Abstract:
It is well known that, under quite general assumptions, fully localized solutions of a homogeneous semilinear elliptic equation must be radially symmetric. Many authors have exhibited the complexity of solutions which are only partially localized (that is, only decaying in some variables).
In this talk I consider solutions which are quasiperiodic in one variable, decaying in all the others. In the first part of the talk I develop a framework for finding such solutions in the nonhomogeneous case; in the second part I show the existence of nonlinearities for which these solutions exist. The latter portion of the talk will also exhibit the difficulties when attempting to obtain the existence of quasiperiodic solutions for nonlinearities such as $f(u)=u^p-u$.