Ground State Solutions for Fractional p-Kirchhoff Equation with Subcritical and Critical Exponential Growth

Abstract

In this paper, we show the existence of nontrivial ground state solutions of fractional p-Kirchhoff problem $$\begin{aligned} \left\{ \begin{array}{ll} m\left( \int _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{d}x\mathrm{d}y\right) (-\Delta )_p^s u=f(x,u) ~&{}\text {in}~\Omega , \\ u=0 ~&{}\text {in}~{\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$where $$(-\Delta )_p^s$$ is the fractional p-Laplacian operator with $$0<s<1<p<\infty $$, $$\Omega $$ is a bounded domain in $${\mathbb {R}}^N$$ with smooth boundary, m is continuous function and the nonlinearity f(x, u) has subcritical or critical exponential growth at $$\infty $$. For the purpose of obtaining our existence results, we used minimax techniques combined with the fractional Moser–Trudinger inequality.