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In this work, we are concerned with a Robin and Neumann problem with (p(•), q(•))-Laplacian. Under some appropriate conditions on the data involved in the elliptic problem, we prove the existence of solutions applying two versions of Mountain Pass theorem, Ekeland's variational principle and Lagrange multiplier rule.

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We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case when the singular set has a smooth structure, and give sufficient conditions for removability of this singularity for the equation in the variable exponent Sobolev space W 1,p(•) (Ω).