Stable and Unstable Manifolds for Non-Linear Elliptic Equations with Parameter

The paper is devoted to investigating the Cauchy problem for non-linear elliptic equations with parameter. It is proved that there exist two analytic manifolds M⁺ and M^- embedded into the phase space E of the equation in question such that the Cauchy problem with initial data on M^± has a unique solution belonging to the Sobolev space. Moreover, these manifolds intersect only at the origin, and the direct sum of the tangent spaces to M⁺ and M^- at this point coincides with E.