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

Liang, Zhanping; Li, Fuyi; Shi, Junping

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

Existence and bifurcation of positive solutions to a Kirchhoff type equation

$$\{\begin{array}{cc}-(a+b\underset{\Omega }{\int }{\left|\nabla u\right|}^2)\Delta u=\nu f(x,u),\hfill & \text{in}\Omega ,\hfill \\ u=0,\hfill & \text{on}\partial \Omega \hfill \end{array}$$

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.

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