Existence and concentration behavior of positive solutions for a quasilinear Schrِdinger equation

In this paper, we study the existence and concentration of positive solutions for the quasilinear Schrödinger equation − h 2 Δ u + V ( x ) u − h 2 Δ ( u 2 ) u = K ( x ) | u | p − 2 u , x ∈ R N , where h > 0 is a small real parameter, N ⩾ 3 , 4 < p < 22 ⁎ = 4 N N − 2 , and V ( x ) and K ( x ) are uniformly continuous functions with V ( x ) having at least one minimum and K ( x ) having at least one maximum, then a positive ground state solution u h ( x ) exists. Moreover, we establish the concentration property of u h ( x ) as h → 0 + .