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

Abstract

This article deals with the following fractional \(\frac{N}{s}\)-Laplace Kichhoff equation involving exponential growth of the form:

$$\begin{aligned} \varepsilon ^{N}K\left( [u]_{s,\frac{N}{s}}^{\frac{N}{s}}\right) (-\Delta )_{{N}/{s}}^{s}u+Z(x)|u|^{\frac{N}{s}-2}u=f(u)\;\text {in}\; \mathbb R^{N}, \end{aligned}$$

where \(\varepsilon >0\) is a parameter, \(s\in (0,1)\) and \((-\Delta )_p^s\) is the fractional p-Laplace operator with \(p=\frac{N}{s}\ge 2\), 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.