## Symmetry of $$C^1$$ solutions of $$p$$-Laplace equations in $$\mathbb{R}^N$$.(English)Zbl 0998.35016

Summary: We consider positive $$C^1$$ solutions of the equation $$-\text{div} (|Du|^{p-2} Du)=f(u)$$ in $$\mathbb{R}^N$$, vanishing at infinity, in the case $$1<p\leq 2$$, $$f$$ locally Lipschitz continuous in $$(0,\infty)$$. We prove that the solutions are radially symmetric under two different sets of assumptions on the behaviour of $$f$$ near zero. In the first case we assume that $$f$$ is nonincreasing in $$(0,s_0)$$, $$s_0>0$$, and improve a previous result of the authors and F. Pacella [Symmetry of ground states of $$p$$-Laplace equations via the moving plane method, Arch. Rat. Mech. Anal. 148, 291-308 (1999; Zbl 0937.35050)] where the symmetry was proved under the hypothesis $$u\in C^1(\mathbb{R}^N)\cap W^{1,p}(\mathbb{R}^N)$$. In the second case, previously studied when $$p=2$$, we assume that $$f(u)= O(u^{\alpha+1})$$ $$(u\to 0)$$, and prove the radial symmetry of the solution $$u$$, provided $$u=O({1\over |x|^m})$$, $$Du(x)= O({1\over|x|^{m+1}})$$ at infinity with $$m(\alpha +2-p)>p$$. These results extend to $$p$$-Laplace equations, $$1<p<2$$, analogous symmetry results previously known in the case of strictly elliptic equations. The proofs exploit some Poincaré and Hardy-Sobolev type inequalities.

### MSC:

 35J60 Nonlinear elliptic equations 35J70 Degenerate elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Zbl 0937.35050
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