Thursday, February 16, 2012

1202.3426 (Vitaly Moroz et al.)

Asymptotic properties of ground states of scalar field equations with a
vanishing parameter

Vitaly Moroz, Cyrill B. Muratov
We study the leading order behaviour of positive solutions of the equation
-\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\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
$pequation 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 behaviour
of the solutions is given by a particular solution of the critical
Emden--Fowler equation, whose choice depends on $\varepsilon$ in a nontrivial
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