On the singularly perturbation fractional Kirchhoff equations: Critical case

Abstract

Abstract This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , \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 > 0 a,b\gt 0 are given constants, ε \varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{\ast }=\frac{2N}{N-2s} with 0 < s < 1 0\lt s\lt 1 and N ≥ 4 s N\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 > 4 s N\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.