Multiple Positive Solutions For Robin Problem Involving Critical Weighted Hardy-sobolev Exponents With Boundary Singularities

In this paper, firstly, the existence of a positive solution is obtained for a class of Robin problem involving critical weighted Hardy-Sobolev exponents with boundary singularities by using Mountain pass lemma, the strong maximum principle, varia-tional methods and some analysis. Secondly, the existence of at least two positive solutions are obtained for a class of nonhomogeneous Robin problem involving criti-cal weighted Hardy-Sobolev exponents with boundary singularities by using Ekeland variational principle, Mountain pass lemma, the strong maximum principle, varia-tional methods and some analysis.Firstly, we consider the following semilinear elliptic problem where Ω is an open bounded domain in RN(N≥3) with C2boundary (?)Ω and O∈(?)Ω,O≤μa<(?),O≤μ<((?)-a)2with-μ=â–³4/(N-2)2, a≤b<a+1, p=p(a,b)=â–³N-2(1+a-b)/2N is the critical weighted Hardy-Sobolev exponent and p(a, a)=â–³N-2/2N is the critical Sobolev exponent, f∈C((?)×R+, R), v denotes the unit outward normal vector to boundary (?)Ω, α∈L∞((?)Ω) is a nonnegative function.In the paper, we give the following assumptions on the function f∈C((?)×R+, R): (f1)There exists a function k∈L∞(Ω),k(x)≤O such that Lim/tâ†'O+f(x.t)/t=k(x) uniformly for x∈(?).(f2) Lim/tâ†'+∞f(x,t)=O uniformly for x∈(?). Then, we can obtain the following main result:Theorem1. Let N>2a+3,O≤a<(?), O≤μ<min a≤b<a+1, α∈L∞(Ω),α≥O,(f1) and (f2) hold. Assume that‖α‖L∞(Ω)is small enough. Then, equation(P1)has at least a positive solution.Remark1.Theorem1extends the results obtained by [1]to a more general case(a,b≠0).Under the assumpation of (f1) and (f2)with a,b≠0used in[1],we get a positive solution by mountain pass lemma without(PS)conditions. Secondly, we consider the following semilinear elliptic problem where Ω is an open bounded domain in RN(N≥3)with C2boundary aQ and0∈(?)Ω,0≤a<(?), O≤μ<((?)-a)2with (?)(N-2)2/4, a≤n<a+1, p=p(a,6)å…¨2N/N-2(1+a-b) is the critical weighted Hardy-Sobolev exponent and p(a,b)(?)2N/N-2is the critical Sobolev exponent, f∈C((?)×R+,R), h∈L-/p-1(Ω) and h(x)≥O, v denotes the unit outward normal vector to boundary (?)Ω, α∈L∞((?)Ω)is a nonnegative function.We begin by assuming the following hypothesis on the function f∈C((?)×R+,R).(f3)There exists a constant d∈[O,p-2/2p)such that where F(x,t)=(?)tOf(x,s)ds.Then we can obtain the following main result: Theorem2. Let N>2a+3, O≤a<(?),O≤μ<min(μ*,(?)-a)2-1/4a≤b<a+1, α∈L∞(Ω), α≥O,(f1),(f2) and (f3) hold,h∈Lp-1/p(Ω) and, Assume thatare small enough. Then, equation (P2) has at least two positive solutions.Remark2. We prove an additional inhomogeneous perturbation of equation (Pi) can produce at least two positives solutions. With the help of (f2), we can prove the boundedness of (PS) sequence in H1(Ω|x|-2a). Providing the condition (f3), we can easily prove the energy functional of equation (P2) satisfies the (PS) condition.