Application Of Variation Principle In Elliptic Equations

At present, the actual problems of physics, biochemistry and other subjects can be studied by equations. So this thesis chooses two elliptic equations with actual meaning: AndUsing variational principle, the paper has the conclusion that the two classes of equations have infinite solutions.The function f(x,u) of the first equation dose not satiety Palais-Smale condition, so this thesis cannot use traditional method to prove the existence of solutions. This thesis makes the function f(x, u) satiety wearer condition—Ceremi condition, using symmetric version of the Mountain Pass Theorem, and then this thesis acquire the conclusion that the first equations have multiple solutions.For the second equation, its form is complex; furthermore, it is wellworthtobestudied. This thesis makes the second function f(x,u) satiety Palais-Smale condition, using the Mountain Pass Theorem, and then this thesis acquire the conclusion that the second equations have infinite number of solutions. Through two different research methods of infinite solutions to the similar elliptic equations this thesis provides different theory basic for the similar subject researches on these two classes of elliptic equations.Finally, this thesis enumerates two elliptic equations about nonlinear quantum mechanics and capillarity, and then this thesis applies the conclusion to practice.