Existence Of Ground State Solution And Sign-changing Solution For A Class Of Quasilinear Schr(?)dinger Equation

This paper mainly use the variational method and some analytical techniques of the critical point theory,we get the existence of ground state solution and least energy sign-changing solution for a class of quasilinear Schrodinger equation.Firstly,we discuss the case where the nonlinear term is hypercubic at infinty.Following is the quasilinear Schrodinger equation:-△u-a(x)△(u2)u+V(x)u=f(x,u),x∈RN,(0.0.1)where N≥3,V is a potential,a(x)is a bounded function,f is the nonlinearity.Following are some assumptions about the V(x),a(x)and f:(V)V ∈C(RN,R)is nonnegative and lim(?)V(x)=∞.(a)There are positive constants a0 and a∞ such that a0≤a(x)≤a∞.(f1)f(x,t)∈C1(RN×R,R)and(?)f(x,t)/t=0.(f2)(?)f(x,t)/t2·2*-1=0.(f3)(?)F(x,t)/t4=+∞,where F(x,t)=∫0t f(x,s)ds.(f4)f(x,t)/|t|3 is increasing for |t|>0.we can get a ground state solution about the equation(0.0.1).The non-Nehari manifold method and deformation lemma are used to obtain the existence of the signchanging solution of equation(0.0.1)which exactly has two nodal domains.Secondly,we discuss the equation(0.0.2)in R3.Following is the quasilinear Schrodinger equation:-△u-a(x)△(u2)u+V(x)u=f(u),x∈R3,(0.0.2)The nonlinear term f(u)is satisfied asymptotically 3-linear at infinity.(f5)(?)f(t)/t3=α,where α ∈(0,+∞).The condition(a),(f1)and(f4)are still valid,by using non-Nehari manifold method,Miranda theorem and deformation arguments,the least energy sign-changing solution of(0.0.2)with only one change of sign is obtained.