| Summary: | This article deals with the following fractional Kirchhoff problem with critical exponent a+b∫RN∣(−Δ)s2u∣2dx(−Δ)su=(1+εK(x))u2s∗−1,inRN,\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a,b>0a,b\gt 0 are given constants, ε\varepsilon is a small parameter, 2s∗=2NN−2s{2}_{s}^{\ast }=\frac{2N}{N-2s} with 0<s<10\lt s\lt 1 and N≥4sN\ge 4s. We first prove the nondegeneracy of positive solutions when ε=0\varepsilon =0. In particular, we prove that uniqueness breaks down for dimensions N>4sN\gt 4s, i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε\varepsilon small.
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