@article {Galewski2005,
 author="Marek Galewski",
 title={A new variational method for the $p(x)$-Laplacian equation},
 journal="Bull. Austral. Math. Soc.",
 fjournal={Bulletin of the Australian Mathematical Society},
 volume="72",
 year="2005",
 number="1",
 pages="53--65",
 issn="0004-9727",
 coden="ALNBAB",
 language="English",
 date="9th February, 2005",
 classmath="35A15, 35J20",
 publisher={AMPAI, Australian Mathematical Society},
 MRnumber="MR2162294",
 ZBLnumber="02212186",
 url="http://www.austms.org.au/Publ/Bulletin/V72P1/721-5041-Galewski/index.shtml",
 acknowledgement={},
 abstract={ Using a dual variational method we shall show the existence of solutions to the Dirichlet problem \begin {eqnarray} -\divv \Bigl ( \bigl | \nabla u( x) \bigr | ^{p( x) -2}\nabla u( x) \Bigr ) &=&F_{u}\bigl ( x,u( x) \bigr ) \text {, }u\in W_{0}^{1,p ( x ) } ( \Omega ) \nonumber \\ x ( y ) | _{\partial \Omega }&=&0. \end {eqnarray} without assuming Palais--Smale condition. }
}