Journal of East China Normal University(Natural Sc ›› 2012, Vol. 2012 ›› Issue (5): 109-119.

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Multiple solutions for ${\bm p}({\bm x})$-Laplacian problems in ${\bf R}^{\bm N}$

CHEN Zi-gao   

  1. Department of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou Henan} 450011, China
  • Received:2011-12-01 Revised:2012-03-01 Online:2012-09-25 Published:2012-09-29

Abstract: By using the fountain theorem and the dual fountain theorem, respectively, the existence  and multiplicity of solutions for $p(x$)-Laplacian equations in $\mathbf{R}^{N}$ were studied, assumed that one of the perturbation terms $f_1(x,u),\, f_2(x,u)$ is superlinear and satisfies the Ambrosetti-Rabinowitz type condition and the other one is sublinear. The discussion was based on variable exponent Lebesgue and Sobolev spaces.

Key words: variable exponent Sobolev spaces, $p(x)$-Laplacian, (PS)$_c^\ast$ condition, fountain theorem, dual fountain theorem

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