Abstract: We discussed the existence of radial symmetric solutions of a Kirchhoff equation of  nonlinearity with gradient term on an annulus in R^N(N≥2) by using the Leray-Schauder fixed point theorem. Under the nonlinearity satisfied certain  conditions which allowed the nonlinearity might be superlinear growth of any order on unknown function term, and quadratic growth on the gradient term of unknown function, the existence results of radial symmetric solutions were obtained.

Key words: Kirchhoff type elliptic equation, radial symmetric solution, existence, Leray-Schauder fixed point theorem