We consider a class of uniformly elliptic second order differential operators and also its maximizing operator. In this paper, we obtain a variational formula for the principal eigenvalue associated with nonlinear semigroup, defined by M.Nisio ([11]), whose infinitesimal generator corresponds to the maximizing operator. Our result is an extension of [1] and [2], in which they considered the problems relative to linear operators. Moreover, as applications, we shall discuss large deviation, rate function and other properties relative to the maximizing operator. Our proofs are almost relied upon stochastic control method which developped by N.V. Krylov [7], [8], W.Fleming [5], P.L.lions [9] and others.