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Published online by Cambridge University Press:
**03 March 2014**

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem

$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$$\mathrm{\left(}{\mathit{P}}_{\mathit{\epsilon}}\mathrm{\right)}\left\{\begin{array}{c}\\ {\mathrm{\mathcal{L}}}_{\mathit{\epsilon}}\mathit{u}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathit{>}\mathrm{0}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathrm{\in}{\mathit{H}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{IR}}^{\mathrm{3}}\mathrm{\right)}\mathit{,}\end{array}\right.$

where ε is a small positive parameter, f : ℝ → ℝ is a continuous function, $$ \mathcal{L}_{\var} $$${\mathrm{\mathcal{L}}}_{\mathit{\epsilon}}$ is a nonlocal operator defined by

$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$${\mathrm{\mathcal{L}}}_{\mathit{\epsilon}}\mathit{u}\mathrm{=}\mathit{M}\left(\frac{\mathrm{1}}{\mathit{\epsilon}}{\mathrm{\int}}_{{\mathrm{IR}}^{\mathrm{3}}}\mathrm{|}\mathrm{\nabla}\mathit{u}{\mathrm{|}}^{\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{{\mathit{\epsilon}}^{\mathrm{3}}}{\mathrm{\int}}_{{\mathrm{IR}}^{\mathrm{3}}}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{2}}\right)\mathrm{[}\mathrm{-}{\mathit{\epsilon}}^{\mathrm{2}}\mathrm{\Delta}\mathit{u}\mathrm{+}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{]}}\mathit{,}$

M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.

- Type
- Research Article
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- ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 389 - 415
- Copyright
- © EDP Sciences, SMAI, 2014

Alves, C.O. and
Corrêa, F.J.S.A., On existence of solutions for a
class of problem involving a nonlinear operator. *Commun.
Appl. Nonlinear Anal.*
8 (2001) 43–56.
Google Scholar

Alves, C.O., Corrêa, F.J.S.A. and Figueiredo, G.M., On a class of nonlocal
elliptic problems with critical growth. *Differ. Equ.
Appl.*
2 (2010) 409–417.
Google Scholar

Alves, C.O., Corrêa, F.J.S.A. and Ma, T.F., Positive solutions for a quasilinear
elliptic equation of Kirchhoff type. *Comput. Math.
Appl.*
49 (2005) 85–93.
Google Scholar

Alves, C.O. and Figueiredo, G.M., Nonlinear perturbations of a
periodic Kirchhoff equation in IR^{N}.
*Nonlinear Anal.*
75 (2012)
2750–2759. Google Scholar

Alves, C.O. and Figueiredo, G.M., Multiplicity of positive
solutions for a quasilinear problem in IR^{N}
*via* penalization method. *Adv. Nonlinear
Stud.*
5 (2005) 551–572.
Google Scholar

Alves, C.O., Figueiredo, G.M. and
Furtado, M.F., Multiple solutions for a
Nonlinear Schrödinger Equation with Magnetic Fields.
*Commun. Partial Differ. Equ.*
36 (2011) 1–22.
Google Scholar

Ambrosetti, A., Badiale, M. and Cingolani, S., Semiclassical stats of
nonlinear Schrodinger equations with potentials. *Arch.
Ration. Mech. Anal.*
140 (1997)
285–300. Google Scholar

Ambrosetti, A., Malchiodi, A. and
Secchi, S., Multiplicity results for some
nonlinear Schorodinger equations with potentials. *Arch.
Ration. Mech. Anal.*
159 (2001)
253–271. Google Scholar

Anelo, G., A uniqueness result for a
nonlocal equation of Kirchhoff equation type and some related open
problem. *J. Math. Anal. Appl.*
373 (2011)
248–251. Google Scholar

G. Anelo, On a pertubed Dirichlet problem for a
nonlocal differential equation of Kirchhoff type. *BVP* (2011) 891430.

Cingolani, S. and
Lazzo, M., Multiple positive solutions to
nonlinear Schrodinger equations with competing potential functions.
*J. Differ. Equ.*
160 (2000)
118–138. Google Scholar

Del Pino, M. and
Felmer, P.L., Local Mountain Pass for
semilinear elliptic problems in unbounded domains. *Calc.
Var.*
4 (1996) 121–137.
Google Scholar

Figueiredo, G.M. and
Santos Junior, J.R., Multiplicity of solutions
for a Kirchhoff equation with subcritical or critical growth.
*Differ. Integral Equ.*
25 (2012)
853–868. Google Scholar

Floer, A. and Weinstein, A., Nonspreading wave packets for
the cubic Schrodinger equation with a bounded potential.
*J. Funct. Anal.*
69 (1986)
397–408. Google Scholar

He, X. and Zou, W., Existence and concentration of
positive solutions for a Kirchhoff equation in IR^{3}.
*J. Differ. Equ.*
252 (2012)
1813–1834. Google Scholar

G. Kirchhoff, Mechanik. Teubner, Leipzig (1883).

Li, Y., Li, F. and Shi, J., Existence of a positive solution to
Kirchhoff type problems without compactness conditions.
*J. Differ. Equ.*
253 (2012)
2285–2294. Google Scholar

Li, G., Some properties of weak solutions of
nonlinear scalar field equations. *Ann. Acad. Sci.
Fenincae Ser. A*
14 (1989) 27–36.
Google Scholar

J.L. Lions, On some questions in boundary value
problems of mathematical physics International Symposium on Continuum, *Mech.
Partial Differ. Equ.* Rio de Janeiro(1977). In vol. 30 of *Math.
Stud.* North-Holland, Amsterdam (1978) 284–346.

Ma, T.F., Remarks on an elliptic equation of
Kirchhoff type. *Nonlinear Anal.*
63 (2005)
1967–1977. Google Scholar

Moser, J., A new proof de Giorgi’s theorem
concerning the regularity problem for elliptic differential equations.
*Commun. Pure Appl. Math.*
13 (1960)
457–468. Google Scholar

Nie, Jianjun and Wu, Xian, Existence and multiplicity of
non-trivial solutions for Schródinger–Kirchhoff equations with radial
potential. *Nonlinear Analysis*
75 (2012)
3470–3479. Google Scholar

Rabinowitz, P.H., On a class of nonlinear
Schrodinger equations. *Z. Angew Math.
Phys.*
43 (1992) 27–42.
Google Scholar

Szulkin, A. and Weth, T., Ground state solutions for some
indefinite variational problems. *J. Funct.
Anal.*
257 (2009)
3802–3822. Google Scholar

A. Szulkin and T. Weth, *The method of
Nehari manifold, Handbook of Nonconvex Analysis and Applications*, edited by
D.Y. Gao and D. Montreanu. International Press, Boston (2010) 597–632.

Wang, J., Tian, L., Xu, J. and Zhang, F., Multiplicity and concentration of
positive solutions for a Kirchhoff type problem with critical growth.
*J. Differ. Equ.*
253 (2012)
2314–2351. Google Scholar

M. Willem, Minimax Theorems.
*Birkhauser* (1996).

Wu, Xian, Existence of nontrivial solutions
and high energy solutions for Schrödinger–Kirchhoff-type equations in R^{N}.
*Nonlinear Anal. RWA*
12 (2011)
1278–1287. Google Scholar

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