ON GROUND STATE OF FRACTIONAL P-KIRCHHOFF EQUATION INVOLVING SUBCRITICAL AND CRITICAL EXPONENTIAL GROWTH

In this paper, we concern the existence of nontrivial ground state solutions of fractional $ p $-Kirchhoff equation

$ \left\{\begin{array}{ll} m\left(\|u\|^p\right) [(-\Delta)_p^su+V(x)|u|^{p-2}u] =f(x,u) \quad\text{in}\, \mathbb{R}^N, \\ \|u\|=\left(\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy +\int_{\mathbb{R}^N}V(x)|u|^pdx\right)^{\frac{1}{p}}, \end{array}\right. $

where $ m:[0,+\infty)\rightarrow [0,+\infty) $ is a continuous function, $ (-\Delta)_p^s $ is the fractional $ p $-Laplacian operator with $ 0<s<1<p<\infty $, $ V:\mathbb{R}^N\rightarrow [0,+\infty) $ is a continuous and $ 1 $ periodic function and $ f\in C (\mathbb{R}^N\times \mathbb{R}) $ is $ 1 $-periodic in $ x_1,\cdot\cdot\cdot,x_N. $ When the nonlinearity $ f(x,u) $ has subcritical or critical exponential growth at $ \infty $ without satisfying the Ambrosetti-Rabinowitz (AR) condition some existence results for nontrivial ground state solutions are obtained by using the minimax techniques, Nehari manifold methods combined with the fractional Moser-Trudinger inequality.