Using variational methods we study the non-existence and multiplicity of non-negative solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions of the form -Div(|∇ u|p^(x)-2∇u)+|u|^p(x)-2u=0 in Ω |∇ u|^p(x)-2\∂u\∂ n =λ g(x,u) on ∂ Ω where Ω is a bounded domain with smooth boundary, n is the outer unit normal to ∂ Ω and λ is a parameter. Furthermore, we want to emphasize that g: ∂ Ω × [0,∞) → ℝ is a continuous function that may or may not satisfy the Ambrosetti-Rabinowitz-type condition. © 2010 Royal Society of Edinburgh.

Chung N.T., Ng Q.-A.

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