Concentrated solution for some non-local and non-variational singularly perturbed problems

Nondegeneracy of solutions to the critical p$p$‐Laplace Kirchhoff equation

“…where 𝑁 ⩾ 3 and 1 < 𝑞 < 2 * , and Li et al [33] proved the uniqueness and nondegeneracy of solutions to (1.7) for 𝑁 = 3. Chen and Dai [10] proved that the uniqueness breaks down for 𝑁 ⩾ 5 and small 𝑏, and they [11] also established analogous conclusions for more general Kirchhoff functions. Furthermore, Yang [45] established the nondegeneracy of solutions to the following fractional Kirchhoff problem:…”

“…[33] proved the uniqueness and nondegeneracy of solutions to (1.7) for $$N=3$$. Chen and Dai [10] proved that the uniqueness breaks down for $$N\geqslant 5$$ and small $$b$$, and they [11] also established analogous conclusions for more general Kirchhoff functions. Furthermore, Yang [45] established the nondegeneracy of solutions to the following fractional Kirchhoff problem: $$$$\begin{equation} -{\left(1+b\int _{\mathbb {R}^N}|(-\Delta )^{\frac{s}{2}} u|^2 dx\right)} (-\Delta )^s u +u=u^{q},\quad u&gt;0\quad \mbox{in}\quad \mathbb {R}^N, \end{equation}$$$$where $$1&lt;q&lt;2^*_s-1$$, combining with the nondegenerate results as those in [28, 29], and the author also proved that the solutions of (1.8) are unique for $$1&lt;N\leqslant 4s$$ while uniqueness breaks down for $$N&gt;4s$$ under some suitable assumptions about $$b&gt;0$$.…”