A positive solution for an asymptotically linear elliptic problem on 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

Abstract

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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.

How to cite

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Jeanjean, 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 -

References

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  8. [8] 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 Linéaire 1 (1984) 109-145 and 223-283. Zbl0704.49004
  9. [9] P.H. Rabinowitz, On a class of nonlinear Shrödinger equations. ZAMP 43 (1992) 270-291. Zbl0763.35087MR1162728
  10. [10] C.A. Stuart, Bifurcation in L p ( ℝ N ) for a semilinear elliptic equation. Proc. London Math. Soc. 57 (1988) 511-541. Zbl0673.35005MR960098
  11. [11] 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. Zbl0862.35123MR1414401
  12. [12] C.A. Stuart and H.S. Zhou, Applying the mountain-pass theorem to an asymtotically linear elliptic equation on ℝ N . Comm. Partial Differential Equations 24 (1999) 1731-1758. Zbl0935.35043MR1708107
  13. [13] A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187 (2001) 25-41. Zbl0984.37072MR1867339

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