Cryptographic Boolean functions play an important role in the design of symmetric ciphers. Many cryptographic criteria such as balancedness, nonlinearity, correlation immunity and transparency order are connected with the Walsh support of a Boolean function. However, we still know little about the possible structure of the Walsh supports of Boolean functions. In 2005, Carlet and Mesnager studied the Walsh supports of Boolean functions and constructed a class of n-variable Boolean functions whose Walsh support is F2n∖{0}, for n≥10. For n≤6, it can be verified using the computer that there is no Boolean function with the Walsh Support F2n∖{0}. However, concerning the values of n=7,8,9, it has been an open problem for many years. In this paper, we construct two classes of balanced Boolean functions with the maximum possible Walsh support F2n∖{0}, and partially solve this problem. The first class of functions are of odd variables with n≥9, and the second class of functions are constructed based on the Maiorana–McFarland bent functions, which are of even variables with n≥8. As a result, the above open problem has been settled for n=8,9, and the only unsolved case is n=7.