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

Moroz, Vitaly, and Muratov, Cyrill B.. "Asymptotic properties of ground states of scalar field equations with a vanishing parameter." Journal of the European Mathematical Society 016.5 (2014): 1081-1109. <http://eudml.org/doc/277599>.

@article{Moroz2014,
abstract = {We study the leading order behaviour of positive solutions of the equation $-\Delta u +\epsilon u-|u|^\{p-2\}u+|u|^\{q-2\}u=0,\qquad x\in \mathbb \{R\}^N$, where $N\ge 3$, $q>p>2$ and when $\epsilon >0$ 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 $2^\ast =\frac\{2N\}\{N-2\}$. For $p<2^\ast $ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>2^\ast $ the solution asymptotically coincides with the solution of the equation with $\epsilon =0$. In the most delicate case $p=2^\ast $ the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on $\epsilon $ in a nontrivial way.},
author = {Moroz, Vitaly, Muratov, Cyrill B.},
journal = {Journal of the European Mathematical Society},
keywords = {critical Sobolev exponent; subcritical; critical and supercritical non-linearity; Pohozaev identity; asymptotic behaviour; critical Sobolev exponent; subcritical; critical and supercritical non-linearity; Pohožaev identity; asymptotic behaviour},
language = {eng},
number = {5},
pages = {1081-1109},
publisher = {European Mathematical Society Publishing House},
title = {Asymptotic properties of ground states of scalar field equations with a vanishing parameter},
url = {http://eudml.org/doc/277599},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Moroz, Vitaly
AU - Muratov, Cyrill B.
TI - Asymptotic properties of ground states of scalar field equations with a vanishing parameter
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 5
SP - 1081
EP - 1109
AB - We study the leading order behaviour of positive solutions of the equation $-\Delta u +\epsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in \mathbb {R}^N$, where $N\ge 3$, $q>p>2$ and when $\epsilon >0$ 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 $2^\ast =\frac{2N}{N-2}$. For $p<2^\ast $ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>2^\ast $ the solution asymptotically coincides with the solution of the equation with $\epsilon =0$. In the most delicate case $p=2^\ast $ the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on $\epsilon $ in a nontrivial way.
LA - eng
KW - critical Sobolev exponent; subcritical; critical and supercritical non-linearity; Pohozaev identity; asymptotic behaviour; critical Sobolev exponent; subcritical; critical and supercritical non-linearity; Pohožaev identity; asymptotic behaviour
UR - http://eudml.org/doc/277599
ER -