Positive solutions to quasilinear p(y)-Laplacian equation with Robin boundary condition

Abstract

This article addresses the following Robin boundary value problem: { − div ( b ( y ) | ∇w | p ( y ) − 2 ∇w ) − F ( y ) | w | η ( y ) − 2 w = λG ( y ) | w | γ ( y ) − 2 w in Ω , b ( y ) | ∇w | p ( y ) − 2 ∂ w ∂ ν + H ( y ) | w | p ( y ) − 2 w = 0 on ∂Ω , here Ω ⊂ R N ( N ≥ 2 ) and p ( y ) , η ( y ) , γ ( y ) ∈ C ( Ω ¯ ) . We establish the existence and multiplicity results of the given problem in generalized Sobolev spaces W 1 , p ( y ) ( Ω ) . The main results of this article are derived using the Nehari manifold method under some appropriate assumptions on functions b, F, G and H.