# A positive solution for an asymptotically linear elliptic problem on ${\mathbb{R}}^{N}$ autonomous at infinity

Louis Jeanjean; Kazunaga Tanaka

ESAIM: Control, Optimisation and Calculus of Variations (2002)

- Volume: 7, page 597-614
- ISSN: 1292-8119

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topJeanjean, Louis, and 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 7 (2002): 597-614. <http://eudml.org/doc/244632>.

@article{Jeanjean2002,

abstract = {In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on $\mathbb \{R\}^N$. 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.},

author = {Jeanjean, Louis, Tanaka, Kazunaga},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {elliptic equations; asymptotically linear problems in $\mathbb \{R\}^N$; lack of compactness; asymptotically linear problems in },

language = {eng},

pages = {597-614},

publisher = {EDP-Sciences},

title = {A positive solution for an asymptotically linear elliptic problem on $\mathbb \{R\}^N$ autonomous at infinity},

url = {http://eudml.org/doc/244632},

volume = {7},

year = {2002},

}

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

PB - EDP-Sciences

VL - 7

SP - 597

EP - 614

AB - In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on $\mathbb {R}^N$. 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.

LA - eng

KW - elliptic equations; asymptotically linear problems in $\mathbb {R}^N$; lack of compactness; asymptotically linear problems in

UR - http://eudml.org/doc/244632

ER -

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- [12] C.A. Stuart and H.S. Zhou, Applying the mountain-pass theorem to an asymtotically linear elliptic equation on ${\mathbb{R}}^{N}$. Comm. Partial Differential Equations 24 (1999) 1731-1758. Zbl0935.35043MR1708107
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