Multiplicity of solutions for differential inclusion problems in {\mathbb{R}^N} involving the p(x)-Laplacian |
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Authors: | Bin Ge Qing-Mei Zhou Xiao-Ping Xue |
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Institution: | 1. Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, Heilongjiang, People’s Republic of China 2. Library, Northeast Forestry University, Harbin, 150040, Heilongjiang, People’s Republic of China 3. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, People’s Republic of China
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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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