## Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems.(English)Zbl 1369.35023

The Kirchhoff-type equation with Hartree-type nonlinearity
$-\bigg(\varepsilon^2a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2dx\bigg)\Delta u+V(x)u = \varepsilon^{\alpha-3}(W_\alpha(x)\ast|u|^p)|u|^{p-2}$
is studied, where $$\varepsilon>0$$ is a parameter, $$a>0$$, $$b\geq0$$ are constants, $$W_\alpha(x)=|x|^{-\alpha}$$, $$\alpha\in(0,3)$$, $$p\in[2,6-\alpha)$$, and $$V\in C(\mathbb R^3,\mathbb R)$$ is an external potential satisfying the condition
$\liminf_{|x|\to\infty}V(x)>V_0=\inf_{\mathbb R^3}V(x)>0.$ Kirchhoff-type equations arise in various models of physical or biological systems, and the Hartree-type nonlinearities $$(w(x)\ast G(u))G'(u)$$ with $$G\in C^1(\mathbb R,\mathbb R)$$ appear naturally in many physical applications.
A solution $$u_0$$ of the problem with $$\varepsilon=0$$ is called ground state solution if its energy is minimal among the energies of all nontrivial solutions to the problem with positive $$\varepsilon$$. The author uses a variational approach to prove that for sufficiently small parameter $$\varepsilon$$ there exist positive ground state solutions $$w_\varepsilon\in H^1(\mathbb R^3)$$ with the following properties:
(i) $$w_\varepsilon$$ has a maximum point $$x_\varepsilon\in\mathbb R^3$$ whose distance from the set $$\{x\in\mathbb R^3: V(x)=V_0\}$$ vanishes for $$\varepsilon\to0^+$$.
(ii) A subsequence of $$\{w_\varepsilon(x_\varepsilon+\varepsilon x)\}$$ converges in $$H^1(\mathbb R^3)$$ to the ground state solution of the problem with $$\varepsilon=0$$ and $$V(x)$$ replaced with $$V_0$$.
(iii) There exist $$C,\xi>0$$ such that $$w_\varepsilon(x)\leq C\exp\bigl(-\frac\xi\varepsilon|x-x_\varepsilon|\bigr)$$ for all $$x\in\mathbb R^3$$.

### MSC:

 35J60 Nonlinear elliptic equations
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### References:

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