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

Abstract

By variational methods and some analysis techniques, the multiplicity of positive solutions is obtained for a class of weighted quasilinear elliptic equations with critical Hardy-Sobolev exponents and concave-convex nonlinearities.

1. Introduction and Main Results

Let Ω be a smooth bounded domain in ℝ^𝑁(𝑁≥3) and 0∈Ω. We will study the multiplicity of positive solutions for the following quasilinear elliptic problem: −div|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2|𝑢|∇𝑢−𝜇^𝑝−2𝑢|𝑥|^𝑝(𝑎+1)=|𝑢|^{𝑝∗(𝑎,𝑏)−2}𝑢|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}|𝑢|+𝜆^𝑞−2𝑢|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}inΩ,𝑢=0on𝜕Ω, (1.1) where 𝜆>0, 1<𝑝<𝑁, 0≤𝜇<𝜇, 𝜇≜((𝑁−𝑝)/𝑝−𝑎)^𝑝, 0≤𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏,𝑑<𝑎+1, 1≤𝑞<𝑝, 𝑝^∗(𝑎,𝑑)≜𝑁𝑝/(𝑁−𝑝(𝑎+1−𝑑)) is the critical Sobolev-Hardy exponent. Note that 𝑝^∗(0,0)=𝑝^∗≜𝑁𝑝/(𝑁−𝑝) is the critical Sobolev exponent.

In this paper, 𝑊≜𝑊_𝑎^1,𝑝(Ω) denotes the space obtained as the completion of 𝐶^∞₀(Ω) with respect to the norm (∫_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥)^1/𝑝. The energy functional of (1.1) is defined on 𝑊 by 𝐽_𝜆(𝑢)=1𝑝_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝|𝑢|−𝜇^𝑝|𝑥|^𝑝(𝑎+1)1𝑑𝑥−𝑝^∗(𝑎,𝑏)_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}𝜆𝑑𝑥−𝑞_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}𝑑𝑥. (1.2) Then 𝐽_𝜆∈𝐶¹(𝑊,ℝ). 𝑢∈𝑊⧵{0} is said to be a solution of (1.1) if ⟨𝐽_𝜆′(𝑢),𝑣⟩=0 for all 𝑣∈𝑊 and a solution of (1.1) is a critical point of 𝐽_𝜆. By the standard elliptic regularity argument, we deduce that 𝑢∈𝐶¹(Ω⧵{0}).

Problem (1.1) is related to the following Hardy inequality [1]: _ℝ𝑁|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}𝑑𝑥^{𝑝/𝑝∗(𝑎,𝑏)}≤𝐶_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥,∀𝑢∈𝐶^∞₀ℝ^𝑁, (1.3) which is also called the (general or weighted) Hardy-Sobolev inequality. For the sharp constants and extremal functions, see [2, 3]. If 𝑏=𝑎+1, then 𝑝^∗(𝑎,𝑏)=𝑝 and the following (general or weighted) Hardy inequality holds [1, 4]: _ℝ𝑁|𝑢|^𝑝|𝑢|^𝑝(𝑎+1)1𝑑𝑥≤𝜇_ℝ^𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥,∀𝑢∈𝐶^∞₀ℝ^𝑁, (1.4) where 𝜇=((𝑁−𝑝)/𝑝−𝑎)^𝑝 is the best Hardy constant.

In the space 𝑊, we employ the following norm if 𝜇<𝜇: ‖𝑢‖=‖𝑢‖_𝑊≜_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝|𝑢|−𝜇^𝑝|𝑥|^𝑝(𝑎+1)𝑑𝑥^1/𝑝. (1.5) By (1.4) it is equivalent to the usual norm (∫_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥)^1/𝑝 of the space 𝑊. According to (1.4), we can define the following best constant for 0≤𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏<𝑎+1 and 𝜇<𝜇: 𝑆_{𝜇,𝑎,𝑏}(Ω)=inf_{{0}𝑢∈𝑊⧵}‖𝑢‖^𝑝∫_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}/|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}𝑑𝑥^{𝑝/𝑝∗(𝑎,𝑏)}. (1.6) From Kang [5, Lemma  2.2], 𝑆_{𝜇,𝑎,𝑏} is independent of Ω⊂ℝ^𝑁. Thus, we will simply denote that 𝑆_{𝜇,𝑎,𝑏}(Ω)=𝑆_{𝜇,𝑎,𝑏}(ℝ^𝑁)=𝑆_{𝜇,𝑎,𝑏}.

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|∇𝑢−𝜇^𝑝−2𝑢|𝑥|^𝑝=|𝑢|^{𝑝∗(𝑡)−2}𝑢|𝑥|^𝑡|𝑢|+𝜆^𝑞−2𝑢|𝑥|^𝑠inΩ,𝑢=0on𝜕Ω, (1.7) where 𝜆>0, 1<𝑝<𝑁, 0≤𝜇<𝜇=((𝑁−𝑝)/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|² with Sobolev-Hardy critical exponents (the case that 𝑡=0) has been considered. Some authors also studied the singular problems with Sobolev-Hardy 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 𝑆_𝜇,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^𝑁 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|∇𝑢−𝜇^𝑝−2𝑢|𝑥|^𝑝(𝑎+1)=|𝑢|^{𝑝∗(𝑎,𝑏)−2}𝑢|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}inℝ^𝑁⧵{0},𝑢>0inℝ^𝑁⧵{0},𝑢∈𝑊_𝑎^1,𝑝ℝ^𝑁, (1.8) where 𝑊_𝑎^1,𝑝(ℝ^𝑁) is the space obtained as the completion of 𝐶^∞₀(ℝ^𝑁) with respect to the norm (∫_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥)^1/𝑝. From [5, Lemma  2.2], we know (1.8) has a unique ground state solution 𝑈_𝑝,𝜇 satisfying 𝑈_𝑝,𝜇(1)=𝑝^∗(𝑎,𝑏)𝜇−𝜇𝑝^{1/(𝑝∗(𝑎,𝑏)−𝑝)} (1.9) and all ground states must be of the form 𝑈_𝜀(𝑥)=𝜀^{−(𝑁−𝑝)/𝑝−𝑎}𝑈_𝑝,𝜇(𝑥/𝜀) for some 𝜀>0, that is, 𝑆_{𝜇,𝑎,𝑏}=inf_{𝑢∈𝑊𝑎1,𝑝ℝ𝑁⧵{0}}∫_ℝ𝑁|∇𝑢|^𝑝−𝜇|𝑢|^𝑝/|𝑥|^𝑝(𝑎+1)𝑑𝑥∫_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}/|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}𝑑𝑥^{𝑝/𝑝∗(𝑎,𝑏)} (1.10) is achieved by 𝑈_𝜀. Moreover, 𝑈_𝑝,𝜇 is radially symmetric and possesses the following properties: lim_𝑟→0𝑟^𝛼(𝜇)𝑈_𝑝,𝜇(𝑟)=𝑐₁>0,lim_𝑟→0𝑟^𝛼(𝜇)+1𝑈^′_𝑝,𝜇(𝑟)=𝑐₁𝛼𝜇≥0,lim_𝑟→+∞𝑟^𝛽(𝜇)𝑈_𝑝,𝜇(𝑟)=𝑐₂>0,lim_𝑟→+∞𝑟^𝛽(𝜇)+1𝑈^′_𝑝,𝜇(𝑟)=𝑐₂𝛽𝜇>0, (1.11) where 𝑐_𝑖(𝑖=1,2) are positive constants and 𝛼(𝜇), 𝛽(𝜇) are the zeros of the function 𝑓(𝜏)=𝑝−1𝜏^𝑝−𝑁−𝑝(𝑎+1)𝜏^𝑝−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≤𝑈_𝑝,𝜇(𝑥)|𝑥|^{𝛼(𝜇)/𝛿}+|𝑥|^{𝛽(𝜇)/𝛿𝛿}≤𝑐₄(),𝛿≜𝑁−𝑝𝑎+1𝑝. (1.13)

Throughout this paper, let 𝑅₀ be the positive constant such that Ω⊂𝐵(0;𝑅₀), where 𝐵(0;𝑅₀)={𝑥∈ℝ^𝑁∶|𝑥|<𝑅₀}. By Hölder and Sobolev-Hardy inequalities, for all 𝑢∈𝑊, we obtain _Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}≤_{𝐵(0;𝑅0)}|𝑥|^{−𝑑𝑝∗(𝑎,𝑑)(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}_Ω|𝑢|^{𝑝∗(𝑎,𝑑)}|𝑥|^{𝑑𝑝∗(𝑎,𝑑)𝑞/𝑝∗(𝑎,𝑑)}≤𝑁𝜔_𝑁^𝑅0₀𝑟^{−𝑑𝑝∗(𝑎,𝑑)+𝑁−1}𝑑𝑟^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}𝑆_{𝜇,𝑎,𝑑}^{−(𝑞/𝑝)}‖𝑢‖^𝑞≤𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}𝑆_{𝜇,𝑎,𝑑}^{−(𝑞/𝑝)}‖𝑢‖^𝑞. (1.14) Set Λ₀=𝑝−𝑞𝑝^∗(𝑎,𝑏)−𝑞^{(𝑝−𝑞)/(𝑝∗(𝑎,𝑏)−𝑝)}𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗(𝑎,𝑏)−𝑞𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{−((𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑))}×𝑆_{𝜇,𝑎,𝑑}^𝑞/𝑝𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)(𝑝−𝑞)/𝑝(𝑝∗(𝑎,𝑏)−𝑝)}, (1.15) where 𝜔_𝑁=2𝜋^𝑁/2/𝑁Γ(𝑁/2) is the volume of the unit ball in ℝ^𝑁.

Furthermore, from 0≤𝑎<(𝑁−𝑝)/𝑝 and 𝑎≤𝑑<𝑎+1, we can deduce that 0<𝑝𝑁−𝑝𝑝−𝑎+𝑝𝑑=𝑁−𝑝(𝑎+1−𝑑)<𝑁, (1.16) which implies 𝑝^∗(𝑎,𝑑)=𝑝𝑁𝑁−𝑝(𝑎+1−𝑑)>𝑝,𝑁−𝑑𝑝^∗(𝑎,𝑑)=𝑁𝑝𝑁−𝑝(𝑎+1−𝑑)𝑁−𝑝𝑝−𝑎>0. (1.17) Combining these with 1≤𝑞<𝑝, we get Λ₀>0.

We are now ready to state our main results.

Theorem 1.1. Assume that 𝑁≥3, 0≤𝜇<𝜇, 0≤𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏,𝑑<𝑎+1, and 1≤𝑞<𝑝<𝑁. Then one has the following results.(i)If 𝜆∈(0,Λ₀), then (1.1) has at least one positive solution in 𝑊.(ii)If 𝜆∈(0,(𝑞/𝑝)Λ₀), then (1.1) has at least two positive solutions in 𝑊.

Remark 1.2. In [5], Kang considered (1.1) with 𝑝-sublinear perturbation of 𝑝≤𝑞<𝑝^∗(𝑎,𝑑). Via variational methods, he proved the existence of positive solutions of (1.1) when the parameters 𝑎,𝑏,𝑑,𝑝,𝑞,𝜆,𝜇 satisfy suitable conditions. But the existence of positive solutions for (1.1) involving the 𝑝-sublinear of 1≤𝑞<𝑝<𝑁 is not considered. In this paper, we will give a complement result.
This paper is organized as follows. In Sections 2 and 3, we give some preliminaries and some properties of Nehari manifold. In Section  4, we complete proofs of Theorem 1.1. At the end of this section, we explain some notations employed. In the following discussions, we will denote various positive constants as 𝐶, 𝐶_𝑖 and omit 𝑑𝑥 in the integral for convenience. We denote 𝐵(0;𝑅) as a ball centered at the origin with radius 𝑅, and 𝜔_𝑁=2𝜋^𝑁/2/𝑁Γ(𝑁/2) is the volume of the unit ball 𝐵(0;1) in ℝ^𝑁. We denote the norm in 𝐿^𝑟(Ω) by |⋅|_𝑟 for 1≤𝑟≤∞, and 𝐿^𝑟(Ω,|𝑥|^−𝑠),1≤𝑟<∞ is the closure of 𝐶^∞₀(Ω) with the norm |⋅|_{𝐿𝑟(Ω,|𝑥|−𝑠)}=(∫_Ω|𝑥|^−𝑠|⋅|^𝑟)^1/𝑟. 𝑊⁻¹ denoting the dual space of 𝑊. 𝑂(𝜀^𝑡) denotes |𝑂(𝜀^𝑡)/𝜀^𝑡|≤𝐶, and 𝑜(𝜀^𝑡) means |𝑜(𝜀^𝑡)/𝜀^𝑡|→0 as 𝜀→0. By 𝑜(1) we always mean it is a generic infinitesimal value.

2. Nehari Manifold

Since the functional 𝐽_𝜆 is not bounded from below on 𝑊, we will work on Nehari manifold. For 𝜆>0 we define 𝒩_𝜆=𝑢∈𝑊⧵{0}∶𝐽^′_𝜆(𝑢),𝑢=0. (2.1) We recall that any nonzero solutions of (1.1) belong to 𝒩_𝜆. Moreover, by definition, we have that 𝑢∈𝒩_𝜆 if and only if ‖𝑢‖≠0,‖𝑢‖^𝑝−1𝑝^∗(𝑎,𝑏)_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}−𝜆𝑞_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}=0. (2.2) The following result is concerned with the behavior of 𝐽_𝜆 on 𝒩_𝜆.

Lemma 2.1. 𝐽_𝜆 is coercive and bounded from below on 𝒩_𝜆.

Proof. If 𝑢∈𝒩_𝜆, then by (1.14) and (2.2), we get 𝐽_𝜆(𝑢)=𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝‖𝑢‖𝑎,𝑏^𝑝𝑝−𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)𝑞_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}≥𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝‖𝑢‖𝑎,𝑏^𝑝𝑝−𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)𝑞𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}×𝑆_{𝜇,𝑎,𝑑}^−𝑞/𝑝‖𝑢‖^𝑞. (2.3)(2.4) Since 0≤𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏,𝑑<𝑎+1 and 1≤𝑞<𝑝<𝑝^∗(𝑎,𝑏), it follows that 𝐽_𝜆 is coercive and bounded from below on 𝒩_𝜆.

Define 𝜓_𝜆∶𝑊→ℝ, by 𝜓_𝜆(𝑢)=⟨𝐽_𝜆′(𝑢),𝑢⟩, that is, 𝜓_𝜆(𝑢)=‖𝑢‖^𝑝−_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}−𝜆_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}. (2.5) Note that 𝜓_𝜆 is of class 𝐶¹ with 𝜓^′_𝜆(𝑢),𝑢=𝑝‖𝑢‖^𝑝−𝑝^∗(𝑎,𝑏)_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}−𝑞𝜆_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}. (2.6) Furthermore, if 𝑢∈𝒩_𝜆, then by (2.2), we have that 𝜓^′_𝜆(𝑢),𝑢=𝑝−𝑞‖𝑢‖^𝑝−𝑝^∗(𝑎,𝑏)−𝑞_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}=𝑝−𝑝^∗(𝑎,𝑏)‖𝑢‖^𝑝−𝑞−𝑝^∗(𝑎,𝑏)𝜆_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}. (2.7)(2.8) Now we split 𝒩_𝜆 into three sets: 𝒩⁺_𝜆=𝑢∈𝒩_𝜆∶𝜓^′_𝜆(𝑢),𝒩,𝑢>0_0𝜆=𝑢∈𝒩_𝜆∶𝜓^′_𝜆(𝑢),𝒩,𝑢=0_−𝜆=𝑢∈𝒩_𝜆∶𝜓^′_𝜆(𝑢),𝑢<0. (2.9)

The following result shows that minimizers on 𝒩_𝜆 are the “usual” critical points for 𝐽_𝜆.

Lemma 2.2. Suppose 𝑢₀ is a local minimizer of 𝐽_𝜆 on 𝒩_𝜆 and 𝑢₀∉𝒩⁰_𝜆. Then, 𝐽^′_𝜆(𝑢₀)=0 in 𝑊⁻¹.

Proof. See [30, Lemma  2.2].

Motivated by the above result, we will get conditions for 𝒩⁰_𝜆=∅.

Lemma 2.3. 𝒩⁰_𝜆=∅ for all 𝜆∈(0,Λ₀).

Proof. We argue by contradiction. Suppose that there exists a 𝜆∈(0,Λ₀) such that 𝒩⁰_𝜆≠∅. Let 𝑢∈𝒩⁰_𝜆 be arbitrary, then by (2.2), (2.7), and (2.8), we have that 0<‖𝑢‖^𝑝=𝑝^∗(𝑎,𝑏)−𝑞𝑝−𝑞_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)},0<‖𝑢‖^𝑝𝑝=𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)−𝑝_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}. (2.10) By (1.14), (2.10), and Sobolev-Hardy inequality, we get ‖𝑢‖≥𝑝−𝑞𝑝^∗(𝑎,𝑏)−𝑞^{1/(𝑝∗(𝑎,𝑏)−𝑝)}𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)/𝑝(𝑝∗(𝑎,𝑏)−𝑝)},‖𝑢‖≤𝜆𝑝^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)−𝑝^{1/(𝑝−𝑞)}𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)(𝑝−𝑞)}𝑆_{𝜇,𝑎,𝑑}^{−(𝑞/𝑝(𝑝−𝑞))}. (2.11) Hence we must have 𝜆≥𝑝−𝑞𝑝^∗(𝑎,𝑏)−𝑞^{(𝑝−𝑞)/(𝑝∗(𝑎,𝑏)−𝑝)}𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗(𝑎,𝑏)−𝑞𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{−((𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑))}×𝑆_{𝜇,𝑎,𝑑}^𝑞/𝑝𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)(𝑝−𝑞)/𝑝(𝑝∗(𝑎,𝑏)−𝑝)}=Λ₀, (2.12) which is a contradiction.

For each 𝑢∈𝑊⧵{0}, let 𝜏_max=𝑝−𝑞‖𝑢‖^𝑝𝑝^∗(𝑎,𝑏)−𝑞∫_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}/|𝑥|^{𝑏𝑝∗(𝑎,𝑏)1/(𝑝∗(𝑎,𝑏)−𝑝)}. (2.13)

Lemma 2.4. If 𝜆∈(0,Λ₀), then for each 𝑢∈𝑊⧵{0}, the set {𝜏𝑢∶𝜏>0} intersects 𝒩_𝜆 exactly twice. More specifically, there exist a unique 𝜏⁻=𝜏⁻(𝑢)>0 such that 𝜏⁻𝑢∈𝒩⁻_𝜆 and a unique 𝜏⁺=𝜏⁺(𝑢)>0 such that 𝜏⁺𝑢∈𝒩⁺_𝜆. Moreover, 𝜏⁺<𝜏_max<𝜏⁻ and 𝐽_𝜆𝜏⁺𝑢=inf_{0≤𝜏≤𝜏max}𝐽_𝜆(𝜏𝑢),𝐽_𝜆(𝜏⁻𝑢)=sup_{𝜏≥𝜏max}𝐽_𝜆(𝜏𝑢). (2.14)

Proof. The proof is similar to that of [29, Lemma  2.7] and is omitted.

We remark that by Lemma 2.3 we have, 𝒩_𝜆=𝒩⁺_𝜆∪𝒩⁻_𝜆 for all 𝜆∈(0,Λ₀). Furthermore, by Lemma 2.4 it follows that 𝒩⁺_𝜆 and 𝒩⁻_𝜆 are non-empty and by Lemma 2.1 we may define 𝛼_𝜆=inf_{𝑢∈𝒩𝜆}𝐽_𝜆(𝑢),𝛼⁺_𝜆=inf_{𝑢∈𝒩+𝜆}𝐽_𝜆(𝑢),𝛼⁻_𝜆=inf_{𝑢∈𝒩−𝜆}𝐽_𝜆(𝑢). (2.15)

Theorem 2.5. (i)If 𝜆∈(0,Λ₀), then one has 𝛼_𝜆≤𝛼⁺_𝜆<0.(ii)If 𝜆∈(0,(𝑞/𝑝)Λ₀), then 𝛼⁻_𝜆>𝑑₀ for some positive constant 𝑑₀.
In particular, for each 𝜆∈(0,(𝑞/𝑝)Λ₀), one has 𝛼_𝜆=𝛼⁺_𝜆<0<𝛼⁻_𝜆.

Proof. (i) Let 𝑢∈𝒩⁺_𝜆. By (2.7), 𝑝−𝑞𝑝^∗()‖𝑢‖𝑎,𝑏−𝑞^𝑝>_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}, (2.16) and so also using (2.2), 𝐽_𝜆(𝑢)=1𝑝−1𝑞‖𝑢‖^𝑝+1𝑞−1𝑝^∗(𝑎,𝑏)_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}<1𝑝−1𝑞+1𝑞−1𝑝^∗(𝑎,𝑏)𝑝−𝑞𝑝^∗()𝑢‖𝑎,𝑏−𝑞‖^𝑝𝑝=−𝑝−𝑞^∗(𝑎,𝑏)−𝑝𝑝𝑞𝑝^∗()‖𝑢‖𝑎,𝑏^𝑝<0. (2.17) Therefore, from the definition of 𝛼_𝜆 and 𝛼⁺_𝜆, we can deduce that 𝛼_𝜆≤𝛼⁺_𝜆<0.
(ii) Let 𝑢∈𝒩⁻_𝜆. By (2.7), 𝑝−𝑞𝑝^∗()‖𝑢‖𝑎,𝑏−𝑞^𝑝<_Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}. (2.18) Moreover, by Sobolev-Hardy inequality, _Ω|𝑢|^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}≤𝑆_{𝜇,𝑎,𝑏}^{−(𝑝∗(𝑎,𝑏)/𝑝)}‖𝑢‖^{𝑝∗(𝑎,𝑏)}. (2.19) This implies ‖𝑢‖>𝑝−𝑞𝑝^∗(𝑎,𝑏)−𝑞^{1/(𝑝∗(𝑎,𝑏)−𝑝)}𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)/𝑝(𝑝∗(𝑎,𝑏)−𝑝)}∀𝑢∈𝒩⁻_𝜆. (2.20) By (2.4) and (2.20), we have 𝐽_𝜆(𝑢)≥‖𝑢‖^𝑞𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝‖𝑢‖𝑎,𝑏^𝑝−𝑞𝑝−𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗()𝑞×𝑎,𝑏𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}𝑆_{𝜇,𝑎,𝑑}^{−(𝑞/𝑝)}>𝑝−𝑞𝑝^∗(𝑎,𝑏)−𝑞^{𝑞/(𝑝∗(𝑎,𝑏)−𝑝)}𝑆_{𝜇,𝑎,𝑏}^{𝑞𝑝∗(𝑎,𝑏)/𝑝(𝑝∗(𝑎,𝑏)−𝑝)}×𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗(𝑎,𝑏)𝑝𝑝−𝑞𝑝^∗(𝑎,𝑏)−𝑞^{(𝑝−𝑞)/(𝑝∗(𝑎,𝑏)−𝑝)}𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)(𝑝−𝑞)/𝑝(𝑝∗(𝑎,𝑏)−𝑝)}𝑝−𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)𝑞𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}𝑆_{𝜇,𝑎,𝑑}^{−(𝑞/𝑝)}=𝑞𝑝Λ₀−𝜆𝑝−𝑞𝑝^∗(𝑎,𝑏)−𝑞^{𝑞/(𝑝∗(𝑎,𝑏)−𝑝)}𝑆_{𝜇,𝑎,𝑏}^{𝑞𝑝∗(𝑎,𝑏)/𝑝(𝑝∗(𝑎,𝑏)−𝑝)}𝑝^∗(𝑎,𝑏)−𝑞𝑝^∗()𝑞×𝑎,𝑏𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}𝑆_{𝜇,𝑎,𝑑}^{−(𝑞/𝑝)}. (2.21) Thus, if 𝜆∈(0,(𝑞/𝑝)Λ₀), then 𝐽_𝜆(𝑢)>𝑑₀∀𝑢∈𝒩⁻_𝜆, (2.22) for some positive constant 𝑑₀.

Remark 2.6. If 𝜆∈(0,(𝑞/𝑝)Λ₀), then by Lemma 2.4 and Theorem 2.5, for each 𝑢∈𝑊⧵{0}, we can easily deduce that 𝜏⁻𝑢∈𝒩⁻_𝜆,𝐽_𝜆(𝜏⁻𝑢)=sup_𝜏≥0𝐽_𝜆(𝜏𝑢)≥𝛼⁻_𝜆>0. (2.23)

3. Proof of the Main Results

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values, and (PS)-conditions in 𝑊 for 𝐽_𝜆 as follows.

Definition 3.1. (i) For 𝑐∈ℝ, a sequence {𝑢_𝑛} is a (PS)_𝑐-sequence in 𝑊 for 𝐽_𝜆 if 𝐽_𝜆(𝑢_𝑛)=𝑐+𝑜(1) and 𝐽^′_𝜆(𝑢_𝑛)=𝑜(1) strongly in 𝑊⁻¹ as 𝑛→∞.
(ii) 𝑐∈ℝ is a (PS)-value in 𝑊 for 𝐽_𝜆 if there exists a (PS)_𝑐-sequence in 𝑊 for 𝐽_𝜆.
(iii) 𝐽_𝜆 satisfies the (PS)_c-condition in 𝑊 if any (PS)_𝑐-sequence {𝑢_𝑛} in 𝑊 for 𝐽_𝜆 contains a convergent subsequence.
Now, we use the Ekeland variational principle [35] to get the following results.

Proposition 3.2. (i) If 𝜆∈(0,Λ₀), then 𝐽_𝜆 has a (PS)_𝛼𝜆-sequence {𝑢_𝑛}⊂𝒩_𝜆.
(ii) If 𝜆∈(0,(𝑞/𝑝)Λ₀), then 𝐽_𝜆 has a (PS)_𝛼⁻_𝜆-sequence {𝑢_𝑛}⊂𝒩⁻_𝜆.

Proof. The proof is similar to [29, Proposition  3.3] and the details are omitted.

Now, we establish the existence of a local minimum for 𝐽_𝜆 on 𝒩_𝜆.

Theorem 3.3. Assume that 𝑁≥3, 0≤𝜇<𝜇, 0≤𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏,𝑑<𝑎+1, and 1≤𝑞<𝑝<𝑁. If 𝜆∈(0,Λ₀), then there exists 𝑢_𝜆∈𝒩⁺_𝜆 such that(i)𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆=𝛼⁺_𝜆,(ii)𝑢_𝜆 is a positive solution of (1.1),(iii)‖𝑢_𝜆‖→0 as 𝜆→0⁺.

Proof. By Proposition 3.2(i), there exists a minimizing sequence {𝑢_𝑛}⊂𝒩_𝜆 such that 𝐽_𝜆𝑢_𝑛=𝛼_𝜆+𝑜(1),𝐽^′_𝜆𝑢_𝑛=𝑜(1)in𝑊⁻¹. (3.1) Since 𝐽_𝜆 is coercive on 𝒩_𝜆 (see Lemma 2.1), we get that {𝑢_𝑛} is bounded in 𝑊. From [5, Lemma  2.1], we deduce that the embedding 𝑊↪𝐿^𝑟(Ω,|𝑥|^{−𝑑𝑝∗(𝑎,𝑑)}) is compact for 1≤𝑟<𝑝^∗(𝑎,𝑑). Thus, there exists 𝑢_𝜆∈𝑊, passing to a subsequence if necessary, using similar arguments found in [27, 36], then one can get that as 𝑛→∞𝑢_𝑛⇀𝑢_𝜆𝑢weaklyin𝑊,_𝑛⟶𝑢_𝜆stronglyin𝐿^𝑞Ω,|𝑥|^{−𝑑𝑝∗(𝑎,𝑑)}𝑢for1≤𝑞<𝑝,_𝑛⟶𝑢_𝜆a.e.inΩ,∇𝑢_𝑛⟶∇𝑢_𝜆𝑢a.e.inΩ,_𝑛|𝑥|^𝑎+1⇀𝑢_𝜆|𝑥|^𝑎+1weaklyin𝐿^𝑝(Ω),_Ω𝑢_𝑛^{𝑝∗(𝑎,𝑏)−2}𝑢_𝑛|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}𝑣⟶_Ω𝑢_𝜆^{𝑝∗(𝑎,𝑏)−2}𝑢_𝜆|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}𝑣,∀𝑣∈𝑊. (3.2) Consequently, passing to the limit in ⟨𝐽_𝜆′(𝑢_𝑛),𝑣⟩, by (3.1) and (3.2), as 𝑛→∞, we have _Ω∇𝑢_𝜆^𝑝−2∇𝑢_𝜆∇𝑣|𝑥|^𝑎𝑝𝑢−𝜇^𝜆𝑝−2𝑢_𝜆𝑣|𝑥|^𝑝(𝑎+1)−_Ω𝑢_𝜆^{𝑝∗(𝑎,𝑏)−2}𝑢_𝜆𝑣|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}−𝜆_Ω𝑢_𝜆^𝑞−2𝑢_𝜆𝑣|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}=0, (3.3) for all 𝑣∈𝑊. That is, ⟨𝐽_𝜆′(𝑢_𝜆),𝑣⟩=0. Thus 𝑢_𝜆 is a weak solution of (1.1). Furthermore, from 𝑢_𝑛∈𝒩_𝜆 and (2.3), we deduce that 𝜆_Ω𝑢_𝑛^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}=𝑞𝑝^∗(𝑎,𝑏)−𝑝𝑝𝑝^∗()𝑢𝑎,𝑏−𝑞^𝑛𝑝−𝑝^∗(𝑎,𝑏)𝑞𝑝^∗()𝐽𝑎,𝑏−𝑞^𝜆𝑢_𝑛. (3.4) Let 𝑛→∞ in (3.4), by (3.1) and (3.2), and since 𝛼_𝜆<0 by (i) of Theorem 2.5, we get 𝜆_Ω𝑢_𝜆^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}𝑝≥−^∗(𝑎,𝑏)𝑞𝑝^∗()𝛼𝑎,𝑏−𝑞^𝜆>0. (3.5) Thus 𝑢_𝜆≢0, and since 𝐽_𝜆′(𝑢_𝜆)=0, it follows that 𝑢_𝜆∈𝒩_𝜆 and, in particular, 𝐽_𝜆(𝑢_𝜆)≥𝛼_𝜆.
Next, we will show, up to a subsequence, that 𝑢_𝑛→𝑢_𝜆 strongly in 𝑊 and 𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆. From the fact 𝑢_𝑛,𝑢∈𝒩_𝜆, (2.3) and the Fatou’s lemma, it follows that 𝛼_𝜆≤𝐽_𝜆𝑢_𝜆=𝑝^∗(𝑎,𝑏)−𝑝𝑝𝑝^∗()𝑢𝑎,𝑏^𝜆𝑝𝑝−𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)𝑞_Ω𝑢_𝜆^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}≤liminf_𝑛→∞𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑢𝑎,𝑏^𝑛𝑝𝑝−𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)𝑞_Ω𝑢_𝑛^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}=liminf_𝑛→∞𝐽_𝜆𝑢_𝑛=𝛼_𝜆, (3.6) which implies that 𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆 and lim_𝑛→∞‖𝑢_𝑛‖^𝑝=‖𝑢_𝜆‖^𝑝. Standard argument shows that 𝑢_𝑛→𝑢_𝜆 strongly in 𝑊. Moreover, 𝑢_𝜆∈𝒩⁺_𝜆. Otherwise, if 𝑢_𝜆∈𝒩⁻_𝜆, by Lemma 2.4, there exist unique 𝜏⁺_𝜆 and 𝜏⁻_𝜆 such that 𝜏⁺_𝜆𝑢_𝜆∈𝒩⁺_𝜆, 𝜏⁻_𝜆𝑢_𝜆∈𝒩⁻_𝜆 and 𝜏⁺_𝜆<𝜏⁻_𝜆=1. Since 𝑑𝐽𝑑𝜏^𝜆𝜏⁺_𝜆𝑢_𝜆𝑑=0,²𝑑𝜏²𝐽_𝜆𝜏⁺_𝜆𝑢_𝜆>0, (3.7) there exists 𝜏∈(𝜏⁺_𝜆,𝜏⁻_𝜆) such that 𝐽_𝜆(𝜏⁺_𝜆𝑢_𝜆)<𝐽_𝜆(𝜏𝑢_𝜆). By Lemma 2.4 we get that 𝐽_𝜆𝜏⁺_𝜆𝑢_𝜆<𝐽_𝜆𝜏𝑢_𝜆≤𝐽_𝜆𝜏⁻_𝜆𝑢_𝜆=𝐽_𝜆𝑢_𝜆, (3.8) which is a contradiction. Since 𝐽_𝜆(𝑢_𝜆)=𝐽_𝜆(|𝑢_𝜆|) and |𝑢_𝜆|∈𝒩⁺_𝜆, by Lemma 2.2, we may assume that 𝑢_𝜆 is a nontrivial nonnegative solution of (1.1). By [5, Lemma  2.3], it follows that 𝑢_𝜆>0 in Ω. Finally, by (1.14) and (2.8), we obtain 𝑢_𝜆^𝑝−𝑞𝑝<𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)−𝑝𝑁𝜔_𝑁𝑅^{𝑁−𝑑𝑝∗}₀^(𝑎,𝑑)𝑁−𝑑𝑝^∗(𝑎,𝑑)^{(𝑝∗(𝑎,𝑑)−𝑞)/𝑝∗(𝑎,𝑑)}𝑆_{𝜇,𝑎,𝑑}^{−(𝑞/𝑝)}, (3.9) which implies that ‖𝑢_𝜆‖→0 as 𝜆→0⁺.

Next, we will establish the existence of the second positive solution of (1.1) by proving that 𝐽_𝜆 satisfies the (PS)_𝛼⁻_𝜆-condition.

Lemma 3.4. Let {𝑢_𝑛} be a bounded sequence in 𝑊. If {𝑢_𝑛} is a (PS)_𝑐-sequence for 𝐽_𝜆 with 𝑝𝑐∈0,^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}, (3.10) then there exists a subsequence of {𝑢_𝑛} converging weakly to a nonzero solution of (1.1).

Proof. Let {𝑢_𝑛}⊂𝑊 be a (PS)_𝑐-sequence for 𝐽_𝜆 with 𝑐∈(0,((𝑝^∗(𝑎,𝑏)−𝑝)/𝑝^∗(𝑎,𝑏)𝑝)(𝑆_{𝜇,𝑎,𝑏})^{𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}). Since {𝑢_𝑛} is bounded in 𝑊 and the embedding 𝑊↪𝐿^𝑟(Ω,|𝑥|^{−𝑑𝑝∗(𝑎,𝑑)}) is compact for 1≤𝑟<𝑝^∗(𝑎,𝑑) (see [5, Lemma  2.1]), thus passing to a subsequence if necessary, we may assume that as 𝑛→∞𝑢_𝑛⇀𝑢₀𝑢weaklyin𝑊,_𝑛⇀𝑢₀weaklyin𝐿^{𝑝∗(𝑎,𝑏)}Ω,|𝑥|^{−𝑏𝑝∗(𝑎,𝑏)}𝑢_𝑛⟶𝑢₀stronglyin𝐿^𝑞Ω,|𝑥|^{−𝑑𝑝∗(𝑎,𝑑)}𝑢for1≤𝑞<𝑝,_𝑛⟶𝑢₀a.e.inΩ. (3.11) Using the same argument in Theorem 3.3, we deduce that 𝐽_𝜆′(𝑢₀)=0 and 𝜆_Ω𝑢_𝑛^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}=𝜆_Ω𝑢₀^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}+𝑜(1). (3.12)
Next we verify that 𝑢₀≢0. Arguing by contradiction, we assume 𝑢₀≡0. Set 𝑙=lim_𝑛→∞Ω𝑢_𝑛^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}. (3.13) Since 𝐽_𝜆′(𝑢₀)=0 and {𝑢_𝑛} is bounded in 𝑊, then by (3.12), we can deduce that 0=lim_𝑛→∞𝐽^′_𝜆𝑢_𝑛,𝑢_𝑛=lim_𝑛→∞𝑢_𝑛^𝑝−_Ω𝑢_𝑛^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}=lim_𝑛→∞𝑢_𝑛^𝑝−𝑙, (3.14) that is, lim_𝑛→∞𝑢_𝑛^𝑝=𝑙. (3.15)
If 𝑙=0, then by (3.12)–(3.15), we get 𝑐=lim_𝑛→∞𝐽_𝜆𝑢_𝑛=lim_𝑛→∞1𝑝𝑢_𝑛^𝑝−_Ω𝑢_𝑛^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}−𝜆_Ω𝑢_𝑛^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}=0, (3.16) which contradicts 𝑐>0. Thus we conclude that 𝑙>0. Furthermore, the Sobolev-Hardy inequality implies that 𝑢_𝑛^𝑝≥𝑆_{𝜇,𝑎,𝑏Ω}𝑢_𝑛^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)𝑝/𝑝∗(𝑎,𝑏)}. (3.17) Then as 𝑛→∞, we have 𝑙=lim_𝑛→∞‖𝑢_𝑛‖^𝑝≥𝑆_{𝜇,𝑎,𝑏}𝑙^{𝑝/𝑝∗(𝑎,𝑏)}, which implies that 𝑙≥𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}. (3.18) Hence, from (3.12)–(3.18) we get 𝑐=lim_𝑛→∞𝐽_𝜆𝑢_𝑛=1𝑝lim_𝑛→∞𝑢_𝑛^𝑝−1𝑝^∗(𝑎,𝑏)lim_𝑛→∞Ω𝑢_𝑛^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}−𝜆𝑞lim_𝑛→∞Ω𝑢_𝑛^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}=1𝑝−1𝑝^∗()𝑙≥𝑝𝑎,𝑏_∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}. (3.19) This contradicts the definition of 𝑐. Therefore, 𝑢₀ is a nontrivial solution of (1.1).

Lemma 3.5. If 𝑁≥3, 0≤𝜇<𝜇, 0≤𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏,𝑑<𝑎+1, and 1≤𝑞<𝑝<𝑁, then for any 𝜆>0, there exists 𝑣_𝜆∈𝑊 such that sup_𝜏≥0𝐽_𝜆𝜏𝑣_𝜆<𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}. (3.20)
In particular, 𝛼⁻_𝜆<((𝑝^∗(𝑎,𝑏)−𝑝)/𝑝^∗(𝑎,𝑏)𝑝)(𝑆_{𝜇,𝑎,𝑏})^{𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)} for all 𝜆∈(0,Λ₀).

Proof. Let 𝑈_𝑝,𝜇 be a ground state solution of (1.8), 𝜌>0 small enough such that 𝐵(0;𝜌)⊂Ω, 𝜂∈𝐶^∞₀(Ω), 0≤𝜂(𝑥)≤1, 𝜂(𝑥)=1 for |𝑥|<𝜌/2,𝜂(𝑥)=0 for |𝑥|≥𝜌. Set 𝑈_𝜀(𝑥)=𝜀^{−((𝑁−𝑝)/𝑝)−𝑎}𝑈_𝑝,𝜇(𝑥/𝜀) and 𝑢_𝜀(𝑥)=𝜂(𝑥)𝑈_𝜀(𝑥),𝜀>0. Then, following the same lines as in [5], we get the following estimates as 𝜀→0: 𝑢_𝜀^𝑝=𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}+𝑂𝜀^{𝛽(𝜇)𝑝+𝑝(𝑎+1)−𝑁}, (3.21)_Ω𝑢_𝜀^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}=𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}+𝑂𝜀^{(𝛽(𝜇)+𝑏)𝑝∗(𝑎,𝑏)−𝑁}, (3.22)_Ω𝑢_𝜀^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}≥𝐶𝜀^{𝑁−𝑑𝑝∗(𝑎,𝑑)−𝑞𝛿},𝑁−𝑑𝑝^∗(𝑎,𝑑)𝛽𝜇<𝑞<𝑝^∗(𝑎,𝑑),𝐶𝜀^{𝑞(𝛽(𝜇)−𝛿)}|ln𝜀|,𝑞=𝑁−𝑑𝑝^∗(𝑎,𝑑)𝛽𝜇,𝐶𝜀^{𝑞(𝛽(𝜇)−𝛿)},1≤𝑞<𝑁−𝑑𝑝^∗(𝑎,𝑑)𝛽𝜇, (3.23) where 𝑏(𝜇) is given in the introduction satisfying 𝛿=(𝑁−𝑝(𝑎+1))/𝑝<𝑏(𝜇)<(𝑁−𝑝(𝑎+1))/(𝑝−1).
Now we consider the following functions: 𝑔(𝜏)=𝐽_𝜆𝜏𝑢_𝜀=𝜏^𝑝𝑝𝑢_𝜀^𝑝−𝜏^{𝑝∗(𝑎,𝑏)}𝑝^∗(𝑎,𝑏)_Ω𝑢_𝜀^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}𝜏−𝜆^𝑞𝑞_Ω𝑢_𝜀^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)},𝑔(𝜏)=𝜏^𝑝𝑝𝑢_𝜀^𝑝−𝜏^{𝑝∗(𝑎,𝑏)}𝑝^∗(𝑎,𝑏)_Ω𝑢_𝜀^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)}. (3.24)
Using the definitions of 𝑔 and 𝑢_𝜀, we get 𝑔(𝜏)=𝐽_𝜆𝜏𝑢_𝜀≤𝜏^𝑝𝑝𝑢_𝜀^𝑝,∀𝜏≥0,𝜆>0. (3.25) Combining this with (3.21), let 𝜀∈(0,1), then there exists 𝜏₀∈(0,1) independent of 𝜀 such that sup_{0≤𝜏≤𝜏0}𝑔(𝜏)<𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)},∀𝜆>0,∀𝜀∈(0,1). (3.26) On the other hand, by the fact for 𝐵₁,𝐵₂>0max_𝜏≥0𝜏^𝑝𝑝𝐵₁−𝜏^{𝑝∗(𝑎,𝑏)}𝑝^∗()𝐵𝑎,𝑏²=𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝐵𝑎,𝑏^{1𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}𝐵₂^{−(𝑝/(𝑝∗(𝑎,𝑏)−𝑝))}, (3.27) and by (3.21) and (3.22), we can get that max_𝜏≥0𝑔(𝜏)=𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑢𝑎,𝑏^{𝜀𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}_Ω𝑢_𝜀^{𝑝∗(𝑎,𝑏)}|𝑥|^{𝑏𝑝∗(𝑎,𝑏)−(𝑝/(𝑝∗(𝑎,𝑏)−𝑝))}=𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}+𝑂𝜀^{𝛽(𝜇)𝑝+𝑝(𝑎+1)−𝑁𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}×𝑆_{𝜇,𝑎,𝑏}^{𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}+𝑂𝜀^{(𝛽(𝜇)+𝑏)𝑝∗(𝑎,𝑏)−𝑁−(𝑝/(𝑝∗(𝑎,𝑏)−𝑝))}=𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}+𝑂𝜀^{𝛽(𝜇)𝑝+𝑝(𝑎+1)−𝑁}. (3.28) Hence as 𝜆>0, 1≤𝑞<𝑝, by (3.28) we have that sup_𝜏≥𝜏0𝑔(𝜏)=sup_𝜏≥𝜏0𝑔(𝜏)𝜏−𝜆^𝑞𝑞_Ω𝑢_𝜀^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}≤𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}+𝑂𝜀^{𝛽(𝜇)𝑝+𝑝(𝑎+1)−𝑁}𝜏−𝜆^𝑞0𝑞_Ω𝑢_𝜀^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}. (3.29)
(i) If 1≤𝑞<(𝑁−𝑑𝑝^∗(𝑎,𝑑))/𝛽(𝜇), then by (3.23) we have that _Ω𝑢_𝜀^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}≥𝐶𝜀^{𝑞(𝛽(𝜇)−𝛿)} (3.30) and since 𝑏(𝜇)>𝛿=(𝑁−𝑝(𝑎+1))/𝑝, then 𝛽𝜇𝑝+𝑝(𝑎+1)−𝑁=𝑝𝛽𝜇−𝛿>𝑞𝛽𝜇−𝛿. (3.31) Combining this with (3.26) and (3.29), for any 𝜆>0, we can choose 𝜀_𝜆 small enough such that sup_𝜏≥0𝐽_𝜆𝜏𝑢_𝜀𝜆<𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}. (3.32)
(ii) If (𝑁−𝑑𝑝^∗(𝑎,𝑑))/𝛽(𝜇)≤𝑞<𝑝, then by (3.23) and 𝑏(𝜇)>𝛿=(𝑁−𝑝(𝑎+1))/𝑝 we have that _Ω𝑢_𝜀^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}≥𝐶𝜀^{𝑁−𝑑𝑝∗(𝑎,𝑑)−𝑞𝛿},𝑁−𝑑𝑝^∗(𝑎,𝑑)𝛽𝜇<𝑞<𝑝^∗(𝑎,𝑑),𝐶𝜀^{𝑞(𝛽(𝜇)−𝛿)}|ln𝜀|,𝑞=𝑁−𝑑𝑝^∗(𝑎,𝑑)𝛽𝜇,𝑁−𝑑𝑝^∗(𝑎,𝑑)−𝑞𝛿≤𝑞𝛽𝜇−𝑞𝛿<𝑝𝛽𝜇−𝛿=𝛽𝜇𝑝+𝑝(𝑎+1)−𝑁. (3.33) Combining this with (3.26) and (3.29), for any 𝜆>0, we can choose 𝜀_𝜆 small enough such that sup_𝜏≥0𝐽_𝜆𝜏𝑢_𝜀𝜆<𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}. (3.34) From (i) and (ii), (3.20) holds by taking 𝑣_𝜆=𝑢_𝜀𝜆.
From Lemma 2.4, the definition of 𝛼⁻_𝜆, and (3.20), for any 𝜆∈(0,Λ₀), we obtain that there exists 𝜏⁻_𝜆>0 such that 𝜏⁻_𝜆𝑣_𝜆∈𝒩⁻_𝜆 and 𝛼⁻_𝜆≤𝐽_𝜆𝜏⁻_𝜆𝑣_𝜆≤sup_𝑡≥0𝐽_𝜆𝜏𝑣_𝜆<𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝𝑆𝑎,𝑏^{𝜇,𝑎,𝑏𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}. (3.35) The proof is thus complete.

Now, we establish the existence of a local minimum of 𝐽_𝜆 on 𝒩⁻_𝜆.

Theorem 3.6. Assume that 𝑁≥3, 0≤𝜇<𝜇, 0≤𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏,𝑑<𝑎+1 and 1≤𝑞<𝑝<𝑁. If 𝜆∈(0,(𝑞/𝑝)Λ₀), then there exists 𝑈_𝜆∈𝒩⁻_𝜆 such that(i)𝐽_𝜆(𝑈_𝜆)=𝛼⁻_𝜆,(ii)𝑈_𝜆 is a positive solution of (1.1).

Proof. If 𝜆∈(0,(𝑞/𝑝)Λ₀), then by Theorem 2.5 (ii), Proposition 3.2 (ii), and Lemma 3.5, there exists a (PS)_𝛼⁻_𝜆-sequence {𝑢_𝑛}⊂𝒩⁻_𝜆 in 𝑊 for 𝐽_𝜆 with 𝛼⁻_𝜆∈(0,((𝑝^∗(𝑎,𝑏)−𝑝)/𝑝^∗(𝑎,𝑏)𝑝)(𝑆_{𝜇,𝑎,𝑏})^{𝑝∗(𝑎,𝑏)/(𝑝∗(𝑎,𝑏)−𝑝)}). Since 𝐽_𝜆 is coercive on 𝒩_𝜆 (see Lemma 2.1), we get that {𝑢_𝑛} is bounded in 𝑊. From Lemma 3.4, there exists a subsequence still denoted by {𝑢_𝑛} and a nontrivial solution 𝑈_𝜆∈𝑊 of (1.1) such that 𝑢_𝑛⇀𝑈_𝜆 weakly in 𝑊.
First, we prove that 𝑈_𝜆∈𝒩⁻_𝜆. Arguing by contradiction, we assume 𝑈_𝜆∈𝒩⁺_𝜆. Since 𝒩⁻_𝜆 is closed in 𝑊, we have ‖𝑈_𝜆‖<liminf_𝑛→∞‖𝑢_𝑛‖. Thus, by Lemma 2.4, there exists a unique 𝜏⁻_𝜆 such that 𝜏⁻_𝜆𝑈_𝜆∈𝒩⁻_𝜆. If 𝑢∈𝒩_𝜆, then it is easy to see that 𝐽_𝜆(𝑢)=𝑝^∗(𝑎,𝑏)−𝑝𝑝^∗()𝑝‖𝑢‖𝑎,𝑏^𝑝𝑝−𝜆^∗(𝑎,𝑏)−𝑞𝑝^∗(𝑎,𝑏)𝑞_Ω|𝑢|^𝑞|𝑥|^{𝑑𝑝∗(𝑎,𝑑)}. (3.36) From Remark 2.6, 𝑢_𝑛∈𝒩⁻_𝜆, ‖𝑈_𝜆‖<liminf_𝑛→∞‖𝑢_𝑛‖, and (3.36), we can deduce that 𝛼⁻_𝜆≤𝐽_𝜆𝜏⁻_𝜆𝑈_𝜆<lim_𝑛→∞𝐽_𝜆𝜏⁻_𝜆𝑢_𝑛≤lim_𝑛→∞𝐽_𝜆𝑢_𝑛=𝛼⁻_𝜆. (3.37) This is a contradiction. Thus, 𝑈_𝜆∈𝒩⁻_𝜆.
Next, by the same argument as that in Theorem 3.3, we get that 𝑢_𝑛→𝑈_𝜆 strongly in 𝑊 and 𝐽_𝜆(𝑈_𝜆)=𝛼⁻_𝜆>0 for all 𝜆∈(0,(𝑞/𝑝)Λ₀). Since 𝐽_𝜆(𝑈_𝜆)=𝐽_𝜆(|𝑈_𝜆|) and |𝑈_𝜆|∈𝒩⁻_𝜆, by Lemma 2.2, we may assume that 𝑈_𝜆 is a nontrivial nonnegative solution of (1.1). Finally, by [5, Lemma  2.3], we obtain that 𝑈_𝜆 is a positive solution of (1.1).

Now, we complete the proof of Theorem 1.1. The part (i) of Theorem 1.1 immediately follows from Theorem 3.3. When 0<𝜆<(𝑞/𝑝)Λ₀<Λ₀, by Theorems 3.3 and 3.6, we obtain (1.1) has two positive solutions 𝑢_𝜆 and 𝑈_𝜆 such that 𝑢_𝜆∈𝒩⁺_𝜆, 𝑈_𝜆∈𝒩⁻_𝜆. Since 𝒩⁺_𝜆∩𝒩⁻_𝜆=∅, this implies that 𝑢_𝜆 and 𝑈_𝜆 are distinct. This completes the proof of Theorem 1.1.