On the classification of solutions of \(-\Delta u= e^u\) on \(\mathbb {R}^N\): Stability outside a compact set and applications. (English) Zbl 1162.35027
Summary: We prove that, for \( 3 \leq N \leq 9\), 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.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35B32 | Bifurcations in context of PDEs |
| 35B35 | Stability in context of PDEs |
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