Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior
Fuyi Li, Junping Shi, Zhanping Liang · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2013
Existence and bifurcation of positive solutions to a Kirchhoff type equation \begin{cases} −\left(a + b\int \limits_{\Omega }|\mathrm{∇}u|^{2}\right)\mathrm{\Delta }u = u f(x,u), & \text{in }\Omega , \\ u = 0, & \text{on }\partial \Omega \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.