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

Vitaly Moroz; Cyrill B. Muratov

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 5, page 1081-1109
- ISSN: 1435-9855

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topMoroz, 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 -

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