# 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 (2010)

- 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 (2010): 597-614. <http://eudml.org/doc/90636>.

@article{Jeanjean2010,

abstract = {
In this paper we establish the existence of a positive solution
for an asymptotically linear elliptic problem on $\xR^N$. 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 $\xR^N$; lack of compactness.; elliptic equations; asymptotically linear problems in ; lack of compactness},

language = {eng},

month = {3},

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/90636},

volume = {7},

year = {2010},

}

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

DA - 2010/3//

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 $\xR^N$. 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 $\xR^N$; lack of compactness.; elliptic equations; asymptotically linear problems in ; lack of compactness

UR - http://eudml.org/doc/90636

ER -

## References

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- P.H. Rabinowitz, On a class of nonlinear Shrödinger equations. ZAMP43 (1992) 270-291. Zbl0763.35087
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