Fractional Kirchhoff equation with a general critical nonlinearity

Abstract

In this paper, we study the fractional Kirchhoff equation with critical nonlinearity \begin{align*} \left(a+b\int_{\mathbb R^N}|(-Δ)^{\frac{s}{2}}u|^2dx\right)(-Δ)^su+u=f(u)\ \ \mbox{in}\ \ \mathbb R^N, \end{align*} where $N>2s$ and $(-\Delta {)}^s$ is the fractional Laplacian with $0<s<1$. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti-Rabinowitz condition when the parameter $b$ small. What's more, we obtain the asymptotic behavior of solutions as $b\to 0$.

References in corpus (3)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension