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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

Citations:

Zbl 0937.35050
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