A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass

Abstract

Abstract In this article, we study the following fractional Kirchhoff-type problems with critical and sublinear nonlinearities: a + b ∬ R N × R N ∣ u ( x ) − u ( y ) ∣ 2 ∣ x − y ∣ N + 2 s d x d y ( − Δ ) s u = λ u q − 1 + u 2 s * − 1 , u > 0 , in Ω , u = 0 , in R N \ Ω , ∫ R N u 2 d x = c 2 , \left\{\begin{array}{l}\left(a+b\mathop{\iint }\limits_{{{\mathbb{R}}}^{N}\times {{\mathbb{R}}}^{N}}\frac{{| u\left(x)-u(y)| }^{2}}{{| x-y| }^{N+2s}}{\rm{d}}x{\rm{d}}y\right){\left(-\Delta )}^{s}u=\lambda {u}^{q-1}+{u}^{{2}_{s}^{* }-1},\hspace{1em}u\gt 0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0\left,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2},\end{array}\right. where ( − Δ ) s {\left(-\Delta )}^{s} is the fractional Laplacian, Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, 0 < s < 1 , 2 s < N < 4 s 0\lt s\lt 1,2s\lt N\lt 4s , 1 < q < 2 1\lt q\lt 2 , λ > 0 , a > 0 , b > 0 , c > 0 \lambda \gt 0,a\gt 0,b\gt 0,c\gt 0 . First, we prove that the bounded Palais-Smale sequence has a profile decomposition in the fractional Laplacian setting. Then, by utilizing decomposition techniques and variational methods, we acquire that there are two positive normalized solutions for the aforementioned problems.