Archive for Rational Mechanics and Analysis

, Volume 148, Issue 4, pp 265–290 | Cite as

Symmetry of Ground States of Quasilinear Elliptic Equations

  • James Serrin
  • Henghui Zou


. We consider the problem of radial symmetry for non‐negative continuously differentiable weak solutions of elliptic equations of the form \( {\rm div}(A(\vert Du\vert)Du) + f(u) = 0,\quad x\in {\vec R}^n, \quad n\geq 2,\eqno(1)\) under the ground state condition \( u(x)\to 0 \mbox{ as } \vert x\vert\to\infty. \eqno(2)\) Using the well‐known moving plane method of Alexandrov and Serrin, we show, under suitable conditions on A and f, that all ground states of (1) are radially symmetric about some origin O in \({\vec R}^n\). In particular, we obtain radial symmetry for compactly supported ground states and give sufficient conditions for the positivity of ground states in terms of the given operator A and the nonlinearity f.


State Condition Weak Solution Elliptic Equation Suitable Condition Plane Method 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • James Serrin
    • 1
  • Henghui Zou
    • 2
  1. 1.Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USAUS
  2. 2.Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama, USAUS

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