Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method

Figueiredo, Giovany M.; Santos, João R.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 2, pp. 389-415.

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

$$\left(P_{\epsilon }\right)\left\{\begin{array}{c}\hfill {ℒ}_{\epsilon }u=f\left(u\right)\text{in}{ℝ}^3,\\ \hfill u>0\text{in}{ℝ}^3,\\ \hfill u\in H^1\left({ℝ}^3\right),\end{array}\right.$$

where $\epsilon$ is a small positive parameter, $f:ℝ\to ℝ$ is a continuous function, ${ℒ}_{\epsilon }$ is a nonlocal operator defined by

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

$M:{ℝ}_{+}\to {ℝ}_{+}$ and $V:{ℝ}^3\to ℝ$ are continuous functions which verify some hypotheses.

Classification : 35J65, 34B15
Mots clés : penalization method, Schrödinger-Kirchhoff type problem, Lusternik-Schnirelmann theory, Moser iteration

[1] C.O. Alves and F.J.S.A. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal. 8 (2001) 43-56. | MR | Zbl

[2] C.O. Alves, F.J.S.A. Corrêa and G.M. Figueiredo, On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2 (2010) 409-417. | MR | Zbl

[3] C.O. Alves, F.J.S.A. Corrêa and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49 (2005) 85-93. | MR | Zbl

[4] C.O. Alves and G.M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in IRN. Nonlinear Anal. 75 (2012) 2750-2759. | MR | Zbl

[5] C.O. Alves and G.M. Figueiredo, Multiplicity of positive solutions for a quasilinear problem in IRN via penalization method. Adv. Nonlinear Stud. 5 (2005) 551-572. | MR | Zbl

[6] C.O. Alves, G.M. Figueiredo and M.F. Furtado, Multiple solutions for a Nonlinear Schrödinger Equation with Magnetic Fields. Commun. Partial Differ. Equ. 36 (2011) 1-22. | MR | Zbl

[7] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical stats of nonlinear Schrodinger equations with potentials. Arch. Ration. Mech. Anal. 140 (1997) 285-300. | MR | Zbl

[8] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schorodinger equations with potentials. Arch. Ration. Mech. Anal. 159 (2001) 253-271. | MR | Zbl

[9] G. Anelo, A uniqueness result for a nonlocal equation of Kirchhoff equation type and some related open problem. J. Math. Anal. Appl. 373 (2011) 248-251. | MR | Zbl

[10] G. Anelo, On a pertubed Dirichlet problem for a nonlocal differential equation of Kirchhoff type. BVP (2011) 891430. | MR | Zbl

[11] S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrodinger equations with competing potential functions. J. Differ. Equ. 160 (2000) 118-138. | MR | Zbl

[12] M. Del Pino and P.L. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains. Calc. Var. 4 (1996) 121-137. | MR | Zbl

[13] I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47 (1974) 324-353. | MR | Zbl

[14] G.M. Figueiredo and J.R. Santos Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differ. Integral Equ. 25 (2012) 853-868. | MR | Zbl

[15] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrodinger equation with a bounded potential. J. Funct. Anal. 69 (1986) 397-408. | MR | Zbl

[16] X. He and W. Zou, Existence and concentration of positive solutions for a Kirchhoff equation in IR3. J. Differ. Equ. 252 (2012) 1813-1834. | MR | Zbl

[17] G. Kirchhoff, Mechanik. Teubner, Leipzig (1883).

[18] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253 (2012) 2285-2294. | MR | Zbl

[19] G. Li, Some properties of weak solutions of nonlinear scalar field equations. Ann. Acad. Sci. Fenincae Ser. A 14 (1989) 27-36. | MR | Zbl

[20] 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. | MR | Zbl

[21] T.F. Ma, Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. 63 (2005) 1967-1977. | Zbl

[22] J. Moser, A new proof de Giorgi's theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13 (1960) 457-468. | MR | Zbl

[23] Jianjun Nie and Xian Wu, Existence and multiplicity of non-trivial solutions for Schródinger-Kirchhoff equations with radial potential. Nonlinear Analysis 75 (2012) 3470-3479. | MR | Zbl

[24] P.H. Rabinowitz, On a class of nonlinear Schrodinger equations. Z. Angew Math. Phys. 43 (1992) 27-42. | MR | Zbl

[25] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257 (2009) 3802-3822. | MR | Zbl

[26] 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. | MR | Zbl

[27] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253 (2012) 2314-2351. | MR

[28] M. Willem, Minimax Theorems. Birkhauser (1996). | MR | Zbl

[29] Xian Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in RN. Nonlinear Anal. RWA 12 (2011) 1278-1287. | MR | Zbl