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

Abstract

In this article, we study the following fractional Kirchhoff-type problems with critical and sublinear nonlinearities: a+b∬RN×RN∣u(x)−u(y)∣2∣x−y∣N+2sdxdy(−Δ)su=λuq−1+u2s*−1,u>0,inΩ,u=0,inRN\Ω,∫RNu2dx=c2,\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⊂RN\Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, 00,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.

Keywords

• fractional laplacian
• kirchhoff-type equation
• normalized solution
• profile decomposition
• 35d30
• 35j20
• 35j60