Semiclassical solutions of perturbed p-Laplacian equations with critical nonlinearity

We consider the perturbed p-Laplacian equation − ε p △ p u + V ( x ) | u | p − 2 u = K ( x ) | u | p ⁎ − 2 u + f ( x , u ) , u ∈ W 1 , p ( R N ) , where △ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplacian operator with 1 < p < N , p ⁎ = p N / ( N − p ) denotes the Sobolev critical exponent, K ( x ) is a bounded positive function. Under some mild conditions on V and f we show that the equation has at least one nontrivial solution provided that ε ⩽ ε 0 , where the bound ε 0 is formulated in terms of p, N, V, K and f.