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

In this short paper we prove that, for 3 ≤ N ≤ 9 3 \le N \le 9 , the problem − Δ u = e u -\Delta u = e^u on the entire Euclidean space R N \mathbb {R}^N does not admit any solution stable outside a compact set of R N \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.