## 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:

 [1] Ackermann, N, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037 [2] Alves, CO; Correa, FJSA; Ma, TF, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49, 85-93, (2005) · Zbl 1130.35045 [3] Alves, CO; Yang, M, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differ. Equ., 257, 4133-4164, (2014) · Zbl 1309.35036 [4] Cingolani, S; Clapp, M; Secchi, S, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63, 233-248, (2012) · Zbl 1247.35141 [5] Cingolani, S; Secchi, S; Squassina, M, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. R. Soc. Edinb. Sect. A, 140, 973-1009, (2010) · Zbl 1215.35146 [6] Clapp, M; Salazar, D, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407, 1-15, (2013) · Zbl 1310.35114 [7] He, Y; Li, G; Peng, S, Concentrating bound states for Kirchhoff type problem in $$\mathbb{R}^{3}$$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14, 441-468, (2014) · Zbl 1305.35033 [8] He, X; Zou, W, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $$\mathbb{R}^{3}$$, J. Differ. Equ., 252, 1813-1834, (2012) · Zbl 1235.35093 [9] Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883) · JFM 08.0542.01 [10] Lieb, E.H., Loss, M.: Analysis. In: Graduate Studies in Mathematics, vol. 14, 2nd edn. AMS, Providence (2001) [11] Lieb, EH, Existence and uniqueness of the minimizing solution of choquard’s nonlinear equation, Stud. Appl. Math., 57, 93-105, (1977) · Zbl 0369.35022 [12] Lieb, EH; Simon, B, The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys., 53, 185-194, (1977) [13] Lions, JL, On some questions in boundary value problems of mathematical physics, North-Holl. Math. Stud., 30, 284-346, (1978) · Zbl 0404.35002 [14] Lions, PL, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072, (1980) · Zbl 0453.47042 [15] Liu, W; He, X, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39, 473-487, (2012) · Zbl 1295.35226 [16] Lü, D, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99, 35-48, (2014) · Zbl 1286.35108 [17] Ma, L; Zhao, L, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455-467, (2010) · Zbl 1185.35260 [18] Moroz, V; Schaftingen, J, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-184, (2013) · Zbl 1285.35048 [19] Moroz, V; Schaftingen, J, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52, 199-235, (2015) · Zbl 1309.35029 [20] Penrose, R, On gravity’s role in quantum state reduction, Gen. Relativ. Gravit., 28, 581-600, (1996) · Zbl 0855.53046 [21] Rabinowitz, PH, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291, (1992) · Zbl 0763.35087 [22] Wang, J; Tian, L; Xu, J; Zhang, F, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253, 2314-2351, (2012) · Zbl 1402.35119 [23] Willem, M.: Minimax theorems. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser, Boston (1996) · Zbl 0856.49001 [24] Wu, X, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $$R^{N}$$, Nonlinear Anal. Real World Appl., 12, 1278-1287, (2011) · Zbl 1208.35034
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