Quasilinear Elliptic Equations Involving Two Critical Hardy-Sobolev Exponents

In this project, we consider the following quasilinear elliptic equation (Eλ ) : −Δpu −μ | u |p−2 u | x |p = u p* (t1 )−2 u | x −ξ1 |t1 + u p* (t2 )−2 u | x −ξ 2 |t2 +λ u q−2 u | x −ξ3 |t3 , in Ω, u ∈W0 1, p (Ω), ⎧ ⎨ ⎪ ⎩ ⎪ where ) 3 ( ≥ ⊂ Ω N RN is a bounded domain with smooth boundary Ω ∂ , 0, ξ i ∈ Ω, 0 ≤ ti ＜ p, i =1, 2,3, Δpu = div(|∇u |p−2 ∇u), 1＜ p ＜ N, 0 ≤μ ＜μ = N − p p ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ p , λ ＞ 0, 1＜ q ＜ p, for all 0 ≤ t ＜ p, p*(t) = p(N − t) N − p is the so-called critical Hardy- Sobolev exponent. When t = 0 , p*(0) = p* = Np N − p is the critical Sobolev exponent. By variational methods and analysis techniques, we hope to obtain the existence and multiplicity results of positive solutions to the equation (Eλ ) .

Project ID:PA10507-0641
External Project ID:MOST105-2115-M182-002