两类拟线性椭圆型方程正解的存在性应用研究.pdf

南京师范大学硕士学位论文两类拟线性椭圆型方程正解的存在性研究姓名:许娟申请学位级别:硕士专业:数学;应用数学指导教师:张吉慧 20110506 r念¨算2一州习皿|口。2%北∥, ()U 0 lim ; ~““1, ㈨一oo <P<q<P幸=番S0<Ⅳ),纵0)=a(x)+A6(z), 9A(z)>0∈C(RN,R)(z)=夕oo>(z),6(z)和入满足适当条件时方程()>【271中Lustcmik. Schnirelman畴数和临界点的关系得到存在入o>0,当A∈(0,Ao)时此类问题至少有两个正解. 第二章研究了下面拟线性椭圆型方程 z∈R垡, () >0,1<r<2<P<q<P+(如果N>P,则p+= Np/(N—p),如果N≤P,则p+=o。)和冗罕={(z7,XN)∈RⅣ一1×R[XN> 0}是RⅣ中的上半空间,本章讨论了如果f,夕是可测函数且满足下面的条件: (A1)y(x)∈c(R垡)nL广(aR罕)(r’=q/(q—r)),在aR罕上f+=max{f,o’≠o; (A2)夕(z)∈c(兄掣)n Loo(R罕)和在月掣中夕+=max{g,o).≠0. 则存在Ao>0,当入∈(0,Ao)时,方程()在w1,p(R掣)中至少有一个正解. 关键词:拟线性椭圆型方程;正解;变分法;山路引理. 一111一气桦“=气 u 卜以叫”圳十V 一= Abstract Abstract Inthispaper weinvestigate some characteristic ofsolutions toquasilinear elliptic equations,including existence andmultiplicity. Inchapter 1,the existence ofmultiple positive solutions of aclassofquasi—linear problem f一△pu+l乱Ip一2让=g(z)Mq一2u, z∈RⅣ, .{lira u:0 (o·0·3) L H一∞ isdiscussed when where2<P<口<P幸=器0<Ⅳ),gx(x)=a(x)+ Ab(x)andg(z)∈C(RⅣ,R)is positive such thatlimIzI--.∞g(z)=900> chapter discusses themultiplepositive solutionswheno(z),b(x)and入satisfy suitable ,We prove theabove problem has aground state solution when A>0 using variational ,By means oftheLusternik-Schnirelman category and variational method,the existence of atleasttwo positive solutions iS obtained forthequasi—linear ellipticequation inRⅣwhen入∈(0,Ao).In chapter 2,we study theexistence ofpositive solutionfor thefollowing elliptic equation: R垡, () where A>0,1<r<2<P<q<P+(p4=Np/(N—P)ifⅣ>P,P+=(20 if N≤p)and R罕={(z7,XN)∈RⅣ一1×RlzⅣ>ot is an upper halfspace inRⅣ, f,g aremeasurable functionsandsatisfy thefollowing conditions: (A1)f(x)∈c(冗牮)nLr(aR掣)(r+=q/(q—r))with,+=maxIf,0)≠0 in ORN+; (A2)g(z)∈c(R筝)nLoo(Rg)and夕+=max{g,o】.≠0 inR掣. Key Words: Quasilinear ellipticequation;positive solutions;variational method