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asymptotically linear problems in $\xr^n$

ESAIM: Control, Optimisation and Calculus of Variations (1)

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A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N$ autonomous at infinity
Louis Jeanjean, Kazunaga Tanaka
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 7 / 2002

Published online by Cambridge University Press:

15 September 2002, pp. 597-614

Print publication:

2002
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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.