an:05563881
Zbl 1171.35035
Souplet, Philippe
The proof of the Lane-Emden conjecture in four space dimensions
EN
Adv. Math. 221, No. 5, 1409-1427 (2009).
00250074
2009
j
35J45 35J60 35B33 35B45 35J50
Nonlinear elliptic systems; Nonexistence; Liouville type theorems; Lame-Emden conjecture
The author considers the following Lam??-Emden system:
\[
\begin{aligned} & -\Delta u = v^p,\\ & - \Delta v = u^q,\end{aligned}
\]
in \(\mathbb R^n\). The author proves that if \(n=3,4\) and \(\frac{1}{p} + \frac{1}{q} > 1 - \frac{2}{n}\), then the system above has no positive classical solutions. In the case \(n \geq 5\) the author obtain a new region of nonexistence. The proof is based on Rellich-Pohozaev type identities, on a comparison property between components via the maximum principle, on Sobolev and interpolation inequalities on \(S^{n-1}\) and on feedback and measure arguments.
Marco Biroli (Milano)