解u p在p→+∞时的渐近性。假设wlog u p(0)< 0，我们证明了对负部分u p−进行适当的重标可收敛于r2中Liouville方程的唯一正则解，而对正部分u p+进行适当的重标可收敛于r2中奇异Liouville方程的（奇异）解。我们还得到了u p -和u p+的L∞范数的精确渐近值，以及能量的渐近估计。最后，我们得到结点线N p:={x∈B:| x|= r p}缩小到一个点，并计算r p的收敛速度。
Abstract We consider the Lane–Emden Dirichlet problem {− Δ u=| u| p− 1 u, in B, u= 0, on∂ B, where p> 1 and B denotes the unit ball in R 2. We study the asymptotic behavior of the least energy nodal radial solution u p, as p→+∞. Assuming wlog that u p (0)< 0, we prove that a suitable rescaling of the negative part u p− converges to the unique regular solution of the Liouville equation in R 2, while a suitable rescaling of the positive part u p+ converges to a (singular) solution of a singular Liouville equation in R 2. We also get exact asymptotic values for the L∞-norms of u p− and u p+, as well as an asymptotic estimate of the energy. Finally, we have that the nodal line N p:={x∈ B:| x|= r p} shrinks to a point and we compute the rate of convergence of r p.