Existence and Asymptotic Behavior of Positive Solutions for a Class of Quasilinear Schrodinger Equations

In this paper, we study the quasilinear Schrodinger equation -Delta u + V(x) u -y/2 (Delta u(2))u = vertical bar u vertical bar(p-2)u, x is an element of R-N, where V(x) : R-N -> R is a given potential, gamma > 0, and either p is an element of (2, 2 *), 2* = 2 N/N-2 for N >= 4 or p is an element of (2, 4) for N = 3. If gamma is an element of (0, gamma(0)) for some gamma(0) > 0, we establish the existence of a positive solution u(gamma) satisfying max(x is an element of R)(N) vertical bar gamma(mu)u(gamma) (x)vertical bar -> 0 as gamma -> 0(+) for any mu > 1/2. Particularly, if V(x) = lambda > 0, we prove the existence of a positive classical radial solution u gamma and up to a subsequence, u(gamma) -> u(0) in H-2(R-N) boolean AND C-2 (R-N) as gamma -> 0(+), where u(0) is the ground state of the problem -Delta u + lambda u = vertical bar u vertical bar(p-2)u, x is an element of R-N.