Existence Of Sign-changing Solutions For The Nonlinear Schr(?)dinger-Poisson System With Zero Mass

Nonlinear partial differential equation usually arises in the natural science and engi-neering areas.Because they can well explain the important natural phenomenon,a large number of science researchers have paid attention to the problems for a long time.In this paper,we use the constraint variational method and the quantitative deformation lemma to study the existence of sign-changing solutions for the nonlinear Schr(?)dinger-Poisson system in R3.Our equation is of the following form:We say that(V,K)?K if the following conditions hold:(H0)V(x)?K(x)>0 for all x?R3 and K?L?(R3);(H1)if {An}(?)R3 is a sequence of Borel sets and for all n and some R>0,|An|?R,then(H2)K/V?L?(R3);or(H3)there exists p?(2,6)such thatAs for the function f,we assume f?C1(R,R)and satisfies the following conditions:(f1)limt?0 f(t)/t=0,if(H2)hold;(f2)limt?0 f(t)/|t|p-1=A?R,if(H3)hold;(f3)f has a "quasicritical growth"?namely,lim|t|??f(t)/t5=0;(f4)lim|t|??F(t)/t4=?,where F(t)=?0tf(s)ds;(f5)the map t?f(t)/|t|3 is nondecreasing on(-?,0)and(0,?)respectively.Our main result is the following theorem:Theorem Suppose that(V,K)?K and f satisfies(f1)-(f5).Then the system(1.1)possesses at least one sign-changing solution.