Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, p. 155-167

Existence and bifurcation of positive solutions to a Kirchhoff type equation $\left\{\begin{array}{cc}-\left(a+b\underset{\Omega }{\int }{|\nabla u|}^{2}\right)\Delta u=\nu f\left(x,u\right),\hfill & \text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\hfill \\ u=0,\hfill & \text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega \hfill \end{array}$ are considered by using topological degree argument and variational method. Here f is a continuous function which is asymptotically linear at zero and is asymptotically 3-linear at infinity. The new results fill in a gap of recent research about the Kirchhoff type equation in bounded domain, and in our results the nonlinearity may be resonant near zero or infinity.

DOI : https://doi.org/10.1016/j.anihpc.2013.01.006
Keywords: Kirchhoff type equation, Topological degree, Variational method, Monotone operator, Bifurcation
@article{AIHPC_2014__31_1_155_0,
author = {Liang, Zhanping and Li, Fuyi and Shi, Junping},
title = {Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {1},
year = {2014},
pages = {155-167},
doi = {10.1016/j.anihpc.2013.01.006},
zbl = {1288.35456},
mrnumber = {3165283},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_1_155_0}
}

Liang, Zhanping; Li, Fuyi; Shi, Junping. Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 155-167. doi : 10.1016/j.anihpc.2013.01.006. http://www.numdam.org/item/AIHPC_2014__31_1_155_0/

[1] C.O. Alves, F.J.S.A. Corrêa, T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 no. 1 (2005), 85-93 | MR 2123187 | Zbl 1130.35045

[2] Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 no. 4 (1976), 620-709 | MR 415432 | Zbl 0345.47044

[3] Ching-Yu Chen, Yueh-Cheng Kuo, Tsung-Fang Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 no. 4 (2011), 1876-1908 | MR 2763559 | Zbl 1214.35077

[4] Bitao Cheng, Xian Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal. 71 no. 10 (2009), 4883-4892 | MR 2548720 | Zbl 1175.35038

[5] Francesca Colasuonno, Patrizia Pucci, Multiplicity of solutions for $p\left(x\right)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 no. 17 (2011), 5962-5974 | MR 2833367 | Zbl 1232.35052

[6] P. D'Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 no. 2 (1992), 247-262 | MR 1161092 | Zbl 0785.35067

[7] Piero D'Ancona, Yoshihiro Shibata, On global solvability of nonlinear viscoelastic equations in the analytic category, Math. Methods Appl. Sci. 17 no. 6 (1994), 477-486 | MR 1274154 | Zbl 0803.35091

[8] Klaus Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985) | MR 787404 | Zbl 0559.47040

[9] B. Gidas, Wei Ming Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 no. 3 (1979), 209-243 | MR 544879 | Zbl 0425.35020

[10] Xiaoming He, Wenming Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 no. 3 (2009), 1407-1414 | MR 2474927 | Zbl 1157.35382

[11] Xiaoming He, Wenming Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser. 26 no. 3 (2010), 387-394 | MR 2657696 | Zbl 1196.35077

[12] Yisheng Huang, Positive solutions of quasilinear elliptic equations, Topol. Methods Nonlinear Anal. 12 no. 1 (1998), 91-107 | MR 1677739 | Zbl 0929.35039

[13] Yisheng Huang, Huan-Song Zhou, Positive solution for $-{\Delta }_{p}u=f\left(x,u\right)$ with $f\left(x,u\right)$ growing as ${u}^{p-1}$ at infinity, Appl. Math. Lett. 17 no. 8 (2004), 881-887 | MR 2082506 | Zbl 1122.35333

[14] Louis Jeanjean, Local conditions insuring bifurcation from the continuous spectrum, Math. Z. 232 no. 4 (1999), 651-664 | MR 1727546 | Zbl 0934.35047

[15] T.F. Ma, J.E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 no. 2 (2003), 243-248 | MR 1962322 | Zbl 1135.35330

[16] Anmin Mao, Zhitao Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. 70 no. 3 (2009), 1275-1287 | MR 2474918 | Zbl 1160.35421

[17] Kanishka Perera, Zhitao Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 no. 1 (2006), 246-255 | MR 2193850 | Zbl 1357.35131 | Zbl 05013580

[18] Paul H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 65, Amer. Math. Soc., Washington, DC (1986) | MR 845785 | Zbl 0609.58002

[19] Michael Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin (1990) | MR 1078018 | Zbl 0746.49010

[20] Ji-Jiang Sun, Chun-Lei Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal. 74 no. 4 (2011), 1212-1222 | MR 2746801 | Zbl 1209.35033

[21] Yang Yang, Jihui Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett. 23 no. 4 (2010), 377-380 | MR 2594846 | Zbl 1188.35084

[22] Eberhard Zeidler, Nonlinear functional analysis and its applications, II/B, Nonlinear Monotone Operators, Springer-Verlag, New York (1990) | MR 1033497 | Zbl 0684.47029

[23] Zhitao Zhang, Kanishka Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 no. 2 (2006), 456-463 | MR 2208932 | Zbl 1100.35008