Multiplicity and Concentration Behavior of Solutions to a Class of Fractional Kirchhoff Equation Involving Exponential Nonlinearity

This article deals with the following fractional \(\frac{N}{s}\) $\frac{N}{s}$ -Laplace Kichhoff equation involving exponential growth of the form: \(\begin{aligned} \varepsilon ^{N}K\left( [u]_{s,\frac{N}{s}}^{\frac{N}{s}}\right) (-\Delta )_{{N}/{s}}^{s}u+Z(x)|u|^{\frac{N}{s}-2}u=f(u)\;\text {in}\; \mathbb R^{N}, \end{aligned}\) $$\begin{array}{c}\hfill {\epsilon }^{N}K\left({[u]}_{s,\frac{N}{s}}^{\frac{N}{s}}\right){(-\mathrm{\Delta })}_{N/s}^{s}u+Z(x){|u|}^{\frac{N}{s}-2}u=f(u)\text{in}{\mathbb{R}}^N,\end{array}$$ where \(\varepsilon >0\) $\epsilon >0$ is a parameter, \(s\in (0,1)\) $s\in (0,1)$ and \((-\Delta )_p^s\) ${(-\mathrm{\Delta })}_p^s$ is the fractional p-Laplace operator with \(p=\frac{N}{s}\ge 2\) $p=\frac{N}{s}\ge 2$ , K is a Kirchhoff function, f is a continuous function with exponential growth and Z is a potential function possessing a local minimum. Under some suitable conditions, we obtain the existence, multiplicity and concentration of solutions to the above problem via penalization methods and Lyusternik-Schnirelmann theory.