In this paper, using the method of boundary layer, a nonlinear singularly perturbed problem with multiple solutions is studied. Under the appropriate assumptions, the asymptotic solutions of the problem with different forms are obtained according to the multiple number of the root of some equation, which is satisfied by the boundary value of the reduced problem, by giving the general expressions for the coefficients of outer solution expansion and the corresponding boundary conditions. In particular, as the multiple number of the root is even, the problem has two solutions. In addition, the relative result is applied into the theory of chemical reactors. And it is illustrated that the asymptotic solutions so constructed possess higher precision by simulating the asymptotic solutions and numerical solutions for an example with multiple solutions.

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