Existence And Multiplicity Of Positive Solutions For Quasilinear Elliptic Equation With Weighted Hardy-Sobolev Critical Exponents

In this thesis,the existence and multiplicity of positive solutions for a class of quasi-linear elliptic problem involving weighted Hardy-Sobolev critical exponent is studied by using Mountain Pass Lemma,the strong maximum principle,variational methods and some analytical techniques.Firstly,we consider the following quasi-linear elliptic problem where Ω is a smooth bounded domain in RN(N>3),and 0∈ Ω 1<p<N,0<a<N-p/p,a≤b<a + 1,0≤μ<μ(?)(N-p/p-a)p,q = p*(a,b)(?)Np/N-p(a+1-b)is the critical weighted Hardy-Sobolev exponent and p*(a,a)= Np/N-p=p*is the critical Sobolev exponent,λ>0.F(x,t)is the primitive function of f(x,t)defined by F(x,t)=∫0t f(x,s)ds for x∈ Ω,t ∈ R.f∈ C(Ω× R+,R)satisfies the following conditions:(f1)lim t→0+f(x,t)tp-1= +∞,limt→+∞f(x,t)tq-1= 0 uniformly for x∈ Ω,(f2)f:Ω ×R+ →R is nondecreasing with respect to the second variable.The main conclusions are as follows:Theorem 1 Suppose that N ≥ 3,0 ≤ a<N-p/p,a≤b<a+1,0≤μ<μ,and(f1)holds.Then there exists λ*>0,such that the problem(P)admits at least one positive solution μλ for all λ ∈(0,λ*).Theorem 2 Suppose N>p2(a + 1)+(1-p)bq,p ≥3N/3(a+1-b)+N,O ≤a<N-p/p,a ≤b<a + 1,0≤μ<μ= N-(p2(a+1)+(1-p)bq/p(N-bq/p)p-1,(f1)and(f2)hold.Then thwre exists λ*>0,such that the problem(P)admits at least two positive solutions for all λ ∈(0,λ*).Then,we consider the situation with one perturbation λh(x)up-1:where λ>0,h(x)is a positive measurable function.G(x,t)is the primitive function of g(x,t)defined by G(x,t)=∫0t g(x,s)ds for x∈ Ω,t∈ R.Make the following assumptions about functions h(x)and g ∈ C(Ω×R+,R):(h1)h∈H,where H = {h:Ω→R+lim|x|→0 |x|p(1+a)h(x)=0,h ∈Lloc∞(Ω\{0})};(g1)lim t→∞g(x,t)tq-1=0 uniformly for x∈Ω;(g2)g∈ C(Ω×R+,R+,limt→0G(x,t)/tp<1/pγ uniformly for all x×Ω,where 0<γ<λ1.λ1 is the first eigenvalue of operator-div(|x|-ap|▽u|p-2▽u)-μ|u|p-2u/|x|p(1+a);(g3)There is a constant ρ>p,such that G(x,t)≥ C|t|ρ uniformly for all(x,t)∈Ω×R+.The main conclusion is as follows:Theorem 3 Suppose N ≥3(1 + a),0≤a<N-p/p,a≤<b<a+1,0≤μ<μ,0<λ<λ1,(h1)holds and g satisfies(g1)-(g3)and following condition where β(μ)is one zero of the function k(t)=(p-1)tp-(N-p(a+1))tp-1+μ,t≥0,0≤μ≤μ,and β(μ)>N-p(a+1)/p.Then there exists at least one positive solution for the problem(P1).