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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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