Abstract: By using the local linking theorem in critical point theory and the theory of variable exponent Sobolev space, the author proved the existence of nontrivial solutions for Kirchhoff-type equation with p(x)-Laplacian operator without  assuming  the  (AR) type superlinearity condition held.

Key words: Kirchhoff-type equation, p(x)-Laplacian operator,  , local linking, nontrivial solution, critical point