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Ground States for Irregular and Indefinite Superlinear Schroedinger Equations

Banff International Research Station for Mathematical Innovation and Discovery

We consider the existence of a ground state for the subcritical stationary semilinear Schrodinger equation $-\Delta u + u=a(x)|{u}|^{p-2}u$ in $H^1$, where $a\in L^\infty(\mathbb{R}^N)$ may change sign. Our focus is on the case where loss of compactness occurs at the ground state energy. By providing a new variant of the Splitting Lemma we do not need to assume the existence of a limit problem at infinity, be it in the form of a pointwise limit for $a$ as $|{x}|\to\infty$ or of asymptotic periodicity. That is, our problem may be \emph{irregular} at infinity. In addition, we allow $a$ to change sign near infinity, a case that has never been treated before.

Mathematics; Partial differential equations; Dynamical systems and ergodic theory

video/mp4

Author affiliation: UNAM

2017-03-28

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61017

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/