Qualitative Analysis of Solutions for Fractional p-Kirchhoff Problems Involving Critical Exponential Growth

In this article, we investigate a class of fractional p-Kirchhoff equations with Trudinger–Moser nonlinearities: \(\begin{aligned} \varepsilon ^{N} (a+b[v]_{s,p}^{p})(-\Delta )_{p}^{s}v+Z(x)|v|^{p-2}v=g(v), \;\;\;x\in {\mathbb {R}}^N, \end{aligned}\) $$\begin{array}{c}\hfill {\epsilon }^N(a+b{[v]}_{s,p}^p){(-\mathrm{\Delta })}_p^{s}v+Z(x){|v|}^{p-2}v=g(v),x\in {\mathbb{R}}^N,\end{array}$$ where \(a>0\) $a>0$ , \(b>0\) $b>0$ , \(N = ps\) $N=ps$ , \(p\geqslant 2\) $p⩾2$ , \(s\in ( 0,1 )\) $s\in (0,1)$ and \(\varepsilon \) $\epsilon$ is a positive parameter, the nonlinear function g has critical exponential growth in the sense of Trudinger–Moser inequality, the continuous potential function Z satisfies appropriate conditions. By using some delicate variational techniques, we first prove the existence of the ground state solutions for above problem, and which are concentrated at a global minimum of Z under some suitable conditions. Then, we prove that these solutions satisfy a decay property. Finally, we get the multiplicity of solutions by the Ljusternik–Schnirelmann theory and the Morse theory, respectively.