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In this paper we study the existence of non-trivial weak so- lutions of the Neumann problem for quasilinear elliptic equations in the form div( h( x) j∇ uj p 2 ∇u) + b( x) j uj p 2 u = f ( x;u) ; p  2 in an unbounded domain Ω  R N ;N  3, with sufficiently smooth bounded boundary @Ω, where h( x) 2 L 1 ( Ω), Ω = Ω ( @Ω, h( x)  1 for all x 2 Ω: The proof of main results rely essentially on the arguments of variational method.

ON EXISTENCE OF WEAK SOLUTIONS OF NEUMANN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING p-LAPLACIAN IN AN UNBOUNDED DOMAIN