# 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

## Access Full Article

top## Abstract

top## How to cite

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

top- A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal.14 (1973) 349-381.
- H. Berestycki and P.L. Lions, Nonlinear scalar field equations I. Arch. Rational Mech. Anal.82 (1983) 313-346.
- H. Berestycki, T. Gallouët and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math.297 (1983) 307-310.
- H. Brezis, Analyse fonctionnelle. Masson (1983).
- V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${}^{N}$. Comm. Pure Appl. Math.XIV (1992) 1217-1269.
- I. Ekeland, Convexity methods in Hamiltonian Mechanics. Springer (1990).
- L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${}^{N}$. Proc. Roy. Soc. Edinburgh Sect. A129 (1999) 787-809.
- P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Parts I and II. Ann. Inst. H. Poincaré Anal. Non Linéaire1 (1984) 109-145 and 223-283.
- P.H. Rabinowitz, On a class of nonlinear Shrödinger equations. ZAMP43 (1992) 270-291.
- C.A. Stuart, Bifurcation in ${L}^{p}{(}^{N})$ for a semilinear elliptic equation. Proc. London Math. Soc.57 (1988) 511-541.
- C.A. Stuart and H.S. Zhou, A variational problem related to self-trapping of an electromagnetic field. Math. Meth. Appl. Sci.19 (1996) 1397-1407.
- C.A. Stuart and H.S. Zhou, Applying the mountain-pass theorem to an asymtotically linear elliptic equation on ${}^{N}$. Comm. Partial Differential Equations24 (1999) 1731-1758.
- A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal.187 (2001) 25-41.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.