Analysis Seminar

Asymptotic Properties of Ground States of Scalar Field Equations with Vanishing Parameter

Speaker: Cyrill Muratov

Location: Warren Weaver Hall 1302

Date: Thursday, April 5, 2012, 11 a.m.

Synopsis:

We study the leading order behavior of positive solutions of the equation $$-\Delta u + \varepsilon u - |u|^{p-2}u + |u|^{q-2}u=0, \qquad x \in \mathbb{R}^N$$ where $N \ge 3$, $q > p > 2$ and when $ \varepsilon > 0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $p^\ast=\frac{2N}{N-2}$. For $p < p^\ast$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p > p^\ast$ the solution asymptotically coincides with the solution of the equation with $\varepsilon = 0$. In the most delicate case $p = p^\ast$ the asymptotic behavior of the solutions is given by a particular solution of the critical Emden-Fowler equation, whose choice depends on $\varepsilon$ in a nontrivial way.