International Journal of Differential Equations

Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

K. Saoudi, M. Kratou, and S. Alsadhan

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We investigate the singular Neumann problem involving the p(x)-Laplace operator: Pλ{-Δpxu+|u|px-2u  =1/uδx+fx,u, in  Ω;u>0,inΩ;upx-2u/ν=λuqx,onΩ}, where ΩRNN2 is a bounded domain with C2 boundary, λ is a positive parameter, and px,qx,δx, and fx,u are assumed to satisfy assumptions (H0)(H5) in the Introduction. Using some variational techniques, we show the existence of a number Λ0, such that problem Pλ has two solutions for λ0,Λ, one solution for λ=Λ, and no solutions for λ>Λ.

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Int. J. Differ. Equ., Volume 2016 (2016), Article ID 3149482, 14 pages.

Received: 5 April 2016
Accepted: 22 June 2016
First available in Project Euclid: 21 December 2016

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Saoudi, K.; Kratou, M.; Alsadhan, S. Multiplicity Results for the $p(x)$ -Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition. Int. J. Differ. Equ. 2016 (2016), Article ID 3149482, 14 pages. doi:10.1155/2016/3149482.

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