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, for3≤N≤93 \le N \le 9, the problem−Δu=eu-\Delta u = e^uon the entire Euclidean spaceRN\mathbb {R}^Ndoes not admit any solution stable outside a compact set ofRN\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.