The Existence Of Positive Solutions For A Class Of Quasilinear Schr(?)dinger Equation

In this paper,the existence and multiplicity of positive solutions of a class of quasilinear Schrodinger equation is studied by means of variable substitution,mono-tonicity trick and Ekeland's variational principle.Firstly,we consider the following quasilinear Schrodinger equation-?u+V(x)u-?[(1+u2)1/2]u/2(1+u2)1/2=?h(c),x?RN,(0.1)where N?3.V?C1(RN,R),h?C(R,R)??0 is a parameter.Under some appropriate assumptions on the potential function V and the nonlinear term h,the existence of a positive solution to the equation(0.1)is obtained by using variable sub-stitution and monotonicity trick.Secondly,we study the following subcritical quasilinear Schrodinger equation with singularity-?u+V(x)u-?[(1+u2)1/2]u/2(1+u2)1/2=a(x)|u|p-2u+?b(x)/us,x?RN,(0.2)whereN? 3,2?p?2*=2N/N-2,??0 is a parameter,0 ?s?1,a(x)?L2*/2*-p(RN)and b(x)?L2*/2*-(1-s)(RN)are two positive functions,and V?C1(RN,R).Making appropriate assumptions about the potential function V,the existence of two positive solutions to the equation(0.2)is obtained by variable substitution,monotonicity trick and Ekeland's variational principle.Finally,we consider the following quasilinear Schrodinger equation with singu-larity and critical exponential growth-?u+v(x)u-?[(1+u2)1/2]u/2(1+u2)1/2=u5+?b(x)/us,x?R3,(0.3)where??0 is a parameter,0?s?1,b(x)?L6/5+s(R3)is a positive function,and V ?C1(R3,R).By appropriate assumptions about the potential function V,the existence of two positive solutions to the equation(0.3)is obtained by variable substitution,monotonicity trick and Ekeland's variational principle.