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Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. (English) Zbl 1135.35058

The aim of the article is to investigate the existence of solutions for the following nonhomogeneous anisotropic eigenvalue problem (quasilinear elliptic equation with a power-like variable reaction term)

- i=1 N x i | x i u| p i (x)-2 x i u=λ|u| q(x)-2 u,xΩ;u| Ω =0,

where Ω N , N3 is a bounded domain with smooth boundary, p i , q are continuous functions on Ω ¯, 2p i (x)<N, q(x)>1 for xΩ ¯, i=1,N ¯, λ is a positive number.

35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
49R50Variational methods for eigenvalues of operators (MSC2000)