On A Quasilinear Elliptic Systems Involving Multiple Critical Nonlinearities

In this paper, a class of on a quasilinear singular elliptic system is investigated,which involves multiple critical nonlinearities. There are five chapters in this paper.First of all, In the first chapter the introduction part, we introduce the system itself andthe relevant research background. And at the same time, we also elaborate some basicpreliminary knowledge and definition in the introduction part, which makes us to knowmore about the problem.Secondly, in the second chapter, because the Mountain-Pass lemma does not yieldcritical points, but only Palais-Smale sequences, so we respectively establish the localPalais-Smale condition of the corresponding functional by the concentrationcompactness principle. Since the problem have multiple critical nonlinearities, and thecorresponding functionals lose global compactness. To address the problem, we to theequations corresponding functional size of the energy value of the appropriate limits, inorder to ensure that the convergence of sequence of critical, so we need construction ofthe appropriate local Palais-Smale condition. And in the process of establishing localPalais-Smale condition, the systems involve multiple singularities, and we need tostudy the problems inR N, so we use the concentration compactness principle to provethe convergence of the critical sequence.Finally, in the four chapter and five chapter, we study the existence of anon-negative weak solution to the system in the caseμ≥0andμ≥0.According to localPalais-Smale convergence of sequences and the strong maximum principle, using theMountain-Pass Theorem prove the existences of a non-negative weak solution to thesystem.