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The proof of the Lane-Emden conjecture in four space dimensions. (English) Zbl 1171.35035
The author considers the following Lamé-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.

MSC:
35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35B45 A priori estimates in context of PDEs
35J50 Variational methods for elliptic systems
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