The present paper is devoted to the investigation of the guided wave problem. This problem is formulated as the eigenvalue problem with a compact self-adjoint operator pencil. Applying the minimax principle for the compact operators in the Hilbert space we obtain a necessary and sufficient condition for the existence of a preassigned number of linearly independent guided modes. As a consequence of this result we also derive simple sufficient conditions, which can be easily applied in practice. We give a statement of the problem in a bounded domain and propose an efficient method for solving the problem.