EXISTENCE AND CONCENTRATION RESULT FOR KIRCHHOFF EQUATIONS WITH CRITICAL EXPONENT AND HARTREE NONLINEARITY

This paper is concerned with the following Kirchhoff-type equations $\left\{ \begin{array}{l} -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x, u), &\quad \mbox{ in }\mathbb{R}^{3}, \\ (-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p}, ~u>0, &\quad \mbox{ in }\mathbb{R}^{3}, \end{array} \right. $ where $f(x, u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0, ~b, ~\mu\geq0$ are constants, $\alpha\in(0, 3)$, $p\in[2, 3), ~q\in[2p, 6)$ and $\varepsilon, ~\lambda>0$ are parameters. Under some mild conditions on $V(x), ~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.