Small divisor problems and divergent formal power series solutions (Q1298037)

We study small divisor phenomenon appearing in a semilinear Goursat problem without assuming any diophantine conditions. We can easily see that there appear divergent formal power series solutions in the class of analytic functions, and that the problem is not well posed in the class of smooth functions. We will show that for any divergent solution there exists a smooth solution whose Taylor expansion (at the origin) is equal to a divergent one under nonresonant condition. Because we assume no diophantine condition the generalized implicit function theorem cannot be applied. Instead of this, we reduce the problem to an elliptic boundary value problem in a corner domain. We can also show the alternatives for non well-posed problems.