Existence And Property Of State Solution For Nonlinear Elliptic Equations With Weight

In this paper, we discuss ground state solutions for semilinear problems with Hardy-Sobolev term. We also study the symmetry and regularity of positive solutions to the integral systems on RN. This paper is constituted with three chapters.In chapter1, we introduce the background and main results.In chapter2, we consider the existence of ground state solutions of nonlinear elliptic equation where Ω is a smooth bounded domain in RN(N≥3), x=(y,z)∈Ω(?)Rk×RN-k=RN,2≤k<N,0<s<2, and2*s=(2(N-s))/(N-2) is the corresponding critical exponent. We show that problem (0.0.3) possesses a ground state solution provided that N=4and λm<λ<λm+1or N≥5and Am≤λ<λm+1for some m∈N, where λm is the m’th eigenvalue of-△with Dirichlet boundary condition.In chapter3, we investigate the symmetry and regularity of positive solutions to the following integral system where N≥3,0<a<N,0<β<N-a,p,q,r>1and Firstly, we show that the positive solution (u, v,w)∈Lp+1[RN) x Lq+1(RN)×Lr+1(RN) is radially symmetric by the moving plane method. Then, by using the regularity lifting theorem I, we obtain the solution pair of the integral system (u, v, w) belongs to L∞(RN)×L∞(RN) x L∞(RN). Moreover, the positive solution is locally Holder continuous by means of the regularity lifting theorem II. Finally, we prove the decay of positive solutions.