On A Class Of Nonlinear Elliptic Equations

We can describe some physical and biological phenomenons via a Kirchhoff type systems,such as the elastic model and the species density problems.However,there are less results on existence and the related properties of the solutions of the the Kirchhoff type system.So,it is necessary for us to discuss the existence of the positive solutions for this kind of elliptic system.Moreover,one of the features of the Kirchhoff type problem is its nonlocal term,it is of great significance to study the effect of the non-local term on the positive solutions.In the present paper,we discuss the existence of positive solutions of a Kirchhoff type elliptic system.We also explore the effect of non-local term b(f?| ?u1|2dx)?)?u1 to the existence,the number and the asymptotic behavior of the positive solutions.Chapter one is preface.In this chapter,we introduce the background and the present status of the problems we are concerned about.We also analyse the methods to achieve our goals and state our results.Moreover,we list some concepts and lemmas.In Chapter two,we discuss the the effect of non-local term to the positive solutions for the following problem where ?(?)Rn(n?3)is a bounded smooth domain,?,a,b,p,q are positive constants,0<pq<1 or pq>1,1/(p+1)+1/(q+1)>(n-2)/n.We represent the solution(u1,u2)of the problem(4)via the suitable solution(v1,v2)of Lane-Emden problem and then,we discuss the existence,the number and the asymptotic properties of the positive solutions of a related algebra equation.By these results,we can attain the effect of non-local term.In Chapter three,we prove the existence of the solution of the following problem where ?(?)Rn(n?3)is a bounded smooth domain,a,? b,p,q are positive constants,t ?[0,1].0<pq<1 or pq>1 and 1/(p+1)+1/(q+1)>(n-2)/n·h1 and h2 satisfy suitable conditions.Due to the fact that problem(6)has no variational structure,we would show the existence of the positive solutions via the a priori estimates and the method of continuity.We would attain the a priori estimates of solutions via the blow-up argument.It is worth mentioning that we attain the existence of positive solutions of the Kirchhoff type non-variational system and that we also attain the maximum norm of positive solutions for the Kirchhoff type system.