Ground State Solutions Of Elliptic Equations Involving Different Hardy-type Terms And Multiple Critical Sobolev Exponents

In this paper,several quasilinear elliptic systems are investigated,which involve different Hardy-type terms and multiple critical Sobolev exponents making the research difficult.In the first chapter,we introduce the problems and their background,establish the definition of used symbols and then display the main research results and the main structure.In the second chapter,a general quasilinear elliptic system is studied,which involves homogeneous critical Sobolev nonlinearities and Hardy-type terms.Due to the critical nonlinearities in the system,the corresponding energy functional lose the global compactness.Applying the Schwartz symmetrization arguments and the concentration compactness principle,the strong convergence of minimizing sequence on the Nehari manifold is verified under certain condition,then the existence of ground state solution to the system and minimizers to related Sobolev constant is verified.In the third chapter,a system of constant coefficient quasilinear elliptic equations is discussed,which involves different Hardy-type terms and multiple critical terms.Applying the conclusion of the second chapter,the existence of the ground state solutions to the system is proved,then the existence conditions about the positive ground state solutions and semi-trivial ground states is obtained,among which the minimizers to related Sobolev constant are found.In the fourth chapter,a system of a variable coefficient quasilinear elliptic equations is investigated which involves Hardy-type terms and critical Sobolev nonlinearities.Firstly preliminary conclusions are established,the concentration compactness principle is used to verify the local PS condition and the strong convergence of critical sequence,then the existence of ground states to the system and minimizers to related Sobolev constant is obtained.Furthermore,by the maximum principle and analytic skills,the existence conditions about the positive and semi-trivial ground state solutions are obtained.In the fifth chapter,by using the method of the second chapter,another system of constant coefficient quasilinear equations is studied,which involves different Hardy-type terms and multiple critical nonlenearities,and prove the existence and nonexistence of ground states under certain conditions.