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.