The Existence Of A Solution For A Class Of (p,q)-Laplaceian Systems With Embossed Terms

Consider the(p,q)-elliptic systems where ?(?)RN(N>3)is a bounded open domain with a smooth boundary ??,1<?+ ?<min{p,q},max{p,g}<?+?<min{p*,q*},where p*=Np/(N-p)and q*=Np/(N-p),a(x)and b(x)are smooth functions which may change signs.The main result can be described as follows:Theorem 1.Assume that 1<?+?<min{p,q},max{p,q}<?+?<min{p*,q*},then there exists AO such that,when 0<?<?0,problem(L)has at least two nontrivial positive solutions.Consider the(p,q)-elliptic systems in ?where ?(?)RN(N>3)is a bounded open domain with a smooth boundary ??,1<?+?<min{p,q},max{p,q}<?+?+ min{p*,q*},where p*=Np/N-p and q*=Nq/N-q,a(x),b(x)>0,a(x),b(x)?L?(?).The main result can be described as follows:Theorem 2.Assume that 1<?+?<min{p,q},max{p,q}<?+?<min{p*,q*},then there exists A such that,when 0<?<?,(LL)has at least two nontrivial solutions(u1,?1)and(u2,?2).