Existence and multiplicity of solutions for $p$-Kirchhoff-type Neumann problems


  • Qin Jiang Department of Mathematics, Huanggang Normal University, Hubei 438000, China
  • Sheng Ma Department of Mathematics, Huanggang Normal University, Hubei 438000, China
  • Daniel Paşca Department of Mathematics and Informatics, University of Oradea, University Street 1, 410087 Oradea, Romania




We establish, based on variational methods, existence theorems for a $p$-Kirchhoff-type Neumann problem under the Landesman–Lazer type condition and under the local coercive condition. In addition, multiple solutions for a $p$-Kirchhoff-type Neumann problem are established using a known three-critical-point theorem proposed by H. Brezis and L. Nirenberg.


Download data is not yet available.


S. Aizicovici, N. S. Papageorgiou, and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Ann. Mat. Pura Appl. (4) 188 no. 4 (2009), 679–719.  DOI  MR  Zbl

S. Aizicovici, N. S. Papageorgiou, and V. Staicu, On a $p$-superlinear Neumann $p$-Laplacian equation, Topol. Methods Nonlinear Anal. 34 no. 1 (2009), 111–130.  DOI  MR  Zbl

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 no. 3 (2010), 409–417.  DOI  MR  Zbl

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 no. 1 (1996), 305–330.  DOI  MR  Zbl

P. A. Binding, P. Drábek, and Y. X. Huang, Existence of multiple solutions of critical quasilinear elliptic Neumann problems, Nonlinear Anal. 42 no. 4, Ser. A: Theory Methods (2000), 613–629.  DOI  MR  Zbl

G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the $p$-Laplacian, Arch. Math. (Basel) 80 no. 4 (2003), 424–429.  DOI  MR  Zbl

G. Bonanno and A. Sciammetta, Existence and multiplicity results to Neumann problems for elliptic equations involving the $p$-Laplacian, J. Math. Anal. Appl. 390 no. 1 (2012), 59–67.  DOI  MR  Zbl

H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 no. 8-9 (1991), 939–963.  DOI  MR  Zbl

F. Cammaroto, A. Chinnì, and B. Di Bella, Some multiplicity results for quasilinear Neumann problems, Arch. Math. (Basel) 86 no. 2 (2006), 154–162.  DOI  MR  Zbl

M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 no. 6 (2001), 701–730.  MR  Zbl

C.-y. Chen, Y.-c. Kuo, and T.-f. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 no. 4 (2011), 1876–1908.  DOI  MR  Zbl

F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Austral. Math. Soc. 74 no. 2 (2006), 263–277.  DOI  MR  Zbl

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 no. 2 (1992), 247–262.  DOI  MR  Zbl

M. Filippakis, L. Gasiński, and N. S. Papageorgiou, Multiplicity results for nonlinear Neumann problems, Canad. J. Math. 58 no. 1 (2006), 64–92.  DOI  MR  Zbl

T. He, C. Chen, Y. Huang, and C. Hou, Infinitely many sign-changing solutions for $p$-Laplacian Neumann problems with indefinite weight, Appl. Math. Lett. 39 (2015), 73–79.  DOI  MR  Zbl

X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 no. 3 (2009), 1407–1414.  DOI  MR  Zbl

E. M. Hssini, M. Massar, and N. Tsouli, Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problems, Bol. Soc. Parana. Mat. (3) 33 no. 2 (2015), 203–217.  DOI  MR  Zbl

R. Iannacci and M. N. Nkashama, Nonlinear two-point boundary value problems at resonance without Landesman-Lazer condition, Proc. Amer. Math. Soc. 106 no. 4 (1989), 943–952.  DOI  MR  Zbl

Q. Jiang and S. Ma, Existence and multiplicity of solutions for semilinear elliptic equations with Neumann boundary conditions, Electron. J. Differential Equations (2015), Paper No. 200, 8 pp.  MR  Zbl Available at https://www.emis.de/journals/EJDE/2015/200/abstr.html.

Q. Jiang, S. Ma, and D. Paşca, Existence and multiplicity of solutions for $p$-Laplacian Neumann problems, Results Math. 74 no. 1 (2019), Paper No. 67, 11 pp.  DOI  MR  Zbl

G. Kirchhoff, Vorlesungen über mathematische Physik. Mechanik, Teubner, Leipzig, 1876.  Zbl Available at https://archive.org/details/vorlesungenberm02kircgoog.

J.-L. Lions, 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, Rio de Janeiro, 1977), North-Holland Math. Stud. 30, North-Holland, Amsterdam-New York, 1978, pp. 284–346.  MR  Zbl

D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Equations 232 no. 1 (2007), 1–35.  DOI  MR  Zbl

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations 257 no. 4 (2014), 1168–1193.  DOI  MR  Zbl

N. S. Papageorgiou and V. D. Rădulescu, Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc. 367 no. 12 (2015), 8723–8756.  DOI  MR  Zbl

N. S. Papageorgiou and V. D. Rădulescu, Neumann problems with indefinite and unbounded potential and concave terms, Proc. Amer. Math. Soc. 143 no. 11 (2015), 4803–4816.  DOI  MR  Zbl

K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 no. 1 (2006), 246–255.  DOI  MR  Zbl

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986.  DOI  MR  Zbl

B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the $p$-Laplacian, Bull. London Math. Soc. 33 no. 3 (2001), 331–340.  DOI  MR  Zbl

C.-L. Tang, Solvability of Neumann problem for elliptic equations at resonance, Nonlinear Anal., Ser. A: Theory Methods 44 no. 3 (2001), 323–335.  DOI  MR  Zbl

C.-L. Tang, Some existence theorems for the sublinear Neumann boundary value problem, Nonlinear Anal., Ser. A: Theory Methods 48 no. 7 (2002), 1003–1011.  DOI  MR  Zbl

C.-L. Tang and X.-P. Wu, Existence and multiplicity for solutions of Neumann problem for semilinear elliptic equations, J. Math. Anal. Appl. 288 no. 2 (2003), 660–670.  DOI  MR  Zbl

X. Wu and K.-K. Tan, On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations, Nonlinear Anal. 65 no. 7 (2006), 1334–1347.  DOI  MR  Zbl

Q.-L. Xie, X.-P. Wu, and C.-L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal. 12 no. 6 (2013), 2773–2786.  DOI  MR  Zbl

K. Yosida, Functional analysis, sixth ed., Grundlehren der Mathematischen Wissenschaften 123, Springer-Verlag, Berlin-New York, 1980.  DOI  MR  Zbl

Z. Yucedag, M. Avci, and R. Mashiyev, On an elliptic system of $p(x)$-Kirchhoff-type under Neumann boundary condition, Math. Model. Anal. 17 no. 2 (2012), 161–170.  DOI  MR  Zbl

J. Zhang, The critical Neumann problem of Kirchhoff type, Appl. Math. Comput. 274 (2016), 519–530.  DOI  MR  Zbl