Symmetry results for the $p(x)$-Laplacian equation
  We consider the Dirichlet problem for the nonlinear $p(x)$-Laplacian equation. For axially symmetric domains we prove that, under suitable assumptions, there exist Mountain-pass solutions which exhibit partial symmetry. Furthermore, we show that Semi-stable or non-degenerate smooth solutions need to be radially symmetric in the ball. 
Luigi Montoro, Berardino Sciunzi, Marco Squassina
math.AP