Existence And Multiplicity Of Solutions For ??₁,?₂?-Laplacian Elliptic Equations

(?₁,?₂)-Laplacian elliptic equations containing ?-Laplacian operators could be used to model many phenomena in the fields of mechanics and physics,such as,plasma phy-sics,nonlinear elasticity,plasticity and generalized Newtonian fluids.So it has important theoretical significance and applied value to study??₁,?₂?-Laplacian elliptic equations.In this dissertation,by using variational methods,the existence and multiplicity of non-trivial solutions and ground state solutions for some kinds of??₁,?₂?-Laplacian elliptic equations are studied in a bounded domain and the whole space.The main research contents and research results of the dissertation are as follows:1.A class of??₁,?₂?-Laplacian elliptic equations containing three parameters and satisfying Dirichlet boundary condition is studied in a bounded domain.When the non-linear term of the equations satisfies?-subcritical and certain?-superlinear growth condi-tions at infinity,by using a three critical points theorem due to Ricceri and a four critical points theorem due to Anello,it proves that the equations have three solutions and four solutions,respectively.2.A class of??₁,?₂?-Laplacian elliptic equations without parameter and satisfying Dirichlet boundary condition is studied in a bounded domain.When the nonlinear term of the equations satisfies?-subcritical and?-superlinear growth conditions at infinity and?-sublinear growth condition near the origin,by using the mountain pass theorem,it proves that the equations have a nontrivial solution.When the nonlinear term possesses an additional symmetry property with respect to origin,by using the symmetric mountain pass theorem,it proves that the equations have infinitely many solutions.The existence results extend and improve those corresponding results in the literature.3.A class of??₁,?₂?-Laplacian elliptic equations containing nonnegative bounded potential functions is studied in the whole space.When the nonlinear term of the equations satisfies?-sublinear growth condition at infinity and local?-superlinear growth condition near the origin,by using the least action principle,it proves that the equations have a nontrivial solution.When the nonlinear term possesses an additional symmetry property with respect to origin,by combing the Clark theorem with genus properties,it proves that the equations have a sequence of solutions whose corresponding energy value series converge to zero.Even if the equations reduce to the scalar case,the existence results are different from those in the literature.4.A class of??₁,?₂?-Laplacian elliptic equations containing nonnegative periodic potential functions is studied in the whole space.When the nonlinear term of the equa-tions satisfies?-subcritical and?-superlinear growth conditions at infinity and?-sublinear growth condition near the origin,by using a generalized mountain pass theorem,it proves that the equations have a nontrivial solution.Based on the fact that equations have a non-trivial solution,it proves that the equations have a ground state solution.The existence results of ground state solution extend those corresponding results in the literature.The obtained results of existence and multiplicity provide theoretical support for solving??₁,?₂?-Laplacian elliptic equations by numerical methods.