Research On Sign-Changing Solutions Of Degenerate Elliptic Equations And Soliton Solutions Of Quasilinear Schrodinger Equations

This doctoral dissertation is devoted to study the sign-changing solutions of degen-erate elliptic equations and soliton solutions for Schrodinger equations. These equations are rich in mathematical physics background. In addition, some Sobolev-Hardy inequali-ties with boundary distance function and general weight function are established. Using these inequalities we get new Sobolev-Hardy spaces and prove the compact theorems. It is made up of five chapters.Chapter one is preface. We first introduce the theory background including the basic methods and theorems. Then we expound the background and research status of all problems of this paper.In Chapter two, we study the following degenerate equation with critical boundary distance weight: Using the Sobolev-Hardy inequalities with critical boundary distance weight da/pp*(x) we get a new Sobolev-Hardy space. By the compactness result and the critical point theo-rem to sign-changing solutions, we obtain infinitely many sign-changing solutions of this equation.In Chapter three, a degenerate elliptic equation with general weight function ψ(x) is concerned: φ is a positive continuous function defined in (0,+∞), ψ=φ(-h’/h)2, where h satisfies rN-1φ(r)(h2(r))’=c0for some constant co,h-1(0)=0. Some embedding inequalities in Sobolev-Hardy space with general weight are established. In the weighted Sobolev-Hardy space, we obtain the existence of infinitely many sign-changing solutions of the degenerate elliptic equation with general weight function. In Chapter four,we study a N-Laplacian quasilinear Shrodinger equation:-LNu+V(x)|u|N-2u=h(u), x∈RN where LNu:=△Nu+△N(u2)u△Nu:=div(|▽u|N-2▽u) is N-Laplacian operator and N≥2,h satisfies the critical growth conditions when N=p.By Mountain Pass lemma,we prove the existence of this equation together with Lions’s compact theory and the Trudinger-Moser inequality in unbounded domain.In Chapter five, we study a quasilinear Schrodinger equation:-△u+V(x)u-α△(|u|2α)|u|2α-2u=μ|u|q-1u+|u|p-1u,x∈RN where V∈C(RN,R+) is periodic in each variable xk(1≤k≤N,N≥3),α>1/2,2≤q+1<p+1<2α2*:=4αN/(N-2) and μ>0.We introduce a new variable replacement,and obtain the soliton solution of this equation by Mountain Pass lemma without(PS) conditions and Lions’s compact theory.We consider two cases including subcritical and critical conditions.In this chapter we also consider the following equation: where V∈C(RN,R+)is periodic in each xk(1≤k≤N,N≥3),and obtain the soliton solution of this equation when12-4(?)<r+1<22*.