On a \(p\)-Kirchhoff problem involving a critical nonlinearity. (English. French summary) Zbl 1298.35096

Author’s abstract: This paper deals with a \(p\)-Kirchhoff type problem involving the critical Sobolev exponent. Under some suitable assumptions, we show the existence of at least one solution.


35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B33 Critical exponents in context of PDEs
Full Text: DOI


[1] Allaoui, M.; El Amrouss, A.; Ourraoui, A., On a class of nonlocal \(p(x)\)-Laplacian Neumann problems, Thai J. Math., (2014), (15 p.), in press · Zbl 1388.35041
[2] Alves, C. O.; Corrêa, F. J.S. A.; Figueiredo, G. M., On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 2, 409-417, (2010) · Zbl 1198.35281
[3] Autuori, G.; Colasuonno, F.; Pucci, P., Lifespan estimates for solutions of polyharmonic Kirchhoff systems, M3AS: Math. Models Methods Appl. Sci., 22, 1150009, (2012), (36 p.) · Zbl 1320.35092
[4] Autuori, G.; Colasuonno, F.; Pucci, P., On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math., (2014), (33 p.), in press · Zbl 1325.35129
[5] Autuori, G.; Pucci, P., Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73, 1952-1965, (2010) · Zbl 1197.35173
[6] Autuori, G.; Pucci, P.; Salvatori, M. C., Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196, 489-516, (2010) · Zbl 1201.35138
[7] Chipot, M.; Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30, 7, 4619-4627, (1997) · Zbl 0894.35119
[8] Colasuonno, F.; Pucci, P., Multiplicity of solutions for \(p(x)\)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74, 5962-5974, (2011) · Zbl 1232.35052
[9] Dreher, M., The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. (Torino), 64, 217-238, (2006) · Zbl 1178.35006
[10] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353, (1974) · Zbl 0286.49015
[11] El Hamidi, A.; Rakotoson, J. M., Compactness and quasilinear problems with critical exponents, Differ. Integral Equ., 18, 1201-1220, (2005) · Zbl 1212.35113
[12] A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a non-local operator, preprint. · Zbl 1283.35156
[13] Hamydy, A.; Massar, M.; Tsouli, N., Existence of solutions for a p-Kirchhoff type problems with critical exponent, Electron. J. Differ. Equ., 2011, 1-8, (2011) · Zbl 1254.35005
[14] Kirchhoff, G., Mechanik, (1883), Teubner Leipzig · JFM 08.0542.01
[15] Lions, J.-L., On some questions in boundary value problems of mathematical physics, (De La Penha, G. M.; Medeiros, L. A.J., Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, 1977, North-Holland Mathematics Studies, vol. 30, (1978)), 284-346
[16] Ma, T. F., Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63, 1967-1977, (2005)
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