The Nontrivial Solutions For Two Nonlinear Elliptic Systems

In this paper,firstly,we study a nonlinear elliptic system with weight functions(?)Where Ω(?)RN(N≥3)is a bounded smooth domain,the weight functions ai,wi∈C(Ω,R0+),R0-=(0,+∞),functions fi,hi(i=1,2)satisfy Caratheodory condition,and there exists a function H(x,u,v),such that▽Ⅱ(x,u,v)=((?)/(?)u)H(x,u,v),(?)/(?)v(x,u,v))=(h1(x,u,v),h2(x,u,v)).By establishing the linking structure in product space,we obtain the existence of three nontrivial solutions by using the minimax principle and the linking theorem in product space.Secondly,we consider the existence of nontrivial nonnegative solutions for a degen-erate elliptic system with weight functions(?)Where Ω(?)RN(N≥3)is an open bounded smooth domain,0Ωo≤a≤b<q+1<N/2,a=bq/α+β,1<q<2,α>1,β>1satisfying 2<α+β<2a,b*(2a,b*=min{2N/N-2,2(N,2b)/N-2(1-a)}),the weight functions f1,f2 and h satisfy some certain conditions.Applying the Nehari manifold and the fibering map,we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters(λ1,λ2)belongs to a certain subset of R2.