On The Existence Of Solutions For Quasilinear Schr(o|")dinger Equations

In this thesis, by using variational methods, we study the existence of nontrivial solutions of a class quasilinear elliptic equations where Ω is a smooth domain (bounded or unbounded) in RN, h(x, u) is continuous on Ω x R. Our results obtained in the thesis are included in the following two parts:In Part 1, when Ω is a bounded domain in RN (N≥ 3), V(x)≡ 0 (x ε Ω), h(x, u)(?) f(x,u)+λg(u), by using the cut-off function technique combining with the result in [21], we show that the equation has a positive solution and a negative solution for λ> 0 small, where f(x, u) ε C(Ω x R) satisfying some restricted growth conditions, λ is a positive parameter, and g(u) is a locally Holder continuous function. We point out that h(x, u) here may not satisfy the classical (AR) condition.In Part 2, when Ω= RN (N≥ 3), V(x)≠ 0 (x ε Ω), h(x,u)(?) K(x)|u|p-2u, we obtain a positive solution for the above equation via the Nehari method and the perturbation technique, where 4< p< 2 ? 2*, V(x) and K(x) are potential functions satisfying some suitable conditions. We note that both the V(x) and K(x) may be sign-changing.