Differential and Integral Equations

Symmetry and nonuniformly elliptic operators

Jean Dolbeault, Patricio Felmer, and Régis Monneau

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The goal of this paper is to study the symmetry properties of nonnegative solutions of elliptic equations involving a nonuniformly elliptic operator. We consider on a ball the solutions of \[ \Delta_pu+f(u)=0 \] with zero Dirichlet boundary conditions, for $p>2$, where $\Delta_p$ is the $p$-Laplace operator and $f$ a continuous nonlinearity. The main tools are a comparison result for weak solutions and a local moving-plane method which has been previously used in the $p=2$ case. We prove local and global symmetry results when $u$ is of class $C^{1,\gamma}$ for $\gamma$ large enough, under some additional technical assumptions.

Article information

Differential Integral Equations, Volume 18, Number 2 (2005), 141-154.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35A30: Geometric theory, characteristics, transformations [See also 58J70, 58J72]


Dolbeault, Jean; Felmer, Patricio; Monneau, Régis. Symmetry and nonuniformly elliptic operators. Differential Integral Equations 18 (2005), no. 2, 141--154. https://projecteuclid.org/euclid.die/1356060226

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