Abstract
In this paper, we consider the differential inclusion in \({\mathbb{R}^N}\) involving the p(x)-Laplacian of the type \({\begin{array}{lll}-\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u\in \partial F(x,u(x)),\;\;{\rm in}\;\;\mathbb{R}^N,\quad\quad\quad\quad\quad\quad ({\rm P})\end{array}}\) where \({p: \mathbb{R}^N \to {\mathbb{R}}}\) is Lipschitz continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. Under suitable oscillatory assumptions on the potential F at zero or at infinity, we show the existence of infinitely many solutions of (P). We also establish a Bartsch-Wang type compact embedding theorem for variable exponent spaces.
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Supported by the National Science Fund (grant 111262861, 10971043, 11001063), China Postdoctoral Science Foundation Funded Project (No. 20110491032), the Fundamental Research Funds for the Central Universities (HEUCF20111134), Heilongjiang Province 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. Infinitely many solutions for a differential inclusion problem in \({\mathbb{R}^N}\) involving p(x)-Laplacian and oscillatory terms. Z. Angew. Math. Phys. 63, 691–711 (2012). https://doi.org/10.1007/s00033-012-0192-1
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DOI: https://doi.org/10.1007/s00033-012-0192-1