The Study Of Some Elliptic Equations On The Whole Space

In this paper,we study the positive solutions and ground state solution of an asymptotically linear Schr(?)dinger equation on RN,and existence and non-existence of solutions of an asymptotically linear Schr(?)dinger-Kirchhoff equation on R3.In the first chapter,we introduce the research background of two kinds of elliptic equations in the whole space.In the second chapter,we study an asymptotically linear Schr(?)dinger equation on RN:Where N? 3,u:RN?R is a positive function,f(x,t)tend to p(x)and q(x)? L?(RN),respectively,as t?0 and t?+?.Nonlinear term f and the potential V satisfy:(V1)V(x)? C(RN,R)and V(x)?? 0 for all x?RN.(V2)limV(x)=V(?)?(0,+?),and V(x)?V(?)for all x?RN.(F1)f(x,t)?C(RN×R+,R),and there exist0?p(x)?(x)?L?(RN)such that lim f(x,s)=p(x)and lim f(x,s)=q(x)uniformly in x?RN.(F2)For any x ?RN,t?R+,p(x)?f(x,t)?q(x).(F3)For all x ?RN,t?R+,there exists ?>0 such that 2F(x,t)t-2<V(?)-?.Define:l=inf{?RN(|?u)2-p(x)u2)dx:u ? H1(RN),?RNu2 dx=1}(2)L=inf{?RN(|?u)2-p(x)u2)dx:u ? H1(RN),?RNu2 dx=1}(3)Using mountain pass lemma and cut-off function,we prove the positive solution and ground state solution of equation(1).In the third chapter,we study an asymptotically linear Schr(?)dinger-Kirchhoff equation on R3:-(a+??R3(|?u|2+V(x)u2)dx)?u+V(x)u=f(x,u),(4)where a is a positive constant,?>0 is a parameter and V(x)(?)constant,f(x,s)/s tend top(x)and q(x),respectively,as s?0 and s??.Nonlinear term f(x,s)and the potential V(x)satisfy the following conditions:(V1)V(x)? C(R3,R)and V(x)?Vor>0 for all x?R3;(V2)lim V(x)=V(?)?(0,+?),and V(x)?V(?)for all x?R3;(F1)f(x,·)? C(R,R)and there exists 0 ?p(x)?q(x)?L?(R3)such that uniformly in x ?R3;(F2)For all x?R3,f(x,s)/s in nondecreasing in s? 0.A definition similar to l,L,redefine l1=inf {?R3(a2 |?u|2-p(x)u2)dx:u ? H1(R3),?R3 u2dx=1}.(5)L1=inf {?R3(a2 |?u|2-p(x)u2)dx:u ? H1(R3),?R3 u2dx=1}.(6)Using variational method and mountain pass lemma,we prove exists ?*>0 and?>0 such that problem(4)has at least one positive solution and no non-trivial solution in H1(R3)for all ??(0,?),???.Finally,this paper is a summary.