Existence Of Solutions For Some Kirchhoff Problems Without Ambrosetti-Rabinowitz Condition

In this paper,we discuss the existence of solutions for some Kirchhoff type problems without Ambrosetti-Rabinowitz condition.Firstly,We study the following Kirchhoff problem where ?(?)R2 is a smooth bounded domain,m:R+?R+ is a Kirchhoff function,f:? ×R?R is a continuous function with critical exponential growth.By using the critical point theory,Trudinger-Moser inequality and Mountain Pass Theorem introduced by Cerami,we prove the existence of positive ground state solutions for the Kirchhoff problem with critical exponential growth when f does not satisfies Ambrosetti-Rabinowitz condition.Then,we study the following polyharmonic Kirchhoff type problem with singular exponential nonlinearity where ?(?)Rn is a bounded domain with smooth boundary,the symbol ?mu,where m is a positive integer,denotes the mth order gradient of u,the symbol(?)u denotes the polyharmonic operator,M:R+?R+is a Kirchhoff function,f(x,u):? × R?R is a continuous function with critical exponential growth without Ambrosetti-Rabinowitz condition.We obtain a ground state solution of the problem by using Adam-Moser inequality with singularity and Mountain Pass Theorem introduced by Cerami.Finally,we study the following Kirchhoff type elliptic systems without the Ambrosetti-Rabinowitz condition where ? is a bounded domain in R2 with smooth boundary,m is a continuous Kirchhoff function,Hu,Hv is the partial derivative of continuous function H:? ×R2?R and satisfy the critical exponential growth.We prove the existence of a positive ground state solution of the system by suitable Trudinger-Moser inequality and Mountain Pass Theorem.