Research On The Existence And Non-existence Of Solutions To The Quasi-linear Schrodinger Equation

In this thesis,we study the existence and nonexistence of standing wave solu-tions for quasilinear Schrodinger equation with various potential.The approaches we used are the variational method,the critical point theory and analytical tech-niques.Firstly,we are concerned with the following quasilinear Schrodinger equation with asymptotically periodic potential-?u + V(x)u-?(u2)u =(x,u),x?RN,where N>3,V,g are asymptotically periodic functions in x,the nonlinear term g is subcritical.By using a change of variables,the quasilinear problem is transformed into a semilinear one.Then we use a Nehari-type constraint to get the existence result.Secondly,we investigate the following quasilinear Schrodinger problem in-volving a critical nonlinearity-?u + V(x)u-?(u2)u = K(x)|u|2·2*-2u + g(x,u),x?RG,whereN ? 3,2*=2N/N-2,V= g are asymptotically periodic functions in x,and V,g satisfy a new reformative condition which unify the asymptotic processes at infinity.By combining variational methods and the concentration-compactness principle,we obtain a ground state solution.Thirdly,we study the following quasilinear Schrodinger equation under a general nonlinear term-?u + V(x)u-?(u2)u = g(u),x?RN(0.2)where V(x)satisfies some geometrical condition,g(u)satisfies the general hy-potheses introduced by Berestycki and Lions,and V(x)tends to zero as |x|??.When V(x)is radially symmetric,we prove the boundedness of the(PS)sequence by the monotonicity trick and we get the compactness by the Strauss lemma.Then,we obtain a positive solution.At last,we also present the existence of a positive ground state solution.Moreover,we prove a nonexistence result by employing the Pohozaev manifold.Finally,we discuss equation(0.2)in R3.The nonlinear term g(u)is nonho-mogeneous and asymptotically 3-linear at infinity.The infimum of the Pohozaev manifold is unreachable,a bound state solution is considered.With the help of the barycenter function and the Pohozaev manifold,we get a link.Then we ob-tain the existence of bound state solutions for equation(0.2)by using the linking theorem.