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Zbl 1149.76692
Mihăilescu, Mihai; Rădulescu, Vicenţiu
A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids.
(English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 462, No. 2073, 2625-2641 (2006); correction 467, No. 2134, 3033-3034 (2011). ISSN 1364-5021; ISSN 1471-2946/e

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.
MSC 2000:
*76W05 Flows in presence of electromagnetic forces

Keywords: $p(x)$-Laplace operator; generalized Lebesgue--Sobolev space; critical point; weak solution; electrorheological fluids

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