We study singular radially symmetric solution to the Lin–Ni–Takagi equation for a supercritical power non-linearity in dimension \(N\ge 3\). It is shown that for any ball and any \(k \ge 0\), there is a singular solution that satisfies Neumann boundary condition and oscillates at least k times around the constant equilibrium. Moreover, we show that the Morse index of the singular solution is finite or infinite if the exponent is respectively larger or smaller than the Joseph–Lundgren exponent.

中文翻译：

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Lin-Ni-Takagi方程的奇异径向解

我们研究尺寸为\（N \ ge 3 \）的超临界功率非线性问题的Lin–Ni–Takagi方程的奇异径向对称解。结果表明，对于任何球和任何\（k \ ge 0 \），都有一个满足Neumann边界条件并在恒定平衡附近至少振荡k次的奇异解。此外，我们表明，如果指数分别大于或小于Joseph-Lundgren指数，则奇异解的Morse指数是有限的或无限的。