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A note on radiality of solutions of \(p\)-Laplacian equation. (English) Zbl 0841.35008

The authors study the problem of radiality of solutions of the following problem: \[ \text{div}(|\nabla u|^{p- 2} \nabla u)+ f(u)= 0\quad\text{in}\quad \Omega,\qquad u(x)> 0 \] in a domain \(\Omega\) which is a ball or \(\mathbb{R}^N\). For such an equation the classical radiality results of Gidas, Ni and Nirenberg do not apply, because the elliptic operator degenerates when \(\nabla u= 0\). The authors prove the following result: any positive solution of this equation, assuming suitable boundary data, is radial if it satisfies the additional condition \(\nabla u(x)\neq 0\) if \(x\neq 0\). Some results on solutions with a singularity are also proved.
Reviewer: M.Badiale (Padova)

MSC:

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