Concentration of solutions for ( N , q )-Laplacian equatio

In this article, we consider the concentration of positive solutions for the following equation with Trudinger–Moser nonlinearity: { − Δ N u − Δ q u + V ( ε x ) ( | u | N − 2 u + | u | q − 2 u ) = f ( u ) , x ∈ R N , u ∈ W 1 , N ( R N ) ∩ W 1 , q ( R N ) , x ∈ R N , where V is a positive continuous function and has a local minimum, ε > 0 is a small parameter, 2 ≤ N < q < + ∞ , f is C 1 with subcritical growth. When V and f satisfy some appropriate assumptions, we construct the solution u ε that concentrates around any given isolated local minimum of V by applying the penalization method for the above equation.