### Abstract

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on 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.

Original language | English |
---|---|

Pages (from-to) | 597-614 |

Number of pages | 18 |

Journal | ESAIM Control, Optimisation and Calculus of Variations |

Issue number | 7 |

Publication status | Published - 2002 |

### Fingerprint

### Keywords

- Asymptotically linear problems in ℝ
- Elliptic equations
- Lack of compactness

### ASJC Scopus subject areas

- Control and Systems Engineering

### Cite this

**A positive solution for an asymptotically linear elliptic problem on ℝ ^{N} autonomous at infinity.** / Jeanjean, Louis; Tanaka, Kazunaga.

Research output: Contribution to journal › Article

^{N}autonomous at infinity',

*ESAIM Control, Optimisation and Calculus of Variations*, no. 7, pp. 597-614.

}

TY - JOUR

T1 - A positive solution for an asymptotically linear elliptic problem on ℝN autonomous at infinity

AU - Jeanjean, Louis

AU - Tanaka, Kazunaga

PY - 2002

Y1 - 2002

N2 - In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on ℝN. 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.

AB - In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on ℝN. 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.

KW - Asymptotically linear problems in ℝ

KW - Elliptic equations

KW - Lack of compactness

UR - http://www.scopus.com/inward/record.url?scp=0036414880&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036414880&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036414880

SP - 597

EP - 614

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

IS - 7

ER -