Fractional Kirchhoff-Schrödinger equation with critical exponential growth in $\mathbb{R}^{N}$

Abstract

In this paper, we consider the following fractional Kirchhoff-Schrödinger equation: \begin{equation*} \begin{cases} \displaystyle \Big(a+b\|u\|_{E}^{p(\theta-1)}\Big) \big[(-\triangle)_{p}^{s}u+V(x)|u|^{p-2}u\big]=f(x, u), & x \in \mathbb{R}^{N},\\ \displaystyle \|u\|_{E}^p:=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy +\int_{\mathbb{R}^{N}}V(x)|u|^{p}dx, \end{cases} \end{equation*} where $a> 0$, $b \geq 0$, $\theta \geq1$, dimension $N=sp$ with $s \in (0, 1)$ and $p\geq 1$. $ V$ is a positive potential and $f$ is a critical nonlinearity with exponential growth. We derive a positive ground state solution by using minimax techniques combined with the fractional Truding-Moser inequality. Moreover, in the particular case of $a=1$ and $b=0$, we also obtain the existence of the ground state solution to the fractional Schrödinger equation.