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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,  in  Ω;  ∇upx-2∂u/∂ν=λuqx,  on  ∂Ω}, where Ω⊂RNN≥2 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 Λ∈0,∞ such that problem Pλ has two solutions for λ∈0,Λ, one solution for λ=Λ, and no solutions for λ>Λ

international journal of differential equations

Multiplicity Results for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>p</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced></mml:math>-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

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