The Existence Of Solutions For Two Classes Of Elliptic Equation With Singularity

In this paper, we are devoted to the study of the existence results of problem:where 0∈Ω(?)R^N(N≥4) is an unbounded domain with smooth boundary.k(x)∈L^∞(R^N)satisfies the following conditions:(A₁)k(x)=k⁺(x)-k^-(x),k^±(x)=max{±k(x),0}≠0.(A₂)k^-(x)=o(|x|^α)(|x|→0), for everyα∈(1,2) and there existsρ＞0 satisfiesk⁺(x)=0 for x∈B(0, 2ρ).and the existence of positive solutions to the class of semilinear elliptic equation:whereΩdenotes an open set containing the origin,bounded or not,of R^N with N≥4.μ≤((N-2-2a)/2)²,a≤b＜a+1,0≤a＜(N-2)/2,p=p(a,b)=2N/(N-2(1+a-b)).Note thatp,(0, 0)=2^*=2N/(N-2) is the critical Sobolev exponent.The thesis consists of four chapters.In chapter one, we introduce some results on existence of the two classes ofsingular elliptic equations.In chapter two, we introduce some basic knowledge of standard Sobolev spacesBesides,we give some basic definition and lemmas. In addition, we give some nota-tions.In chapter three, we obtain several results on the existence of nontrivial solu-tions to the problem (1.1).In chapter four, we consider the semilinear elliptic problem (1.2).