The Existence And Multiplicity Of Solutions To Semilinear Elliptic ?_?-Laplace Equations

This theise mainly uses nonlinear analysis tools such as symmetric mountain pass theorem,saddle point theorem and pseudo-index theory to study the existence and multi-plicity solutions involving ?_? operator.Here the ?_? operator is in the specific form ?_? :=(?),where (?),function (?) are continuous,strictly positive and of C~1 outside the coordinate hyper-plane.Firstly,let us discuss a class of semi-linear elliptic equations:(?),and its characteristic equations:(?),here ? is a smooth bounded domain in R^N(N?2),and ? is a parameter.At first,we give the eigenvalue properties of the characteristic equation and prove that,the exist-ence of the solution by using saddle point theorem and pseudo-index theory.Here,non-linear f is an asymptotically linear growth.Secondly,we study the semi-linear ?_? -Laplaceequation:(?),Here V?C(R^N,R)and f is a function with more general superliner or asymptoti-cally linear conditions.We mainly use the Nehari manifold method to prove its existence.Then we study a class of nonlinear Kirchhoff equations in R^N(N?1):(?),here a,b?0 is constants,f satisfies superlinear and sublinear conditions.We find the multiplicity of solutions by using symmetric mountain pass theorem.Finally,we study the ground state solution of the above equation,where V?C(R^N,R)is the coercive potential.The Nehari manifold method is also used to find the existence of ground state solution.