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 ? is a smooth bounded domain in RN(N>3),and 0? ? 1<p<N,0<a<N-p/p,a?b<a + 1,0??<?(?)(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,?>0.F(x,t)is the primitive function of f(x,t)defined by F(x,t)=?0t f(x,s)ds for x? ?,t ? R.f? C(?× 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? ?,(f2)f:? ×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??<?,and(f1)holds.Then there exists ?*>0,such that the problem(P)admits at least one positive solution ?? for all ? ?(0,?*).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??<?= N-(p2(a+1)+(1-p)bq/p(N-bq/p)p-1,(f1)and(f2)hold.Then thwre exists ?*>0,such that the problem(P)admits at least two positive solutions for all ? ?(0,?*).Then,we consider the situation with one perturbation ?h(x)up-1:where ?>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? ?,t? R.Make the following assumptions about functions h(x)and g ? C(?×R+,R):(h1)h?H,where H = {h:??R+lim|x|?0 |x|p(1+a)h(x)=0,h ?Lloc?(?\{0})};(g1)lim t??g(x,t)tq-1=0 uniformly for x??;(g2)g? C(?×R+,R+,limt?0G(x,t)/tp<1/p? uniformly for all x×?,where 0<?<?1.?1 is the first eigenvalue of operator-div(|x|-ap|?u|p-2?u)-?|u|p-2u/|x|p(1+a);(g3)There is a constant ?>p,such that G(x,t)? C|t|? uniformly for all(x,t)??×R+.The main conclusion is as follows:Theorem 3 Suppose N ?3(1 + a),0?a<N-p/p,a?<b<a+1,0??<?,0<?<?1,(h1)holds and g satisfies(g1)-(g3)and following condition where ?(?)is one zero of the function k(t)=(p-1)tp-(N-p(a+1))tp-1+?,t?0,0????,and ?(?)>N-p(a+1)/p.Then there exists at least one positive solution for the problem(P1).