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A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. (English) Zbl 1149.76692
Summary: We study the boundary value problem $-{\text{div}}(a(x,\bar{v}u))=\lambda(u\gamma-1-u\beta-1)$ in $\Omega,u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in $RN$ and ${\text{div}}(a(x,\bar{v}u))$ is $a$ is a $p(x)$-Laplace type operator, with $1<\beta<\gamma<{\text{inf}}x \in \Omega p(x)$. We prove that if $\lambda$ is large enough then there exist at least two non-negative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue--Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

76W05Magnetohydrodynamics and electrohydrodynamics
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