## A refined approach for non-negative entire solutions of $$\Delta u + u^p = 0$$ with subcritical Sobolev growth.(English)Zbl 1375.35024

Summary: Let $$N\geq2$$ and $$1<p<(N+2)/(N-2)_{+}$$. Consider the Lane-Emden equation $$\Delta u+u^{p}=0$$ in $$\mathbb{R}^{N}$$ and recall the classical Liouville type theorem: if $$u$$ is a non-negative classical solution of the Lane-Emden equation, then $$u\equiv0$$. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of J. Serrin and H. Zou [Differ. Integral Equ. 9, No. 4, 635–653 (1996; Zbl 0868.35032); Atti Semin. Mat. Fis. Univ. Modena 46, 369–380 (1998; Zbl 0917.35031)], originally used for the Lane-Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.

### MSC:

 35B08 Entire solutions to PDEs 35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs 35J60 Nonlinear elliptic equations

### Citations:

Zbl 0868.35032; Zbl 0917.35031
Full Text:

### References:

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