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

Dancer, E.; Farina, Alberto

doi:10.1090/S0002-9939-08-09772-4

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

Proc. Amer. Math. Soc. 137 (2009), 1333-1338

DOI: https://doi.org/10.1090/S0002-9939-08-09772-4

Published electronically: December 4, 2008

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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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