Several Types Of The Alternating Elliptical Equations Existence

In this paper, we will use the variational method and the theory of critical points tostudy the existence and multiplicity of the sign-changing solutions for three classes of ellipticequations.Firstly, we will study the following class of elliptic equations with non-homogeneousboundary conditions:wheounded domain with the smooth boundary, g(x) is a con-tinuous function defined on, the nonlinearities f (x, u) and h(x) are continuous functionsdefined respectively on×R and, λ>0is a parameter. In Chapters2and3, we shallrespectively discuss the existence of sign-changing solutions for the above equations underconditions that f (x, u) is or is not odd in u. In Chapter2, by using a basic propositionestablished in this chapter (proposition A in chapter two), the method of invariant sets of thedescend flow and the method of perturbation, we will show that Equation (I) has infinitelymany sign-changing solutions if f (x, u) is odd in u and λ=1. In Chapter3, by using themethod of sub-and supper-solution, the method of invariant sets of the descend flow and themaximum principle, we will prove that Equation (I) has one or two sign-changing solutionswhen λ small enough, h(x)=0and f (x, u) may not be odd in u.Next, we will discuss existence of sign-changing solutions for the following class of ellipticequations:where RNounded domain with smooth boundary, pu=div(|u|p2u)is the p-Laplacian,1<p <N, λ>0is a parameter and p=pN/(N p) is the criticalSobelev exponent. In Chapter4, by establishing a new deformation lemma, we show thatEquation (II) has infinitely many sign-changing solutions.Finally, we shall discuss the following class of Schr¨odinger equations in Chapters5and6:where V (x) and K(x) are the potentials,2<p <2=2N/(N2) and λ>0is a parameter.In Chapter5, we shall study the existence of sign-changing solutions for the above Schr¨odinger equations when V (x) is sign-changing, K(x)>0and both V (x) and K(x) may vanish at∞(that is, both V (x) and K(x) go to0as|x|tends to∞§. By using the method of invariantsets of the descend flow, we prove that Equation (III) has at least a sign-changing solutionand a positive solution for λ small enough. In chapter6, we consider the existence of sign-changing solutions of Equation (III) with λ=1and both V (x) and K(x) being sign-changing.By designing a new method of perturbation and using the Nihari method, we show that thiskind of Schr¨odinger equations have a ground state solution and a least energy sign-changingsolution.