Study On The Existence And Multiplicity Of Solutions Of Elliptic Partial Differential Equations With Exponential Weighting

In this paper,the existence and multiplicity of solutions of three classes of elliptic partial differential equations with exponential weights are studied by means of variational method.The main content is as follows:Firstly,we consider the existence of ground state solutions for a class of elliptic partial differential equations with general nonlinear items:-(a+b∫RNK(x)|▽u|2 dx)div(K(x)▽u)=μK(x)g(x,u)+K(x)f(x,u),x∈RN Where N=2,3,a,b>0 is positive constant,K(x)=exp｛|x|2/4}is the weight and f∈C(RN,R).When N=3,we assume that nonlinear items f(x,u)is sub critical growth in R3,superlinear growth at the origin,and its primitive function F(x,u)is superquartic growth and the global(AR)conditions or local(AR)conditions.Utilizing the mountain pass theorem,we prove that the equation(0.1)has an nontrivial solution under local(AR)conditions and has an ground state solution under the global(AR)conditions.When N=2,we assume that nonlinear item f(x,u)is critical growth,and f(x,u)satisfies some assumptions.Using Trudinger-Moser inequation and mountain pass theorem,we prove that the equation(0.1)has a ground state solution.Secondly,we consider the existence of multiple solutions for a class of elliptic partial differential equations with perturbed terms:-(a+b∫R3K(x)|▽u|2 dx)div(K(x)▽u)=K(x)f(x,u)+K(x)g(x),x∈R3,where N=2,3,a,b>0is constant,K(x)=exp{|x|2/4} is the weight,f∈C(RN,R).When N=3,g(x)∈Lk2(R3),g(?)0 and the antifunction F is superquartic growth.Using ekeland variational principle and mountain pass theorem,we prove that the equation(0.2)has two different solutions,namely a positive energy solution and a negative energy solution.When N=3,we assume that nonlinear items f(x,u)is critical growth,and satisfies some appropriate assumptions.Using the test function,the Trudings-Moser inequality,the mountain pass theorem,and a new method and so on,we prove that the equation(0.2)has two different solutions,namely a positive energy solution and a negative energy solution.Finally,we consider the existence of infinitely many solutions for a class of elliptic partial differential equations with concave-convex nonlinear terms:-(a+b∫R3K(x)|▽u|2 dx)div(K(x)▽u)=μK(x)g(x,u)+K(x)f(x,u),x∈R3.where a,b>0is positive constant,g,f∈C(R3×R,R),K(x)=exp{|x|2/4} is the weight Assuming that the antifunction F is superquartic growth.Using Z2-mountain pass theorem and spatial values and decomposition,we have an infinite many solutions to the equation(0.3).