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Multiplicity of solutions for differential inclusion problems in \({\mathbb{R}^N}\) involving the p(x)-Laplacian

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Abstract

In this paper we consider the differential inclusion problem in \({\mathbb{R}^N}\) involving the p(x)-Laplacian of the type

$$ -\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u\in \partial F(x,u)\,\,\,{\rm in}\, \mathbb{R}^N. $$

The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, based on the Weirstrass Theorem and Mountain Pass Theorem, we get there exist at least two nontrivial solutions. We also establish a Bartsch–Wang type compact embedding theorem for variable exponent spaces.

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Authors and Affiliations

  1. Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, Heilongjiang, People’s Republic of China

    Bin Ge

  2. Library, Northeast Forestry University, Harbin, 150040, Heilongjiang, People’s Republic of China

    Qing-Mei Zhou

  3. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, People’s Republic of China

    Xiao-Ping Xue

Authors
  1. Bin Ge
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  2. Qing-Mei Zhou
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  3. Xiao-Ping Xue
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Correspondence to Bin Ge.

Additional information

Communicated by Adrian Constantin.

This work is supported by the National Science Found of China (Grand. 10971043,11001063), the Fundamental Research Funds for the Central Universities (No. HEUCF 20111134), China Postdoctoral Science Foundation Funded Project (No. 20110491032), Heilongjiang Provincial Science Foundation for distinguished young scholars (JC200810), Program of excellent team in Harbin Institute of Technology and the Natural Science Foundation of Heilongjiang province (No. A200803).

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Ge, B., Zhou, QM. & Xue, XP. Multiplicity of solutions for differential inclusion problems in \({\mathbb{R}^N}\) involving the p(x)-Laplacian. Monatsh Math 168, 363–380 (2012). https://doi.org/10.1007/s00605-011-0342-0

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  • DOI: https://doi.org/10.1007/s00605-011-0342-0

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