Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior

Liang, Zhanping; Li, Fuyi; Shi, Junping

doi:10.1016/j.anihpc.2013.01.006

Liang, Zhanping ; Li, Fuyi ; Shi, Junping

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 155-167

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Résumé

Existence and bifurcation of positive solutions to a Kirchhoff type equation

{\begin{matrix} - (a + b \int_{Ω} {| \nabla u |}^{2}) Δ u = ν f (x, u), & in Ω, \\ u = 0, & on \partial Ω \end{matrix}

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.

MR Zbl

DOI : 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},
     pages = {155--167},
     publisher = {Elsevier},
     volume = {31},
     number = {1},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.01.006},
     mrnumber = {3165283},
     zbl = {1288.35456},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.01.006/}
}

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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, Tome 31 (2014) no. 1, pp. 155-167. doi: 10.1016/j.anihpc.2013.01.006

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