Multiple solutions of the Kirchhoff-type problem in R^N

[1]T. Bartsch, Z. Wang. (1995), Existence and multiple results for some superlinear elliptic problems on R^N, Commun. Partial. Differ. Equ. 20(9-10):1725-1741. 10.1080/03605309508821149BartschT.WangZ.1995Existence and multiple results for some superlinear elliptic problems on R^N, Commun. Partial. Differ. Equ209-101725174110.1080/03605309508821149Open DOISearch in Google Scholar

[2]G. Kirchhoff. (1883), Mechanik, Teubner, leipzig.KirchhoffG.1883Mechanik, Teubner, leipzigSearch in Google Scholar

[3]C.O. Alves, F.J.S.A. Corrêa, T.F. Ma. (2005), Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49:85-93. 10.1016/j.camwa.2005.01.008AlvesC.O.CorrêaF.J.S.A.MaT.F.2005Positive solutions for a quasilinear elliptic equation of Kirchhoff typeComput. Math. Appl49859310.1016/j.camwa.2005.01.008Open DOISearch in Google Scholar

[4]J. Lions. (1978), On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos. Inst. Mat, Univ. Fed. Rio de Janeiro, 1997, in: North-Holland Math. Stud. 30:284-346. 10.1016/S0304-0208(08)70870-3LionsJ.1978On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. InternatSympos. Inst. Mat, Univ. Fed. Rio de Janeiro1997North-Holland Math. Stud3028434610.1016/S0304-0208(08)70870-3Open DOISearch in Google Scholar

[5]S. Bernstein. (1940), Sur une class d’zquations fonctionnelles aux dzrivzes partielles, Bull. Acad. Sci. URSS. Szr. Math. [Izv. Akad. Nauk SSSR] 4:17-26.BernsteinS.1940Sur une class d’zquations fonctionnelles aux dzrivzes partielles, Bull. Acad. SciURSS. Szr. Math. [Izv. Akad. Nauk SSSR]41726Search in Google Scholar

[6]B. Cheng, X. Wu. (2009), Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal. 71:4883-4892. 10.1016/j.na.2009.03.065ChengB.WuX.2009Existence results of positive solutions of Kirchhoff type problemsNonlinear Anal714883489210.1016/j.na.2009.03.065Open DOISearch in Google Scholar

[7]X. He, W. Zou. (2009), Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70(3):1407-1414. 10.1016/j.na.2008.02.021HeX.ZouW.2009Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal7031407141410.1016/j.na.2008.02.021Open DOISearch in Google Scholar

[8]T.F. Ma, J.E. Muñoz Rivera. (2003), Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16:243-248. 10.1016/S0893-9659(03)80038-1MaT.F.RiveraJ.E. Muñoz2003Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett1624324810.1016/S0893-9659(03)80038-1Open DOISearch in Google Scholar

[9]A. Mao, Z. Zhang. (2009), Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition, Nonlinear Anal. 70:1275-1287. 10.1016/j.na.2008.02.011MaoA.ZhangZ.2009Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition, Nonlinear Anal701275128710.1016/j.na.2008.02.011Open DOISearch in Google Scholar

[10]Z. Zhang, K. Perera. (2006), Sign changing solutions of Kirchhoff type problems via invarint sets of descent flow, J. Math. Anal. Appl. 317:456-463. 10.1016/j.jmaa.2005.06.102ZhangZ.PereraK.2006Sign changing solutions of Kirchhoff type problems via invarint sets of descent flow, J. Math. Anal. Appl31745646310.1016/j.jmaa.2005.06.102Open DOISearch in Google Scholar

[11]P. H. Rabinowitz. (1986), Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math. No. 65, Amer. Math. Soc., Providence, RI.RabinowitzP. H.1986Minimax methods in critical point theory with applications to differential equations, CBMS RegConf. Ser. in Math. No. 65, Amer. Math. SocProvidence, RI10.1090/cbms/065Search in Google Scholar

[12]A. Ambrosetti, P.H. Rabinowitz. (1973), Dual variational methods in critical point theory and applications, J. Funct. Anal. 14:349-381. 10.1016/0022-1236(73)90051-7AmbrosettiA.RabinowitzP.H.1973Dual variational methods in critical point theory and applications, J. Funct. Anal1434938110.1016/0022-1236(73)90051-7Open DOISearch in Google Scholar

[13]M. Willem. (1996), Minimax Theorems, progress in Nonlinear Differential Equations and Their Applications Volume 24, Brikhäauser.WillemM.1996Minimax Theorems, progress in Nonlinear Differential Equations and Their Applications Volume24Brikh ¨auserSearch in Google Scholar

[14]W. Zou. (2001), Variant fountain theorem and their applications, Manuscr. Math. 104:343-358. 10.1007/s002290170032ZouW.2001Variant fountain theorem and their applications, Manuscr. Math10434335810.1007/s002290170032Open DOISearch in Google Scholar

[15]X. He, W. Zou. (2012), Existence and concentration behavior of positive solutions for a Kirchhoff equation in R³, J. Differ. Equ. 252:1813-1834. 10.1016/j.jde.2011.08.035HeX.ZouW.2012Existence and concentration behavior of positive solutions for a Kirchhoff equation in R³, J. Differ. Equ2521813183410.1016/j.jde.2011.08.035Open DOISearch in Google Scholar

[16]X. Wu. (2011), Existence of nontrivial solutions and high energy solutions for Schröinger-Kirchhoff-type equations in R^N, Nonlinear Anal. RWA 12:1278-1287. 10.1016/j.nonrwa.2010.09.023WuX.2011Existence of nontrivial solutions and high energy solutions for Schröinger-Kirchhoff-type equations in R^N, Nonlinear Anal. RWA121278128710.1016/j.nonrwa.2010.09.023Open DOISearch in Google Scholar

[17]W. L., X. He. (2012), Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput. 39:473-487. 10.1007/s12190-012-0536-1HeW. L. X.2012Multiplicity of high energy solutions for superlinear Kirchhoff equationsJ. Appl. Math. Comput3947348710.1007/s12190-012-0536-1Open DOISearch in Google Scholar

[18]J. Jin, X. Wu. (2010), Infinitely many radial solutions for Kirchhoff-type problems in R^N, J. Math. Anal. Appl. 369:564-574. 10.1016/j.jmaa.2010.03.059JinJ.WuX.2010Infinitely many radial solutions for Kirchhoff-type problems in R^NJ. Math. Anal. Appl36956457410.1016/j.jmaa.2010.03.059Open DOISearch in Google Scholar

[19]A. Azzollini, P. d’Avenia, A. Pomponio. (2011), Multiple critical points for a class of nonlinear functionals, Annali di Matematica 190:507-523. 10.1007/s10231-010-0160-3AzzolliniA.d’AveniaP.PomponioA.2011Multiple critical points for a class of nonlinear functionalsAnnali di Matematica19050752310.1007/s10231-010-0160-3Open DOISearch in Google Scholar

[20]C.O. Alves, G.M. Figueiredo. (2012), Nonlinear perturbations of a periodic Kirchhoff equation in R^N, Nonlinear Anal. 75(5):2750-2759. 10.1016/j.na.2011.11.017AlvesC.O.FigueiredoG.M.2012Nonlinear perturbations of a periodic Kirchhoff equation in R^NNonlinear Anal7552750275910.1016/j.na.2011.11.017Open DOISearch in Google Scholar

[21]L. Jeanjean. (1999), On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on R^N, Proc. Roy. Soc. Edinburgh, 129A:787-809. 10.1017/S0308210500013147JeanjeanL.1999On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on R^NProc. Roy. Soc. Edinburgh129A78780910.1017/S0308210500013147Open DOISearch in Google Scholar

[22]Y. Li, F. Li, J. Shi. (2012), Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ. 253(7):2285-2294. 10.1016/j.jde.2012.05.017LiY.LiF.ShiJ.2012Existence of a positive solution to Kirchhoff type problems without compactness conditionsJ. Differ. Equ25372285229410.1016/j.jde.2012.05.017Open DOISearch in Google Scholar

[23]C. Ji. (2012), Infinitely many radial solutions for the p(x)-Kirchhoff-type equation with oscillatory nonlinearities in R^N, J. Math. Anal. Appl. 388:727-738. 10.1016/j.jmaa.2011.09.065JiC.2012Infinitely many radial solutions for the p(x)-Kirchhoff-type equation with oscillatory nonlinearities in R^NJ. Math. Anal. Appl38872773810.1016/j.jmaa.2011.09.065Open DOISearch in Google Scholar

[24]H.L. Elliott, L. Loss. (2001), Analysis, Published by AMS.ElliottH.L.LossL.2001AnalysisPublished by AMSSearch in Google Scholar

[25]J. Su, Z. Wang. (2007), M. Willem, Nonlinear Schröinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9:571-583. 10.1142/S021919970700254XSuJ.WangZ.2007M. Willem, Nonlinear Schröinger equations with unbounded and decaying radial potentials, CommunContemp. Math957158310.1142/S021919970700254XOpen DOISearch in Google Scholar

[26]J. Nie, X. Wu. (2012), Existence and multiplicity of non-trivial solutions for Schröinger-Kirchhoff-type equations with radial potential, Nonlinear Anal. 75:3470-3479. 10.1016/j.na.2012.01.004NieJ.WuX.2012Existence and multiplicity of non-trivial solutions for Schröinger-Kirchhoff-type equations with radial potentialNonlinear Anal753470347910.1016/j.na.2012.01.004Open DOISearch in Google Scholar

[27]W. Zou, M. Schechter. (2006), Critical Point Theory and its Application, Springer, New York.ZouW.SchechterM.2006Critical Point Theory and its ApplicationSpringerNew YorkSearch in Google Scholar

[28]H. Brezis and L. Nirenberg. (1991), Remarks on finding critical points, Comm. Pure Appl. Math. 44:939-963. 10.1002/cpa.3160440808BrezisH.NirenbergL.1991Remarks on finding critical points, CommPure Appl. Math4493996310.1002/cpa.3160440808Open DOISearch in Google Scholar