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Using variational methods, we study the non-existence and multi- plicity of non-negative solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions of the form div(jruj p(x) 2 ru) +juj p(x) 2 u = 0 in , jruj p(x) 2 ¶u ¶n = g(x, u) on ¶ , where is a bounded domain with smooth boundary, n the outer unit normal to ¶ and a parameter. Besides, we want to emphasize that g : ¶ (0,1)! R is just a continuous function which may satisfy the Ambrosetti-Rabinowitz type condition or not. The operator div(jruj p(x) 2 ru) is called the p(x)-Laplace operator and it is a nat- ural generalization of the p-Laplace operator in which p(x) = p > 1 is a constant. For this reason the equations studied in the case in which the p(x)-Laplace opera- tor is involved are, in general, extensions of p-Laplacian problems. However, we point out that such generalizations are not trivial since the p(x)-Laplace operator possesses more complicated nonlinearity; for example it is inhomogeneous.