Abstract
Abstract In this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity { - ( ε 2 a + ε b ∫ ℝ 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = ε μ - 3 ( ∫ ℝ 3 K ( y ) F ( u ( y ) ) | x - y | μ d y ) K ( x ) f ( u ) , u ∈ H 1 ( ℝ 3 ) , \left\{\begin{aligned} \displaystyle-&\displaystyle\biggl{(}\varepsilon^{2}a+% \varepsilon b\int_{\mathbb{R}^{3}}\lvert\nabla u\rvert^{2}\mathop{}\!dx\biggr{% )}\Delta u+V(x)u=\varepsilon^{\mu-3}\biggl{(}\int_{\mathbb{R}^{3}}\frac{K(y)F(% u(y))}{\lvert x-y\rvert^{\mu}}\mathop{}\!dy\biggr{)}K(x)f(u),\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right. where ε > 0 {\varepsilon>0} is a small parameter, a , b > 0 {a,b>0} , μ ∈ ( 0 , 3 ) {\mu\in(0,3)} , V , K {V,K} are two positive continuous function and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of the Hardy–Littlewood–Sobolev inequality. We show that the equation admits a positive ground state solution for ε > 0 {\varepsilon>0} sufficiently small. Furthermore, we prove that these ground state solutions concentrate around such points which are both the minima points of the potential V and the maximum points of the potential K as ε → 0 {\varepsilon\to 0} .
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