In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on . The main difficulties to overcome are the lack of a priori bounds for Palais-Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order to regain compactness.
Mots clés : elliptic equations, asymptotically linear problems in $\mathbb {R}^N$, lack of compactness
@article{COCV_2002__7__597_0,
author = {Jeanjean, Louis and Tanaka, Kazunaga},
title = {A positive solution for an asymptotically linear elliptic problem on $\mathbb {R}^N$ autonomous at infinity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {597--614},
publisher = {EDP-Sciences},
volume = {7},
year = {2002},
doi = {10.1051/cocv:2002068},
mrnumber = {1925042},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2002068/}
}
TY - JOUR
AU - Jeanjean, Louis
AU - Tanaka, Kazunaga
TI - A positive solution for an asymptotically linear elliptic problem on $\mathbb {R}^N$ autonomous at infinity
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
SP - 597
EP - 614
VL - 7
PB - EDP-Sciences
UR - http://www.numdam.org/articles/10.1051/cocv:2002068/
DO - 10.1051/cocv:2002068
LA - en
ID - COCV_2002__7__597_0
ER -
%0 Journal Article
%A Jeanjean, Louis
%A Tanaka, Kazunaga
%T A positive solution for an asymptotically linear elliptic problem on $\mathbb {R}^N$ autonomous at infinity
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 597-614
%V 7
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2002068/
%R 10.1051/cocv:2002068
%G en
%F COCV_2002__7__597_0
Jeanjean, Louis; Tanaka, Kazunaga. A positive solution for an asymptotically linear elliptic problem on $\mathbb {R}^N$ autonomous at infinity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 597-614. doi : 10.1051/cocv:2002068. http://www.numdam.org/articles/10.1051/cocv:2002068/
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