Normalized solutions to the fractional Kirchhoff equations with combined nonlinearities

Abstract

In this paper, we study the existence and asymptotic properties of solutions to the following fractional Kirchhoff equation \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u=λu+μ|u|^{q-2}u+|u|^{p-2}u \quad \hbox{in ${\mathbb{R}}^3$,} \end{equation*} with a prescribed mass \begin{equation*} \int_{\mathbb{R}^{3}}|u|^{2}dx=c^{2}, \end{equation*} where $s\in (0,1)$, $a,b,c>0$, $2<q<p<2_s^{\ast }=\frac{6}{3-2s}$, $\mu >0$ and $\lambda \in \mathbb{R}$ as a Lagrange multiplier. Under different assumptions on $q<p$, $c>0$ and $\mu >0$, we prove some existence results about the normalized solutions. Our results extend the results of Luo and Zhang (Calc. Var. Partial Differential Equations 59, 1-35, 2020) to the fractional Kirchhoff equations. Moreover, we give some results about the behavior of the normalized solutions obtained above as $\mu \to 0^{+}$.

References in corpus (1)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian