Multiplicity of Positive Solutions for Weighted Quasilinear Elliptic Equations Involving Critical Hardy-Sobolev Exponents and Concave-Convex Nonlinearities

and Applied Analysis 3 When a 0, we set s dp∗ 0, d and t bp∗ 0, b , then 1.1 is equivalent to the following quasilinear elliptic equations: −div ( |∇u|p−2∇u ) − μ |u| p−2u |x| |u|p t −2u |x| λ |u|q−2u |x| in Ω, u 0 on ∂Ω, 1.7 where λ > 0, 1 < p < N, 0 ≤ μ < μ N − p /p , 0 ≤ s, t < p, 1 ≤ q < p and p∗ t p N − t / N − p . Such kind of problem relative with 1.7 has been extensively studied by many authors. When p 2, people have paid much attention to the existence of solutions for singular elliptic problems see 6–16 and their references therein , besides, in the most of these papers, the operator −Δ − μ/|x| with Sobolev-Hardy critical exponents the case that t 0 has been considered. Some authors also studied the singular problems with SobolevHardy critical exponents the case that t / 0 see 17–22 and their references therein . In 23, 24 , the authors deal with doubly-critical exponents. When p / 2. The quasilinear problems related to Hardy inequality and Sobolev-Hardy inequality have been studied by some authors 25–32 . Here we recall the work in 25 , where the extremal functions for the best Sobolev constant Sμ,0,0 were studied. The results can be employed to study the problems with critical Sobolev exponents and Hard terms, see 25, 28 . In 26 it is investigated in R a quasilinear elliptic equation involving doubly critical exponents by the concentration compactness principle 33, 34 . We should note that the nonlinearities of problems studied in 11–14, 26, 28, 31 are all not sublinear or p-sublinear near the origin. To the best of our knowledge, there are few results of problem 1.7 with nonlinearities being p-sublinear near the origin for 1 < p < N. We are only aware of the works 20, 30, 32 which studied the existence and multiplicity of positive solution of problem 1.7 with 1 ≤ q < p < N. In this paper, we study 1.1 and extend the results of 20, 30, 32 to the case a/ 0 and 1 ≤ q < p < N. For 0 ≤ a < N − p /p, a ≤ b < a 1, and 0 ≤ μ < μ, consider the following limiting problem: −div ( |x|−ap|∇u|p−2∇u ) − μ |u| p−2u |x| a 1 |u|p a,b −2u |x|bp a,b in R \ {0}, u > 0 in R \ {0}, u ∈ W a ( R N ) , 1.8 where W a R is the space obtained as the completion of C∞ 0 R N with respect to the norm ∫ RN |x|−ap|∇u|pdx . From 5, Lemma 2.2 , we know 1.8 has a unique ground state solution Up,μ satisfying Up,μ 1 ( p∗ a, b ( μ − μ) p )1/ p∗ a,b −p 1.9 4 Abstract and Applied Analysis and all ground states must be of the form Ũε x ε− N−p /p−aUp,μ x/ε for some ε > 0, that is, Sμ,a,b inf u∈W a RN \{0} ∫ RN ( |∇u| − μ|u|/|x| a 1 ) dx (∫ Ω |u| ∗ a,b /|x|bp a,b dx )p/p∗ a,b 1.10 is achieved by Ũε. Moreover, Up,μ is radially symmetric and possesses the following properties: lim r→ 0 r μ Up,μ r c1 > 0, lim r→ 0 r μ 1 ∣∣U′ p,μ r ∣∣ c1α ( μ ) ≥ 0, lim r→ ∞ r μ Up,μ r c2 > 0, lim r→ ∞ r μ 1 ∣∣U′ p,μ r ∣∣ c2β ( μ ) > 0, 1.11 where ci i 1, 2 are positive constants and α μ , β μ are the zeros of the function f τ ( p − 1)τp − (N − p a 1 )τp−1 μ, τ ≥ 0, 0 ≤ μ < μ, 1.12 which satisfy 0 ≤ α μ < N − p a 1 /p < β μ < N − p a 1 / p − 1 . Furthermore, there exist the positive constants c3 c3 μ, p, a, b and c4 c4 μ, p, a, b such that c3 ≤ Up,μ x ( |x| μ /δ |x| μ /δ )δ ≤ c4, δ N − p a 1 p . 1.13 Throughout this paper, let R0 be the positive constant such that Ω ⊂ B 0;R0 , where B 0;R0 {x ∈ R : |x| < R0}. By Hölder and Sobolev-Hardy inequalities, for all u ∈ W , we obtain