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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. [ABSTRACT FROM AUTHOR]