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Zbl 1135.35058
Mihăilescu, Mihai; Pucci, Patrizia; Rădulescu, Vicenţiu
Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent.
(English)
[J] J. Math. Anal. Appl. 340, No. 1, 687-698 (2008). ISSN 0022-247X

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) $$-\sum_{i=1}^N\partial_{x_i}\left( \vert \partial_{x_i}u\vert ^{p_i(x)-2}\partial_{x_i}u \right)=\lambda \vert u\vert ^{q(x)-2}u,\quad x\in\Omega;\ u\vert_{\partial\Omega}=0,$$ where $\Omega\subset\mathbb{R}^N$, $N\geq 3$ is a bounded domain with smooth boundary, $p_i$, $q$ are continuous functions on $\overline{\Omega}$, $2\leq p_i(x)<N$, $q(x)>1$ for $x\in\overline{\Omega}$, $i=\overline{1,N}$, $\lambda$ is a positive number.
MSC 2000:
*35P30 Nonlinear eigenvalue problems for PD operators
49R50 Variational methods for eigenvalues of operators

Keywords: nonhomogeneous differential operators; anisotropic Sobolev spaces; continuous spectrum; variational methods

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