Existence of Solutions to Quasilinear Schrodinger Equations Involving Critical Sobolev Exponent

By using variational approaches, we study a class of quasilinear Schrodinger equations involving critical Sobolev exponents -Delta u + V(x)u + 1/2 kappa[Delta(u(2))]u = vertical bar u vertical bar(p-2) u + vertical bar u vertical bar(2*-2)u, x is an element of R-N, where V(x) is the potential function, kappa > 0, max{(N + 3)/(N - 2), 2} < p < 2* := 2N/(N - 2), N >= 4. If kappa is an element of [0, (kappa) over bar) for some (kappa) over bar > 0, we prove the existence of a positive solution u(x) satisfying max(x is an element of RN) vertical bar u(x)vertical bar <= root/1/(2 kappa).