Liang, Zhanping; Li, Fuyi; Shi, Junping Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. (English) Zbl 1288.35456 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 1, 155-167 (2014). Summary: Existence and bifurcation of positive solutions to a Kirchhoff type equation \[ \begin{cases} -\left(a+b\int\limits_{\varOmega}|\nabla u|^2\right)\Delta u=\nu f(x,u),\quad &\text{in }\varOmega ,\\u=0,\quad &\text{on }\partial\varOmega\end{cases} \] 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. Cited in 89 Documents MSC: 35Q74 PDEs in connection with mechanics of deformable solids 74K05 Strings 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35A15 Variational methods applied to PDEs 74Q10 Homogenization and oscillations in dynamical problems of solid mechanics Keywords:Kirchhoff type equation; topological degree; variational method; monotone operator; bifurcation PDFBibTeX XMLCite \textit{Z. Liang} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 1, 155--167 (2014; Zbl 1288.35456) Full Text: DOI References: [1] Alves, C. O.; Corrêa, F. J.S. A.; Ma, T. F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. 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