Sign-changing Solutions For Dirichlet Problem In Weighted Sobolev Space

In this dissertation,we discuss the existence of sign-changing solutions for Dirichlet problem on manifolds with conical singu-larity.We get the positive results both for subcritical and critical nonlinearities.We divide the dissertation into four chapters.In Chapter1,we recall the development on semilinear elliptic Dirichlet equations on Ω(?)Rm with-â–³and the background of PDEs on singular manifolds.Moreover,we show the recent results on semilinear Dirichlet problems in cone Sobolev space H2,01,n/2(B).In Chapter2,we introduce the weighted Sobolev space Hpm,γ(B) and some basic properties of those Sobolev spaces.Then,we list some useful inequalities and results which will be used for next chapters.In Chapter3,we prove the existence of multiple sign-changing solutions in cone Soblev space H2,01,n/2(B) for with0≤μ<1,where V is a singular potential satisfying Hardy inequality,and0<λ<λ1,λ1is the first eigenvalue of operator一△B-μV,2<p<2*,2*=2n/n-2is the critical Sobolev exponent. For the follow problem with f(x,u):B×Râ†'R be a Caratheodory function satisfying some subcritical and superlinear conditions,we also obtain the similar results.In Chapter4,we prove the existence of sign-changing solution in cone Sobolev space F2,01,n/2(B) for with λ>0,2*=2n/n-2be the critical Sobolev exponent.