Nodal solutions for fractional Kirchhoff problems involving critical exponential growth

Abstract

In this paper we discuss the existence of least energy nodal solutions for a class of fractional Kirchhoff problems $(a+b{\left[u\right]}_{1/2}^2){(-\Delta )}^{1/2}u$ + $V\left(x\right)u$ = $f(x,u)$ in $R$, where $a>0$, $b\ge 0$ and $f(x,u)$ is a nonlinear term with critical exponential growth. By using the deformation lemma, we obtain a least energy nodal solution $u_b$ for this class of problems. Furthermore, the study of the asymptotic behavior of $u_b$ as $b\to 0$ allows us to prove the existence of nodal solutions for the equation in the absence of the Kirchhoff term. To the best of our knowledge, this is the first result proving the existence of nodal solutions for this type of equations.