References
- Bartsch T, Wang ZQ. Existence and multiplicity results for superlinear elliptic problems on ℝN. Commun. Part. Differ. Equ. 1995;20:1725–1741.
- Bartsch T, Pankov A, Wang ZQ. Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 2001;3:549–569.
- Bartsch T, Tang Z. Multibump solutions of nonlinear Schröinger equations with steep potential well and indefinite potential. Discrete Contin. Dyn. Syst. 2013;33:7–26.
- Ding YH, Szulkin A. Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. Part. Differ. Equ. 2007;29:397–419.
- Stuart C, Zhou H. Global branch of solutions for nonlinear Schrödinger equations with deepening potential well. Proc. Lond. Math. Soc. 2006;92:655–681.
- Wang Z, Zhou H. Positive solutions for nonlinear Schrödinger equations with deepening potential well. J. Eur. Math. Soc. 2009;11:545–573.
- Kirchhoff G. Mechanik. Leipzig: Teubner; 1883.
- Bernstein S. Sur une classe d’équations fonctionelles aux dérivées partielles. Bull. Acad. Sci. URSS. Sér. 1940;4:17–26.
- Pohožaev SI. A certain class of quasilinear hyperbolic equations. Mat. Sb. (N.S.). 1975;96:152–166, 168. Russian.
- Lions JL. On some questions in boundary value problems of mathematical physics. In: Contemporary development in continuum mechanics and partial differential equations. Vol. 30, North-Holland mathematical studies. Amsterdam: North-Holland; 1978. p. 284–346.
- Alves CO, Corrêa FJSA. On existence of solutions for a class of problem involving a nonlinear operator. Comm. Appl. Nonlinear Anal. 2001;8:43–56.
- Alves CO, Figueiredo GM. On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in ℝN. J. Differ. Equ. 2009;246:1288–1311.
- Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 1996;348:305–330.
- Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA. Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 2001;6:701–730.
- D’Ancona P, Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 1992;108:247–262.
- Alves CO, Corrêa FJSA, Ma TF. Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 2005;49:85–93.
- Chen CY, Kuo YC, Wu TF. The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 2011;250:1876–1908.
- Chipot M, Lovat B. Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. 1997;30:4619–4627.
- Ma TF, Muñoz Rivera JE. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 2003;16:243–248.
- Perera K, Zhang Z. Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 2006;221:246–255.
- He XM, Zou WM. Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 2009;70:1407–1414.
- Li GB, Ye HY. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ3. J. Differ. Equ. 2014;257:566–600.
- Alves CO, Corrêa FJSA, Figueiredo GM. On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2010;2:409–417.
- Bensedik A, Bouchekif M. On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity. Math. Comput. Modell. 2009;49:1089–1096.
- D’Ancona P, Shibata Y. On global solvability of non-linear viscoelastic equations in the analytic category. Math. Methods Appl. Sci. 1994;17:477–489.
- He XM, Zou WM. Existence and concentration behavior of positive solutions for a Kirchhoff equation in ℝ3. J. Differ. Equ. 2012;2:1813–1834.
- Jin J, Wu X. Infinitely many radial solutions for Kirchhoff-type problems in ℝN. J. Math. Anal. Appl. 2010;369:564–574.
- Li Y, Li F, Shi J. Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 2012;253:2285–2294.
- Mao A, Zhang Z. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 2009;70:1275–1287.
- Nishihara K. On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 1984;7:437–459.
- 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. 2012;253:2314–2351.
- Wu X. Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in ℝN. Nonlinear Anal. Real World Appl. 2011;12:1278–1287.
- Zhang Z, Perera K. Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 2006;317:456–463.
- Jiang Y, Zhou H. Schrödinger–Poisson system with steep potential well. J. Differ. Equ. 2011;251:582–608.
- Zhao L, Liu H, Zhao F. Existence and concentration of solutions for the Schrödinger–Poisson equations with steep well potential. J. Differ. Equ. 2013;255:1–23.
- Ding YH, Wei JC. Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. J. Funct. Anal. 2007;251:546–572.
- Sun JT, Wu TF. Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J. Differ. Equ. 2014;256:1771–1792.
- Alves CO, Figueiredo GM. Nonlinear perturbations of a periodic Kirchhoff equation in ℝN. Non. Anal. 2012;75:2750–2759.
- He Y, Li GB, Peng SJ. Concentrating bound states for Kirchhoff type problems in ℝ3 involving critical Sobolev exponents. Adv. Nonlinear Stud. 2014;14:483–510.
- Szulkin A, Weth T. The method of Nehari manifold. In: Gao DY, Motreanu D, editors. Handbook of nonconvex analysis and applications. Boston: International Press; 2010. p. 597–632.
- Lions JL. The concentration-compactness principle in the calculus of variations. The locally compact case, part I. Ann. Inst. H. Poincaré Anal. Non Linéaire. 1984;1:109–145.
- Willem M. Minimax theorems. Boston (MA): Birkhäuser; 1996.