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ISSN 1088-6826(e) ISSN 0002-9939(p)

On the classification of solutions of $-\Delta u= e^u$ on $\mathbb{R}^N$ : Stability outside a compact set and applications

Author(s): E. N. Dancer; Alberto Farina
Journal: Proc. Amer. Math. Soc. 137 (2009), 1333-1338.
MSC (2000): Primary 35J60, 35B05, 35J25, 35B32
Posted: December 4, 2008
MathSciNet review: 2465656
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Abstract: In this short paper we prove that, for $3 \le N \le 9$ , the problem $-\Delta u = e^u$ on the entire Euclidean space $\mathbb{R}^N$ does not admit any solution stable outside a compact set of $\mathbb{R}^N$ . This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.

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