The Existence And Multiplicity Of Solutions To Semilinear Elliptic Δ_α-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 Δ_α operator.Here the Δ_α operator is in the specific form Δ_α :=（?）,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 Ω is a smooth bounded domain in R^N（N＞2）,and λ 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 Δ_α -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.