A new solution to $\Delta u+u^p-u=0$ on the entire space.

Abstract: In this talk we develop some techniques to construct solutions of certain semilinear elliptic equations which are periodic in some variables, decaying in others, and quasiperiodic in one variable. These solutions, which are found near ground states of a lower-dimensional problem, are constructed using spatial dynamics and results from the KAM theory. The use of a suitable KAM-type theorem provides hypotheses for homogeneous equations which rely on simple scaling arguments, applying in particular to $\Delta u+u^p-u=0$ with $p>1$ Sobolev subcritical.

Posted on Sep 2, 2020 in Differential Equations, Seminars