The Eigenvalue Problem for $p(x)$-Laplacian Equations Involving Robin Boundary Condition
Received:May 01, 2017  Revised:October 28, 2017
Key Word: variable exponents   eigenvalue   Robin boundary condition   $p(x)$-Laplacian equations  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11571057).
Author NameAffiliation
Lujuan YU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
Fengquan LI School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
Fei XU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
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Abstract:
      This paper studies the eigenvalue problem for $p(x)$-Laplacian equations involving Robin boundary condition. We obtain the Euler-Lagrange equation for the minimization of the Rayleigh quotient involving Luxemburg norms in the framework of variable exponent Sobolev space. Using the Ljusternik-Schnirelman principle, for the Robin boundary value problem, we prove the existence of infinitely many eigenvalue sequences and also show that, the smallest eigenvalue exists and is strictly positive, and all eigenfunctions associated with the smallest eigenvalue do not change sign.
Citation:
DOI:10.3770/j.issn:2095-2651.2018.01.006
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