Multiplicity Results for p-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

Abstract

The multiple results of positive solutions for the following quasilinear elliptic equation: −Δ_𝑝𝑢=𝜆𝑓(𝑥)|𝑢|^𝑞−2𝑢+𝑔(𝑥)|𝑢|^𝑝∗−2𝑢 in Ω, 𝑢=0 on 𝜕Ω, are established. Here, 0∈Ω is a bounded smooth domain in ℝ^𝑁,Δ_𝑝 denotes the 𝑝-Laplacian operator, 1≤𝑞<𝑝<𝑁,𝑝^∗=𝑁𝑝/(𝑁−𝑝),𝜆 is a positive real parameter, and 𝑓,𝑔 are continuous functions on Ω which are somewhere positive but which may change sign on Ω. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.

1. Introduction

In this paper, we study the multiple results of positive solutions for the following quasilinear elliptic equation:

−Δ_𝑝𝑢=𝜆𝑓(𝑥)𝑢|𝑢=|0^𝑞−2𝑢+𝑔(𝑥)|𝑢|^𝑝∗−2𝑢inΩ,on𝜕Ω,(𝐸_{𝜆𝑓,𝑔}) where 𝜆>0, Δ_𝑝𝑢=div(|∇𝑢|^𝑝−2∇𝑢) is the 𝑝-Laplacian, 0∈Ω is a bounded domain in ℝ^𝑁 with smooth boundary 𝜕Ω, 1<𝑞<𝑝<𝑁, 𝑝^∗=𝑁𝑝/(𝑁−𝑝) is the so-called critical Sobolev exponent and the weight functions 𝑓,𝑔 are satisfying the following conditions:

(𝑓1)𝑓∈𝐶(Ω) and 𝑓⁺=max{𝑓,0}≢0;(𝑓2) there exist 𝛽₀,𝜌₀>0 and 𝑥₀∈Ω such that 𝐵(𝑥₀,2𝜌₀)⊂Ω and 𝑓(𝑥)≥𝛽₀ for all 𝑥∈𝐵(𝑥₀,2𝜌₀). Without loss of generality, we assume that 𝑥₀=0,(𝑔1)𝑔∈𝐶(Ω) and 𝑔⁺=max{𝑔,0}≢0;(𝑔2)|𝑔⁺|_∞=𝑔(0)=max_𝑥∈Ω𝑔(𝑥);(𝑔3)𝑔(𝑥)>0 for all 𝑥∈𝐵(0,2𝜌₀);(𝑔4) there exists 𝛽>𝑁/(𝑝−1) such that

𝑔(𝑥)=𝑔(0)+𝑜|𝑥|^𝛽as𝑥⟶0.(1.1)

For the weight functions 𝑓≡𝑔≡1, (𝐸_{𝜆𝑓,𝑔}) has been studied extensively. Historically, the role played by such concave-convex nonlinearities in producing multiple solutions was investigated first in the work [1]. They studied the following semilinear elliptic equation:

−Δ𝑢=𝑢𝜆𝑢>0^𝑞−1+𝑢^2∗−1inΩ,𝑢=0inΩ,on𝜕Ω,(1.2) for 1<𝑞<2 and showed the existence of 𝜆₀>0 such that (1.2) admits at least two solutions for all 𝜆∈(0,𝜆₀) and no solution for 𝜆>𝜆₀. Subsequently, in the work [2, 3], the corresponding quasilinear version has been studied

−Δ_𝑝𝑢=𝑢𝜆𝑢>0^𝑞−1+𝑢^𝑝∗−1inΩ,𝑢=0inΩ,on𝜕Ω,(1.3) where 1<𝑝<𝑁 and 1<𝑞<𝑝. They obtained results similar to the results of [1] above, but only for some ranges of the exponents 𝑝 and 𝑞. We summarize their results in what follows.

Theorem 1.1 (see [2, 3]). Assume that either 2𝑁/(𝑁+2)<𝑝<3 or 𝑝>3,𝑝>𝑞>𝑝^∗−2/(𝑝−1). Then there exists 𝜆₀>0 such that (1.3) admits at least two solutions for all 𝜆∈(0,𝜆₀) and no solution for 𝜆>𝜆₀.

It is possible to get complete multiplicity result for problem (1.3) if Ω is taken to be a ball in ℝ^𝑁. Prashanth and Sreenadh [4] have studied (1.3) in the unit ball 𝐵^𝑁(0;1) in ℝ^𝑁 and obtained the following results.

Theorem 1.2 (see [4]). Let Ω=𝐵^𝑁(0;1),1<𝑝<𝑁,1<𝑞<𝑝. Then there exists 𝜆₀>0 such that (1.3) admits at least two solutions for all 𝜆∈(0,𝜆₀) and no solution for 𝜆>𝜆₀. Additionally, if 1<𝑝<2, then (1.3) admits exactly two solutions for all small 𝜆>0.

For 𝑝=2, Tang [5] has studied the exact multiplicity about the following semilinear elliptic equation:

−Δ𝑢=𝑢𝜆𝑢>0^𝑞−1+𝑢^𝑟−1in𝐵^𝑁(0;1),𝑢=0in𝐵^𝑁(0;1),on𝜕𝐵^𝑁(0;1),(1.4) where 1<𝑞<2<𝑟≤2𝑁/(𝑁−2) and 𝑁≥3. We also mention his result below.

Theorem 1.3 (see [5]). There exists 𝜆₀>0 such that (1.4) admits exactly two solutions for 𝜆∈(0,𝜆₀), exactly one solution for 𝜆=𝜆₀, and no solution for 𝜆>𝜆₀.

To proceed, we make some motivations of the present paper. Recently, in [6] the author has considered (1.2) with subcritical nonlinearity of concave-convex type, 𝑔≡1, and 𝑓 is a continuous function which changes sign in Ω, and showed the existence of 𝜆₀>0 such that (1.2) admits at least two solutions for all 𝜆∈(0,𝜆₀) via the extraction of Palais-Smale sequences in the Nehair manifold. In a recent work [7], the author extended the results of [6] to the quasilinear case with the more general weight functions 𝑓,𝑔 but also having subcritical nonlinearity of concave-convex type. In the present paper, we continue the study of [7] by considering critical nonlinearity of concave-convex type and sign-changing weight functions 𝑓,𝑔.

In this paper, we use a variational method involving the Nehari manifold to prove the multiplicity of positive solutions. The Nehari method has been used also in [8] to prove the existence of multiple for a singular elliptic problem. The existence of at least one solution can be obtained by using the same arguments as in the subcritical case [7]. The existence of a second solution needs different arguments due to the lack of compactness of the Palais-Smale sequences. For what, we need addtional assumptions (𝑓2) and (𝑔2) to prove the compactness of the extraction of Palais-Smale sequences in the Nehari manifold (see Theorem 4.4). The multiplicity result is proved only for the parameter 𝜆∈(0,(𝑞/𝑝)Λ₁) (see Theorem 1.5) but for all 1<𝑝<𝑁 and 1≤𝑞<𝑝. This is not the case in the papers referred [2, 3] where the multiplicity is global but not with the full range of 𝑝, 𝑞 and with the weight functions 𝑓≡𝑔≡1. Finally, we mention a recent contribution on 𝑝-Laplacian equation with changing sign nonlinearity by Figuereido et al. [9] which gives the global multiplicity but not with the full range of 𝑝 and 𝑞. The method used in the paper by Figuereido et al. is similar to the method introduced in [1].

In order to represpent our main results, we need to define the following constant Λ₁. Set

Λ₁=𝑝−𝑞(𝑝^∗−𝑞)||𝑔⁺||_∞^{(𝑝−𝑞)/(𝑝∗−𝑝)}𝑝^∗−𝑝(𝑝^∗−𝑞)||𝑓⁺||_∞||Ω||^{(𝑞−𝑝∗)/𝑝∗}𝑆^{(𝑁/𝑝)−(𝑁/𝑝2)𝑞+(𝑞/𝑝)}>0,(1.5) where |Ω| is the Lebesgue measure of Ω and 𝑆 is the best Sobolev constant (see (2.2)).

Theorem 1.4. Assume (𝑓1) and (𝑔1) hold. If 𝜆∈(0,Λ₁), then (𝐸_{𝜆𝑓,𝑔}) admits at least one positive solution 𝑢_𝜆∈𝐶^1,𝛼(Ω) for some 𝛼∈(0,1).

Theorem 1.5. Assume that (𝑓1)-(𝑓2) and (𝑔1)-(𝑔4) hold. If 𝜆∈(0,(𝑞/𝑝)Λ₁), then (𝐸_{𝜆𝑓,𝑔}) admits at least two positive solutions 𝑢_𝜆,𝑈_𝜆∈𝐶^1,𝛼(Ω) for some 𝛼∈(0,1).

This paper is organized as follows. In Section 2, we give some preliminaries and some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorems 1.4 and 1.5.

2. Preliminaries and Nehari Manifold

Throughout this paper, (𝑓1) and (𝑔1) will be assumed. The dual space of a Banach space 𝐸 will be denoted by 𝐸⁻¹. 𝑊^01,𝑝(Ω) denotes the standard Sobolev space with the following norm:

‖𝑢‖^𝑝=_Ω||∇𝑢||^𝑝𝑑𝑥.(2.1)𝑊^01,𝑝(Ω) with the norm ‖⋅‖ is simply denoted by 𝑊. We denote the norm in 𝐿^𝑝(Ω) by |⋅|_𝑝 and the norm in 𝐿^𝑝(ℝ^𝑁) by |⋅|_𝐿^𝑝_(ℝ^𝑁₎ . |Ω| is the Lebesgue measure of Ω. 𝐵(𝑥,𝑟) is a ball centered at 𝑥 with radius 𝑟. 𝑂(𝜀^𝑡) denotes |𝑂(𝜀^𝑡)|/𝜀^𝑡≤𝐶, 𝑜(𝜀^𝑡) denotes |𝑜(𝜀^𝑡)|/𝜀^𝑡→0 as 𝜀→0, and 𝑜_𝑛(1) denotes 𝑜_𝑛(1)→0 as 𝑛→∞. 𝐶, 𝐶_𝑖 will denote various positive constants; the exact values of which are not important. 𝑆 is the best Sobolev embedding constant defined by

𝑆=inf_{𝑢∈𝑊⧵{0}}||∇𝑢||^𝑝𝑝|𝑢|^𝑝𝑝_∗.(2.2)

Definition 2.1. Let 𝑐∈ℝ, 𝐸 be a Banach space and 𝐼∈𝐶¹(𝐸,ℝ).
(i){𝑢_𝑛} is a (PS)𝑐-sequence in 𝐸 for 𝐼 if 𝐼(𝑢_𝑛)=𝑐+𝑜_𝑛(1) and 𝐼^(𝑢_𝑛)=𝑜_𝑛(1) strongly in 𝐸⁻¹ as 𝑛→∞.(ii) We say that 𝐼 satisfies the (PS)𝑐 condition if any (PS)𝑐-sequence {𝑢_𝑛} in 𝐸 for 𝐼 has a convergent subsequence.

Associated with (𝐸_{𝜆𝑓,𝑔}), we consider the energy functional 𝐽_𝜆 in 𝑊, for each 𝑢∈𝑊,

𝐽_𝜆(𝑢)=1𝑝‖𝑢‖^𝑝−𝜆𝑞_Ω𝑓|𝑢|^𝑞𝑑𝑥−1𝑝^∗_Ω𝑔|𝑢|^𝑝∗𝑑𝑥.(2.3) It is well known that 𝐽_𝜆 is of 𝐶¹ in 𝑊 and the solutions of (𝐸_{𝜆𝑓,𝑔}) are the critical points of the energy functional 𝐽_𝜆 (see Rabinowitz [10]).

As the energy functional 𝐽_𝜆 is not bounded below on 𝑊, it is useful to consider the functional on the Nehari manifold

𝒩_𝜆=𝑢∈𝑊⧵{0}∶𝐽^𝜆(𝑢),𝑢=0.(2.4) Thus, 𝑢∈𝒩_𝜆 if and only if

𝐽^𝜆(𝑢),𝑢=‖𝑢‖^𝑝−𝜆_Ω𝑓|𝑢|^𝑞𝑑𝑥−_Ω𝑔|𝑢|^𝑝∗𝑑𝑥=0.(2.5) Note that 𝒩_𝜆 contains every nonzero solution of (𝐸_{𝜆𝑓,𝑔}). Moreover, we have the following results.

Lemma 2.2. The energy functional 𝐽_𝜆 is coercive and bounded below on 𝒩_𝜆.

Proof. If 𝑢∈𝒩_𝜆, then by (𝑓1), (2.5), and the Hölder inequality and the Sobolev embedding theorem we have 𝐽_𝜆(𝑢)=𝑝^∗−𝑝𝑝^∗𝑝‖𝑢‖^𝑝−𝜆𝑝^∗−𝑞𝑝^∗𝑞_Ω𝑓|𝑢|^𝑞𝑑𝑥(2.6)≥1𝑁‖𝑢‖^𝑝−𝜆𝑝^∗−𝑞𝑝^∗𝑞𝑆^−𝑞/𝑝||Ω||^{(𝑝∗−𝑞)/𝑝∗}‖𝑢‖^𝑞||𝑓⁺||_∞.(2.7) Thus, 𝐽_𝜆 is coercive and bounded below on 𝒩_𝜆.

Define

𝜓_𝜆(𝑢)=𝐽^𝜆(𝑢),𝑢.(2.8) Then for 𝑢∈𝒩_𝜆,

𝜓^𝜆(𝑢),𝑢=𝑝‖𝑢‖^𝑝−𝜆𝑞_Ω𝑓|𝑢|^𝑞𝑑𝑥−𝑝^∗_Ω𝑔|𝑢|^𝑝∗𝑑𝑥(2.9)=(𝑝−𝑞)‖𝑢‖^𝑝−𝑝^∗−𝑞_Ω𝑔|𝑢|^𝑝∗𝑑𝑥(2.10)=𝜆𝑝^∗−𝑞_Ω𝑓|𝑢|^𝑞𝑑𝑥−𝑝^∗−𝑝‖𝑢‖^𝑝.(2.11) Similar to the method used in Tarantello [11], we split 𝒩_𝜆 into three parts:

𝒩𝒩^+𝜆=𝑢∈𝒩_𝜆∶𝜓^𝜆(𝑢),𝑢>0_0𝜆,𝒩=𝑢∈𝒩_𝜆∶𝜓^𝜆(𝑢),𝑢=0_−𝜆,=𝑢∈𝒩_𝜆∶𝜓^𝜆(𝑢),𝑢<0.(2.12) Then, we have the following results.

Lemma 2.3. Assume that 𝑢_𝜆 is a local minimizer for 𝐽_𝜆 on 𝒩_𝜆 and 𝑢_𝜆∉𝒩^0𝜆. Then 𝐽^𝜆(𝑢_𝜆)=0 in 𝑊⁻¹.

Proof. Our proof is almost the same as that in Brown and Zhang [12, Theorem 2.3] (or see Binding et al. [13]).

Lemma 2.4. One has the following.
(i) If 𝑢∈𝒩^+𝜆, then ∫_Ω𝑓|𝑢|^𝑞𝑑𝑥>0.
(ii) If 𝑢∈𝒩^0𝜆, then ∫_Ω𝑓|𝑢|^𝑞𝑑𝑥>0 and ∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥>0.
(iii) If 𝑢∈𝒩^−𝜆, then ∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥>0.

Proof. The proof is immediate from (2.10) and (2.11).

Moreover, we have the following result.

Lemma 2.5. If 𝜆∈(0,Λ₁), then 𝒩^0𝜆=∅ where Λ₁ is the same as in (1.5).

Proof. Suppose otherwise that there exists 𝜆∈(0,Λ₁) such that 𝒩^0𝜆≠∅. Then by (2.10) and (2.11), for 𝑢∈𝒩^0𝜆, we have ‖𝑢‖^𝑝=𝑝^∗−𝑞𝑝−𝑞_Ω‖𝑢‖_𝑝𝑔|𝑢|^𝑝∗𝑑𝑥,=𝜆𝑝^∗−𝑞𝑝^∗−𝑝_Ω𝑓|𝑢|^𝑞𝑑𝑥.(2.13) Moreover, by (𝑓1), (𝑔1), and the Hölder inequality and the Sobolev embedding theorem, we have ‖𝑢‖≥𝑝−𝑞(𝑝^∗−𝑞)||𝑔⁺||_∞𝑆^𝑝∗/𝑝^{1/(𝑝∗−𝑝)}‖𝑢‖≤𝜆𝑝_∗,−𝑞𝑝^∗−𝑝𝑆^−𝑞/𝑝||Ω||^{(𝑝∗−𝑞)/𝑝∗}||𝑓⁺||_∞^{1/(𝑝−𝑞)}.(2.14) This implies 𝜆≥𝑝−𝑞(𝑝^∗−𝑞)||𝑔⁺||_∞^{(𝑝−𝑞)/(𝑝∗−𝑝)}𝑝^∗−𝑝(𝑝^∗−𝑞)||𝑓⁺||_∞||Ω||^{(𝑞−𝑝∗)/𝑝∗}𝑆^{(𝑁/𝑝)−(𝑁/𝑝2)𝑞+(𝑞/𝑝)}=Λ₁,(2.15) which is a contradiction. Thus, we can conclude that if 𝜆∈(0,Λ₁), we have 𝒩^0𝜆=∅.

By Lemma 2.5, we write 𝒩_𝜆=𝒩^+𝜆∪𝒩^−𝜆 and 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).(2.16) Then we get the following result.

Theorem 2.6. (i) If 𝜆∈(0,Λ₁) and 𝑢∈𝒩^+𝜆, then one has 𝐽_𝜆(𝑢)<0 and 𝛼_𝜆≤𝛼^+𝜆<0.
(ii) If 𝜆∈(0,(𝑞/𝑝)Λ₁), then 𝛼^−𝜆>𝑑₀ for some positive constant 𝑑₀ depending on 𝜆,𝑝,𝑞,𝑁,𝑆,|𝑓⁺|_∞,|𝑔⁺|_∞, and |Ω|.

Proof. (i) Let 𝑢∈𝒩^+𝜆. By (2.10), we have 𝑝−𝑞𝑝^∗−𝑞‖𝑢‖^𝑝>_Ω𝑔|𝑢|^𝑝∗𝑑𝑥,(2.17) and so 𝐽_𝜆(𝑢)=1𝑝−1𝑞‖𝑢‖^𝑝+1𝑞−𝑝1^∗_Ω<1𝑔|𝑢|^𝑝∗𝑑𝑥𝑝−1𝑞+1𝑞−𝑝1^∗𝑝𝑝^∗−−𝑞𝑞‖𝑢‖^𝑝=−𝑝−𝑞𝑞𝑁‖𝑢‖^𝑝<0.(2.18) Therefore, from the definition of 𝛼_𝜆, 𝛼^+𝜆, we can deduce that 𝛼_𝜆≤𝛼^+𝜆<0.
(ii) Let 𝑢∈𝒩^−𝜆. By (2.10), we have 𝑝−𝑞𝑝^∗−𝑞‖𝑢‖^𝑝<_Ω𝑔|𝑢|^𝑝∗𝑑𝑥.(2.19) Moreover, by (𝑔1) and the Sobolev embedding theorem, we have _Ω𝑔|𝑢|^𝑝∗𝑑𝑥≤𝑆^{−𝑝∗/𝑝}‖𝑢‖^𝑝∗||𝑔⁺||_∞.(2.20) This implies ‖𝑢‖>𝑝−𝑞(𝑝^∗−𝑞)||𝑔⁺||_∞^{1/(𝑝∗−𝑝)}𝑆^𝑁/𝑝2,∀𝑢∈𝒩^−𝜆.(2.21) By(2.7) in the proof of Lemma 2.2, we have 𝐽_𝜆(𝑢)≥‖𝑢‖^𝑞𝑝^∗−𝑝𝑝^∗𝑝‖𝑢‖^𝑝−𝑞−𝜆𝑆^−𝑞/𝑝𝑝^∗−𝑞𝑝^∗𝑞||Ω||^{(𝑝∗−𝑞)/𝑝∗}||𝑓⁺||_∞>𝑝−𝑞(𝑝^∗−𝑞)||𝑔⁺||_∞^{𝑞/(𝑝∗−𝑝)}𝑆^{𝑞𝑁/𝑝2}×⎡⎢⎢⎣𝑝^∗−𝑝𝑝^∗𝑝𝑆^{(𝑝−𝑞)𝑁/𝑝2}(𝑝^∗−𝑝−𝑞)||𝑞𝑔⁺||_∞^{(𝑝−𝑞)/(𝑝∗−𝑝)}−𝜆𝑆^−𝑞/𝑝𝑝^∗−𝑞𝑝^∗𝑞||Ω||^{(𝑝∗−𝑞)/𝑝∗}||𝑓⁺||_∞⎤⎥⎥⎦.(2.22) Thus, if 𝜆∈(0,(𝑞/𝑝)Λ₁), then 𝐽_𝜆(𝑢)>𝑑₀,∀𝑢∈𝒩^−𝜆,(2.23) for some positive constant 𝑑₀=𝑑₀(𝜆,𝑝,𝑞,𝑁,𝑆,|𝑓⁺|_∞,|𝑔⁺|_∞,|Ω|). This completes the proof.

For each 𝑢∈𝑊 with ∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥>0, we write

𝑡_max=(𝑝−𝑞)‖𝑢‖^𝑝(𝑝^∗−𝑞)∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥^{1/(𝑝∗−𝑝)}>0.(2.24) Then the following lemma holds.

Lemma 2.7. Let 𝜆∈(0,Λ₁). For each 𝑢∈𝑊 with ∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥>0, one has the following:
(i) if ∫_Ω𝑓|𝑢|^𝑞𝑑𝑥≤0, then there exists a unique 𝑡⁻>𝑡_max such that 𝑡⁻𝑢∈𝒩^−𝜆 and 𝐽_𝜆(𝑡⁻𝑢)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢),(2.25)
(ii) if ∫_Ω𝑓|𝑢|^𝑞𝑑𝑥>0, then there exists unique 0<𝑡⁺<𝑡_max<𝑡⁻ such that 𝑡⁺𝑢∈𝒩^+𝜆, 𝑡⁻𝑢∈𝒩^−𝜆, and 𝐽_𝜆𝑡⁺𝑢=inf_{0≤𝑡≤𝑡max}𝐽_𝜆(𝑡𝑢);𝐽_𝜆(𝑡⁻𝑢)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢).(2.26)

Proof. Fix 𝑢∈𝑊 with ∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥>0. Let 𝑘(𝑡)=𝑡^𝑝−𝑞‖𝑢‖^𝑝−𝑡^{𝑝∗−𝑞}_Ω𝑔|𝑢|^𝑝∗𝑑𝑥for𝑡≥0.(2.27) It is clear that 𝑘(0)=0, 𝑘(𝑡)→−∞ as 𝑡→∞. From 𝑘^(𝑡)=(𝑝−𝑞)𝑡^{𝑝−𝑞−1}‖𝑢‖^𝑝−𝑝^∗−𝑞𝑡^{𝑝∗−𝑞−1}_Ω𝑔|𝑢|^𝑝∗𝑑𝑥,(2.28) we can deduce that 𝑘^(𝑡)=0 at 𝑡=𝑡_max, 𝑘^(𝑡)>0 for 𝑡∈(0,𝑡_max) and 𝑘^(𝑡)<0 for 𝑡∈(𝑡_max,∞). Then 𝑘(𝑡) that achieves its maximum at 𝑡_max is increasing for 𝑡∈[0,𝑡_max) and decreasing for 𝑡∈(𝑡_max,∞). Moreover, 𝑘𝑡_max=(𝑝−𝑞)‖𝑢‖^𝑝(𝑝^∗−𝑞)∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥^{(𝑝−𝑞)/(𝑝∗−𝑝)}‖𝑢‖^𝑝−(𝑝−𝑞)‖𝑢‖^𝑝(𝑝^∗−𝑞)∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥^{(𝑝∗−𝑞)/(𝑝∗−𝑝)}_Ω=‖𝑢‖_𝑞𝑔|𝑢|^𝑝∗𝑑𝑥𝑝𝑝^∗−−𝑞𝑞^{(𝑝−𝑞)/(𝑝∗−𝑝)}−𝑝−𝑞𝑝^∗−𝑞^{(𝑝∗−𝑞)/(𝑝∗−𝑝)}∫_Ω‖𝑔𝑢|𝑢‖|^{𝑝∗𝑝∗}𝑑𝑥^{(𝑝−𝑞)/(𝑝∗−𝑝)}≥‖𝑢‖^𝑞𝑝^∗−𝑝𝑝^∗−𝑞(𝑝^∗−𝑝𝑞−)𝑞||𝑔⁺||_∞𝑆^𝑝∗/𝑝^{(𝑝−𝑞)/(𝑝∗−𝑝)}.(2.29) We have ∫_Ω𝑓|𝑢|^𝑞𝑑𝑥≤0. There exists a unique 𝑡⁻>𝑡_max such that 𝑘(𝑡⁻)=𝜆∫_Ω𝑓|𝑢|^𝑞𝑑𝑥 and 𝑘^(𝑡⁻)<0. Now, (𝑝−𝑞)(𝑡⁻)^𝑝‖𝑢‖^𝑝−𝑝^∗−𝑞(𝑡⁻)^𝑝_Ω=(𝑡₋𝑔|𝑢|^𝑝∗𝑑𝑥)^1+𝑞(𝑝−𝑞)(𝑡⁻)^{𝑝−𝑞−1}‖𝑢‖^𝑝−𝑝^∗−𝑞(𝑡⁻)^{𝑝∗−𝑞−1}_Ω=(𝑡₋𝑔|𝑢|^𝑝∗𝑑𝑥𝐽)^1+𝑞𝑘^(𝑡⁻)<0,_𝜆(𝑡⁻𝑢),𝑡⁻𝑢=(𝑡⁻)^𝑝‖𝑢‖^𝑝−(𝑡⁻)^𝑝∗_Ω=(𝑡₋𝑔|𝑢|^𝑝∗𝑑𝑥−(𝑡⁻)^𝑞𝜆_Ω𝑓|𝑢|^𝑞𝑑𝑥)^𝑞𝑘(𝑡⁻)−𝜆_Ω𝑓|𝑢|^𝑞𝑑𝑥=0.(2.30) Then we have that 𝑡⁻𝑢∈𝒩^−𝜆. For 𝑡>𝑡_max, we have (𝑝−𝑞)‖𝑡𝑢‖^𝑝−𝑝^∗−𝑞_Ω𝑔|𝑡𝑢|^𝑝∗<0,𝑑²𝑑𝑡²𝑑𝐽_𝜆(𝑡𝑢)<0,𝑑𝑡𝐽^𝜆(𝑡𝑢)=𝑡^𝑝−1‖𝑢‖^𝑝−𝑡^𝑝∗−1_Ω=0𝑔|𝑢|^𝑝∗𝑑𝑥−𝑡^𝑞−1𝜆_Ω𝑓|𝑢|^𝑞𝑑𝑥for𝑡=𝑡⁻.(2.31) Thus, 𝐽_𝜆(𝑡⁻𝑢)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢).
(ii) We have ∫_Ω𝑓|𝑢|^𝑞𝑑𝑥>0. By (2.29) and 𝑘(0)=≤𝜆0𝑆<𝜆_−𝑞/𝑝_Ω𝑓|𝑢|^𝑞𝑑𝑥||Ω||^{(𝑝∗−𝑞)/𝑝∗}‖𝑢‖^𝑞||𝑓⁺||_∞<‖𝑢‖^𝑞𝑝^∗−𝑝𝑝^∗−𝑞(𝑝^∗−𝑝𝑞−)𝑞||𝑔⁺||_∞𝑆^𝑝∗/𝑝^{(𝑝−𝑞)/(𝑝∗−𝑝)}≤𝑘𝑡_maxfor𝜆∈0,Λ₁,(2.32) there are unique 𝑡⁺ and 𝑡⁻ such that 0<𝑡⁺<𝑡_max<𝑡⁻, 𝑘𝑡⁺𝑘_=𝜆_Ω𝑓|𝑢|^𝑞𝑑𝑥=𝑘(𝑡⁻),𝑡⁺>0>𝑘^(𝑡⁻).(2.33) We have 𝑡⁺𝑢∈𝒩^+𝜆, 𝑡⁻𝑢∈𝒩^−𝜆, and 𝐽_𝜆(𝑡⁻𝑢)≥𝐽_𝜆(𝑡𝑢)≥𝐽_𝜆(𝑡⁺𝑢) for each 𝑡∈[𝑡⁺,𝑡⁻] and 𝐽_𝜆(𝑡⁺𝑢)≤𝐽_𝜆(𝑡𝑢) for each 𝑡∈[0,𝑡⁺]. Thus, 𝐽_𝜆𝑡⁺𝑢=inf_{0≤𝑡≤𝑡max}𝐽_𝜆(𝑡𝑢),𝐽_𝜆(𝑡⁻𝑢)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢).(2.34) This completes the proof.

3. Proof of Theorem 1.4

First, we will use the idea of Tarantello [11] to get the following results.

Lemma 3.1. If 𝜆∈(0,Λ₁), then for each 𝑢∈𝒩_𝜆, there exist 𝜖>0 and a differentiable function 𝜉∶𝐵(0;𝜖)⊂𝑊→ℝ⁺ such that 𝜉(0)=1, the function 𝜉(𝑣)(𝑢−𝑣)∈𝒩_𝜆, and 𝜉^(0),𝑣=𝑝∫^Ω||∇𝑢||^𝑝−2∇𝑢∇𝑣𝑑𝑥−𝜆𝑞∫_Ω𝑓|𝑢|^𝑞−2𝑢𝑣𝑑𝑥−𝑝^∗∫_Ω𝑔|𝑢|^𝑝∗−2𝑢𝑣𝑑𝑥(𝑝−𝑞)‖𝑢‖^𝑝−(𝑝^∗−𝑞)∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥(3.1) for all 𝑣∈𝑊.

Proof. For 𝑢∈𝒩_𝜆, define a function 𝐹∶ℝ×𝑊→ℝ by 𝐹_𝑢(𝜉,𝑤)==𝜉_𝑝𝐽^𝜆(𝜉(𝑢−𝑤)),𝜉(𝑢−𝑤)_Ω−𝜉_𝑝∗||∇(𝑢−𝑤)||^𝑝𝑑𝑥−𝜉^𝑞𝜆_Ω𝑓|𝑢−𝑤|^𝑞𝑑𝑥_Ω𝑔|𝑢−𝑤|^𝑝∗𝑑𝑥.(3.2) Then 𝐹_𝑢(1,0)=⟨𝐽^𝜆(𝑢),𝑢⟩=0 and 𝑑𝑑𝜉𝐹^𝑢(1,0)=𝑝‖𝑢‖^𝑝−𝜆𝑞_Ω𝑓|𝑢|^𝑞𝑑𝑥−𝑝^∗_Ω=(𝑝−𝑞)‖𝑢‖_𝑝𝑔|𝑢|^𝑝∗𝑑𝑥−𝑝^∗−𝑞_Ω𝑔|𝑢|^𝑝∗𝑑𝑥≠0.(3.3) According to the implicit function theorem, there exist 𝜖>0 and a differentiable function 𝜉∶𝐵(0;𝜖)⊂𝑊→ℝ such that 𝜉(0)=1, 𝜉^(0),𝑣=𝑝∫^Ω||∇𝑢||^𝑝−2∇𝑢∇𝑣𝑑𝑥−𝜆𝑞∫_Ω𝑓|𝑢|^𝑞−2𝑢𝑣𝑑𝑥−𝑝^∗∫_Ω𝑔|𝑢|^𝑝∗−2𝑢𝑣𝑑𝑥𝐹(𝑝−𝑞)‖𝑢‖^𝑝−(𝑝^∗−𝑞)∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥_𝑢,(𝜉(𝑣),𝑣)=0,∀𝑣∈𝐵(0;𝜖),(3.4) which is equivalent to 𝐽^𝜆(𝜉(𝑣)(𝑢−𝑣)),𝜉(𝑣)(𝑢−𝑣)=0,∀𝑣∈𝐵(0;𝜖),(3.5) that is, 𝜉(𝑣)(𝑢−𝑣)∈𝒩_𝜆.

Lemma 3.2. Let 𝜆∈(0,Λ₁), then for each 𝑢∈𝒩^−𝜆, there exist 𝜖>0 and a differentiable function 𝜉⁻∶𝐵(0;𝜖)⊂𝑊→ℝ⁺ such that 𝜉⁻(0)=1, the function 𝜉⁻(𝑣)(𝑢−𝑣)∈𝒩^−𝜆, and (𝜉⁻)^(0),𝑣=𝑝∫^Ω||∇𝑢||^𝑝−2∇𝑢∇𝑣𝑑𝑥−𝜆𝑞∫_Ω𝑓|𝑢|^𝑞−2𝑢𝑣𝑑𝑥−𝑝^∗∫_Ω𝑔|𝑢|^𝑝∗−2𝑢𝑣𝑑𝑥(𝑝−𝑞)‖𝑢‖^𝑝−(𝑝^∗−𝑞)∫_Ω𝑔|𝑢|^𝑝∗𝑑𝑥(3.6) for all 𝑣∈𝑊.

Proof. Similar to the argument in Lemma 3.1, there exist 𝜖>0 and a differentiable function 𝜉⁻∶𝐵(0;𝜖)⊂𝑊→ℝ such that 𝜉⁻(0)=1 and 𝜉⁻(𝑣)(𝑢−𝑣)∈𝒩_𝜆 for all 𝑣∈𝐵(0;𝜖). Since 𝜓^𝜆(𝑢),𝑢=(𝑝−𝑞)‖𝑢‖^𝑝−𝑝^∗−𝑞_Ω𝑔|𝑢|^𝑝∗𝑑𝑥<0.(3.7) Thus, by the continuity of the function 𝜉⁻, we have 𝜓^𝜆(𝜉⁻(𝑣)(𝑢−𝑣)),𝜉⁻(𝑣)(𝑢−𝑣)=(𝑝−𝑞)‖𝜉⁻(𝑣)(𝑢−𝑣)‖^𝑝−𝑝^∗−𝑞_Ω𝑔||𝜉⁻(𝑣)(𝑢−𝑣)||^𝑝𝑑𝑥<0,(3.8) if 𝜖 sufficiently small, this implies that 𝜉⁻(𝑣)(𝑢−𝑣)∈𝒩^−𝜆.

Proposition 3.3. (i) If 𝜆∈(0,Λ₁), then there exists a (PS)𝛼_𝜆-sequence {𝑢_𝑛}⊂𝒩_𝜆 in 𝑊 for 𝐽_𝜆.
(ii) If 𝜆∈(0,(𝑞/𝑝)Λ₁), then there exists a (PS)𝛼^−𝜆-sequence {𝑢_𝑛}⊂𝒩^−𝜆 in 𝑊 for 𝐽_𝜆.

Proof. (i) By Lemma 2.2 and the Ekeland variational principle [14], there exists a minimizing sequence {𝑢_𝑛}⊂𝒩_𝜆 such that 𝐽_𝜆𝑢_𝑛<𝛼_𝜆+1𝐽_𝜆𝑛,𝑢_𝑛<𝐽_𝜆(𝑤)+1𝑛‖‖𝑤−𝑢^𝑛‖‖foreach𝑤∈𝒩_𝜆.(3.9) By 𝛼_𝜆<0 and taking 𝑛 large, we have 𝐽_𝜆𝑢_𝑛=1𝑝−𝑝1^∗‖‖𝑢^𝑛‖‖^𝑝−1𝑞−𝑝1^∗𝜆_Ω<𝛼_𝜆𝑓||𝑢_𝑛||^𝑞𝑑𝑥+1𝑛<𝛼𝑝^𝜆.(3.10) From (2.7), (3.10), 𝛼_𝜆<0, and the Hölder inequality, we deduce that ||𝑓⁺||_∞𝜆𝑆^−𝑞/𝑝||Ω||^{(𝑝∗−𝑞)/𝑝∗}‖‖𝑢_𝑛‖‖^𝑞≥𝜆_Ω𝑓||𝑢_𝑛||^𝑞𝑑𝑥>−𝑝^∗𝑞𝑝(𝑝^∗−𝑞)𝛼^𝜆>0.(3.11) Consequently, 𝑢_𝑛≠0 and putting together (3.10), (3.11), and the Hölder inequality, we obtain ‖‖𝑢_𝑛‖‖>−𝑝^∗𝑞𝑝𝜆(𝑝^∗−𝑞)||𝑓⁺||_∞𝛼_𝜆𝑆^𝑞/𝑝||Ω||^{(𝑞−𝑝∗)/𝑝∗}^1/𝑞‖‖𝑢_𝑛,‖‖<𝑝𝑝^∗−𝑞𝑞(𝑝^∗−𝑝)𝜆𝑆^−𝑞/𝑝||Ω||^{(𝑝∗−𝑞)/𝑝∗}||𝑓⁺||_∞^{1/(𝑝−𝑞)}.(3.12) Now, we show that ‖‖𝐽^𝜆𝑢_𝑛‖‖_𝑊⁻¹⟶0,as𝑛⟶∞.(3.13) Apply Lemma 3.1 with 𝑢_𝑛 to obtain the functions 𝜉_𝑛∶𝐵(0;𝜖_𝑛)→ℝ⁺ for some 𝜖_𝑛>0, such that 𝜉_𝑛(𝑤)(𝑢_𝑛−𝑤)∈𝒩_𝜆. Choose 0<𝜌<𝜖_𝑛. Let 𝑢∈𝑊 with 𝑢≢0 and let 𝑤_𝜌=𝜌𝑢/‖𝑢‖. We set 𝜂_𝜌=𝜉_𝑛(𝑤_𝜌)(𝑢_𝑛−𝑤_𝜌). Since 𝜂_𝜌∈𝒩_𝜆, we deduce from (3.9) that 𝐽_𝜆𝜂_𝜌−𝐽_𝜆𝑢_𝑛≥−1𝑛‖‖𝜂^𝜌−𝑢_𝑛‖‖,(3.14) and by the mean value theorem, we have 𝐽^𝜆𝑢_𝑛,𝜂_𝜌−𝑢_𝑛+𝑜‖‖𝜂_𝜌−𝑢_𝑛‖‖≥−1𝑛‖‖𝜂^𝜌−𝑢_𝑛‖‖.(3.15) Thus, 𝐽^𝜆𝑢_𝑛,−𝑤_𝜌+𝜉_𝑛𝑤_𝜌−1𝐽^𝜆𝑢_𝑛,𝑢_𝑛−𝑤_𝜌≥−1𝑛‖‖𝜂^𝜌−𝑢_𝑛‖‖+𝑜‖‖𝜂_𝜌−𝑢_𝑛‖‖.(3.16) Since 𝜉_𝑛(𝑤_𝜌)(𝑢_𝑛−𝑤_𝜌)∈𝒩_𝜆 and (3.16) it follows that −𝜌𝐽^𝜆𝑢_𝑛,𝑢‖𝑢‖+𝜉^𝑛𝑤_𝜌−1𝐽^𝜆𝑢_𝑛−𝐽^𝜆𝜂_𝜌,𝑢_𝑛−𝑤_𝜌≥−1𝑛‖‖𝜂^𝜌−𝑢_𝑛‖‖+𝑜‖‖𝜂_𝜌−𝑢_𝑛‖‖.(3.17) Thus, 𝐽^𝜆𝑢_𝑛,‖𝑢𝑢‖≤‖‖𝜂^𝜌−𝑢_𝑛‖‖𝑛𝜌+𝑜‖‖𝜂^𝜌−𝑢_𝑛‖‖+𝜉_𝑛𝜌𝑤_𝜌−1𝜌𝐽^𝜆𝑢_𝑛−𝐽^𝜆𝜂_𝜌,𝑢_𝑛−𝑤_𝜌.(3.18) Since ‖𝜂_𝜌−𝑢_𝑛‖≤𝜌𝜉_𝑛(𝑤_𝜌)+|𝜉_𝑛(𝑤_𝜌)−1|‖𝑢_𝑛‖ and lim_𝜌→0||𝜉_𝑛𝑤_𝜌−1||𝜌≤‖‖𝜉^𝑛(0)‖‖,(3.19) if we let 𝜌→0 in (3.18) for a fixed 𝑛, then by (3.12) we can find a constant 𝐶>0, independent of 𝜌, such that 𝐽^𝜆𝑢_𝑛,‖𝑢𝑢‖≤𝐶𝑛1+‖‖𝜉^𝑛(0)‖‖.(3.20) The proof will be complete once we show that ‖𝜉^𝑛(0)‖ is uniformly bounded in 𝑛. By (3.1), (3.12), (𝑓1), (𝑔1), and the Hölder inequality and the Sobolev embedding theorem, we have 𝜉^𝑛(0),𝑣≤𝑏‖𝑣‖||(𝑝−𝑞)‖‖𝑢_𝑛‖‖^𝑝−(𝑝^∗−𝑞)∫_Ω𝑔||𝑢_𝑛||^𝑝∗𝑑𝑥||forsome𝑏>0.(3.21) We only need to show that ||||(𝑝−𝑞)‖‖𝑢_𝑛‖‖^𝑝−𝑝^∗−𝑞_Ω𝑔||𝑢_𝑛||^𝑝∗𝑑𝑥||||>𝐶(3.22) for some 𝐶>0 and 𝑛 large enough. We argue by contradiction. Assume that there exists a subsequence {𝑢_𝑛} such that (𝑝−𝑞)‖‖𝑢_𝑛‖‖^𝑝−𝑝^∗−𝑞_Ω𝑔||𝑢_𝑛||^𝑝∗𝑑𝑥=𝑜_𝑛(1).(3.23) By (3.23) and the fact that 𝑢_𝑛∈𝒩_𝜆, we get ‖‖𝑢_𝑛‖‖^𝑝=𝑝^∗−𝑞𝑝−𝑞_Ω‖‖𝑢𝑔||𝑢_𝑛||^𝑝∗𝑑𝑥+𝑜_𝑛(1),_𝑛‖‖^𝑝=𝜆𝑝^∗−𝑞𝑝^∗−𝑝_Ω𝑓||𝑢_𝑛||^𝑞𝑑𝑥+𝑜_𝑛(1).(3.24) Moreover, by (𝑓1), (𝑔1), and the Hölder inequality and the Sobolev embedding theorem, we have ‖‖𝑢_𝑛‖‖≥𝑝−𝑞(𝑝^∗−𝑞)||𝑔⁺||_∞𝑆^𝑝∗/𝑝^{1/(𝑝∗−𝑝)}‖‖𝑢+𝑜_𝑛(1),_𝑛‖‖≤𝜆𝑝^∗−𝑞||𝑓⁺||_∞𝑝^∗−𝑝𝑆^−𝑞/𝑝||Ω||^{(𝑝∗−𝑞)/𝑝∗}^{1/(𝑝−𝑞)}+𝑜_𝑛(1).(3.25) This implies 𝜆≥Λ₁ which is a contradiction. We obtain 𝐽^𝜆𝑢_𝑛,‖𝑢𝑢‖≤𝐶𝑛.(3.26) This completes the proof of (i).
(ii) Similarly, by using Lemma 3.2, we can prove (ii). We will omit detailed proof here.

Now, we establish the existence of a local minimum for 𝐽_𝜆 on 𝒩^+𝜆.

Theorem 3.4. If 𝜆∈(0,Λ₁), then 𝐽_𝜆 has a minimizer 𝑢_𝜆 in 𝒩^+𝜆 and it satisfies that
(i)𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆=𝛼^+𝜆;
(ii)𝑢_𝜆 is a positive solution of (𝐸_{𝜆𝑓,𝑔}) in 𝐶^1,𝛼(Ω) for some 𝛼∈(0,1).

Proof. By Proposition 3.3(i), there exists a minimizing sequence {𝑢_𝑛} for 𝐽_𝜆 on 𝒩_𝜆 such that 𝐽_𝜆𝑢_𝑛=𝛼_𝜆+𝑜_𝑛(1),𝐽^𝜆𝑢_𝑛=𝑜_𝑛(1)in𝑊⁻¹.(3.27) Since 𝐽_𝜆 is coercive on 𝒩_𝜆 (see Lemma 2.2), we get that {𝑢_𝑛} is bounded in 𝑊. Going if necessary to a subsequence, we can assume that there exists 𝑢_𝜆∈𝑊 such that 𝑢𝑢_𝑛⇀𝑢_𝜆weaklyin𝑊,_𝑛𝑢⟶𝑢_𝜆almosteverywhereinΩ,_𝑛⟶𝑢_𝜆stronglyin𝐿^𝑠(Ω)∀1≤𝑠<𝑝^∗.(3.28) First, we claim that 𝑢_𝜆 is a nontrivial solution of (𝐸_{𝜆𝑓,𝑔}). By (3.27) and (3.28), it is easy to see that 𝑢_𝜆 is a solution of (𝐸_{𝜆𝑓,𝑔}). From 𝑢_𝑛∈𝒩_𝜆 and (2.6), we deduce that 𝜆_Ω𝑓||𝑢_𝑛||^𝑞𝑑𝑥=𝑞𝑝^∗−𝑝𝑝(𝑝^∗−𝑞)‖‖𝑢^𝑛‖‖^𝑝−𝑝^∗𝑞𝑝^∗−𝑞𝐽^𝜆𝑢_𝑛.(3.29) Let 𝑛→∞ in (3.29), by (3.27), (3.28), and 𝛼_𝜆<0, we get _Ω𝑓||𝑢_𝜆||^𝑞𝑑𝑥≥−𝑝^∗𝑞𝑝^∗−𝑞𝛼^𝜆>0.(3.30) Thus, 𝑢_𝜆∈𝒩_𝜆 is a nontrivial solution of (𝐸_{𝜆𝑓,𝑔}). Now we prove that 𝑢_𝑛→𝑢_𝜆 strongly in 𝑊 and 𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆. By (3.29), if 𝑢∈𝒩_𝜆, then 𝐽_𝜆(𝑢)=𝑝^∗−𝑝𝑝^∗𝑝‖𝑢‖^𝑝−𝑝^∗−𝑞𝑝^∗𝑞𝜆_Ω𝑓|𝑢|^𝑞𝑑𝑥.(3.31) In order to prove that 𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆, it suffices to recall that 𝑢_𝜆∈𝒩_𝜆, by (3.31), and applying Fatou's lemma to get 𝛼_𝜆≤𝐽_𝜆𝑢_𝜆=𝑝^∗−𝑝𝑝^∗𝑝‖‖𝑢^𝜆‖‖^𝑝−𝑝^∗−𝑞𝑝^∗𝑞𝜆_Ω≤liminf_𝑛→∞𝑓||𝑢_𝜆||^𝑞𝑑𝑥𝑝^∗−𝑝𝑝^∗𝑝‖‖𝑢^𝑛‖‖^𝑝−𝑝^∗−𝑞𝑝^∗𝑞𝜆_Ω≤liminf_𝑛→∞𝑓||𝑢_𝑛||^𝑞𝑑𝑥𝐽_𝜆𝑢_𝑛=𝛼_𝜆.(3.32) This implies that 𝐽_𝜆(𝑢_𝜆)=𝛼_𝜆 and lim_𝑛→∞‖𝑢_𝑛‖^𝑝=‖𝑢_𝜆‖^𝑝. Let 𝑣_𝑛=𝑢_𝑛−𝑢_𝜆, then Brézis and Lieb lemma [15] implies that ‖‖𝑣_𝑛‖‖^𝑝=‖‖𝑢_𝑛‖‖^𝑝−‖‖𝑢_𝜆‖‖^𝑝+𝑜_𝑛(1).(3.33) Therefore, 𝑢_𝑛→𝑢_𝜆 strongly in 𝑊. Moreover, we have 𝑢_𝜆∈𝒩^+𝜆. On the contrary, if 𝑢_𝜆∈𝒩^−𝜆, then by Lemma 2.7, there are unique 𝑡⁺⁰ and 𝑡⁻⁰ such that 𝑡⁺⁰𝑢_𝜆∈𝒩^+𝜆 and 𝑡⁻⁰𝑢_𝜆∈𝒩^−𝜆. In particular, we have 𝑡⁺⁰<𝑡⁻⁰=1. Since 𝑑𝑑𝑡𝐽^𝜆𝑡⁺⁰𝑢_𝜆=0,𝑑²𝑑𝑡²𝐽_𝜆𝑡⁺⁰𝑢_𝜆>0,(3.34) there exists 𝑡⁺⁰<𝑡≤𝑡⁻⁰ such that 𝐽_𝜆(𝑡⁺⁰𝑢_𝜆)<𝐽_𝜆(𝑡𝑢_𝜆). By Lemma 2.7, 𝐽_𝜆𝑡⁺⁰𝑢_𝜆<𝐽_𝜆𝑡𝑢_𝜆≤𝐽_𝜆𝑡⁻⁰𝑢_𝜆=𝐽_𝜆𝑢_𝜆,(3.35) which is a contradiction. Since 𝐽_𝜆(𝑢_𝜆)=𝐽_𝜆(|𝑢_𝜆|) and |𝑢_𝜆|∈𝒩^+𝜆, by Lemma 2.3 we may assume that 𝑢_𝜆 is a nontrivial nonnegative solution of (𝐸_{𝜆𝑓,𝑔}). Moreover, from 𝑓,𝑔∈𝐿^∞(Ω), then using the standard bootstrap argument (see, e.g., [16]) we obtain 𝑢_𝜆∈𝐿^∞(Ω); hence by applying regularity results [17, 18] we derive that 𝑢_𝜆∈C^1,𝛼(Ω) for some 𝛼∈(0,1) and finally, by the Harnack inequality [19] we deduce that 𝑢_𝜆>0. This completes the proof.

Now, we begin the proof of Theorem 1.4. By Theorem 3.4, we obtain (𝐸_{𝜆𝑓,𝑔}) that has a positive solution 𝑢_𝜆 in 𝐶^1,𝛼(Ω) for some 𝛼∈(0,1).

4. Proof of Theorem 1.5

Next, we will establish the existence of the second positive solution of (𝐸_{𝜆𝑓,𝑔}) by proving that 𝐽_𝜆 satisfies the (PS)𝛼^−𝜆 condition.

Lemma 4.1. Assume that (𝑓1) and (𝑔1) hold. If {𝑢_𝑛}⊂𝑊 is a (PS)𝑐-sequence for 𝐽_𝜆, then {𝑢_𝑛} is bounded in 𝑊.

Proof. We argue by contradiction. Assume that ‖𝑢_𝑛‖→∞. Let 𝑢_𝑛̂=𝑢_𝑛/‖𝑢_𝑛‖. We may assume that 𝑢_𝑛̂⇀𝑢̂ in 𝑊. This implies that 𝑢_𝑛̂→𝑢̂ strongly in 𝐿^𝑠(Ω) for all 1≤𝑠<𝑝^∗ and 𝜆𝑞_Ω𝑓||𝑢_𝑛̂||^𝑞𝑑𝑥=𝜆𝑞_Ω𝑓||𝑢||̂^𝑞𝑑𝑥+𝑜_𝑛(1).(4.1) Since {𝑢_𝑛} is a (PS)𝑐-sequence for 𝐽_𝜆 and ‖𝑢_𝑛‖→∞, there hold 1𝑝_Ω||∇𝑢_𝑛̂||^𝑝𝑑𝑥−𝜆‖‖𝑢^𝑛‖‖^𝑞−𝑝𝑞_Ω𝑓||𝑢_𝑛̂||^𝑞𝑑𝑥−‖‖𝑢^𝑛‖‖^{𝑝∗−𝑝}𝑝^∗_Ω𝑔||𝑢_𝑛̂||^𝑝∗𝑑𝑥=𝑜_𝑛(1),_Ω||∇𝑢_𝑛̂||^𝑝𝑑𝑥−𝜆‖‖𝑢_𝑛‖‖^𝑞−𝑝_Ω𝑓||𝑢_𝑛̂||^𝑞𝑑𝑥−‖‖𝑢_𝑛‖‖^{𝑝∗−𝑝}_Ω𝑔||𝑢_𝑛̂||^𝑝∗𝑑𝑥=𝑜_𝑛(1).(4.2) From (4.1)-(4.2), we can deduce that _Ω||∇𝑢_𝑛̂||^𝑝𝑑𝑥=𝑝𝑝^∗−𝑞𝑞(𝑝^∗−𝑝)‖‖𝑢^𝑛‖‖^𝑞−𝑝𝜆_Ω𝑓||𝑢||̂^𝑞𝑑𝑥+𝑜_𝑛(1).(4.3) Since 1≤𝑞<2 and ‖𝑢_𝑛‖→∞, (4.3) implies _Ω||∇𝑢_𝑛̂||^𝑝𝑑𝑥⟶0,as𝑛⟶∞,(4.4) which is contrary to the fact ‖𝑢_𝑛̂‖=1 for all 𝑛.

Lemma 4.2. Assume that (𝑓1) and (𝑔1) hold. If {𝑢_𝑛}⊂𝑊 is a (PS)𝑐-sequence for 𝐽_𝜆 with 𝑐∈(0,(1/𝑁)|𝑔⁺|^{∞−(𝑁−𝑝)/𝑝}𝑆^𝑁/𝑝), then there exists a subsequence of {𝑢_𝑛} converging weakly to a nontrivial solution of (𝐸_{𝜆𝑓,𝑔}).

Proof. Let {𝑢_𝑛}⊂𝑊 be a (PS)𝑐-sequence for 𝐽_𝜆 with 𝑐∈(0,(1/𝑁)|𝑔⁺|^{∞−(𝑁−𝑝)/𝑝}𝑆^𝑁/𝑝). We know from Lemma 4.1 that {𝑢_𝑛} is bounded in 𝑊, and then there exists a subsequence of {𝑢_𝑛} (still denoted by {𝑢_𝑛} and 𝑢₀∈𝑊 such that 𝑢𝑢_𝑛⇀𝑢₀weaklyin𝑊,_𝑛𝑢⟶𝑢₀almosteverywhereinΩ,_𝑛⟶𝑢₀stronglyin𝐿^𝑠(Ω)∀1≤𝑠<𝑝^∗.(4.5)
It is easy to see that 𝐽^𝜆(𝑢₀)=0 and 𝜆_Ω𝑓(𝑥)||𝑢_𝑛||^𝑞𝑑𝑥=𝜆_Ω𝑓(𝑥)||𝑢₀||^𝑞𝑑𝑥+𝑜_n(1).(4.6)
Next we verify that 𝑢₀≢0. Arguing by contradiction, we assume 𝑢₀≡0. Setting 𝑙=lim_𝑛→∞_Ω𝑔||𝑢_𝑛||^𝑝∗𝑑𝑥.(4.7) Since 𝐽^𝜆(𝑢_𝑛)=𝑜_𝑛(1) and {𝑢_𝑛} is bounded, then by (4.6), we can deduce that 0=lim_𝑛→∞𝐽^𝜆𝑢_𝑛,𝑢_𝑛=lim_𝑛→∞‖‖𝑢^𝑛‖‖^𝑝−_Ω𝑔||𝑢_𝑛||^𝑝∗=lim_𝑛→∞‖‖𝑢_𝑛‖‖^𝑝−𝑙,(4.8) that is, lim_𝑛→∞‖‖𝑢_𝑛‖‖^𝑝=𝑙.(4.9)
If 𝑙=0, then we get 𝑐=lim_𝑛→∞𝐽_𝜆(𝑢_𝑛)=0, which contradicts with 𝑐>0. Thus we conclude that 𝑙>0. Furthermore, the Sobolev inequality implies that ‖‖𝑢_𝑛‖‖^𝑝≥𝑆_Ω||𝑢_𝑛||^𝑝∗^𝑝/𝑝∗≥𝑆_Ω𝑔||𝑔⁺||_∞||𝑢_𝑛||^𝑝∗^𝑝/𝑝∗=𝑆||𝑔⁺||^{∞−(𝑁−𝑝)/𝑁}_Ω𝑔||𝑢_𝑛||^𝑝∗^𝑝/𝑝∗.(4.10) Then as 𝑛→∞ we have 𝑙=lim_𝑛→∞‖‖𝑢_𝑛‖‖^𝑝≥𝑆||𝑔⁺||^{−(𝑁−𝑝)/𝑁}lim_𝑛→∞_Ω𝑔||𝑢_𝑛||^𝑝∗^𝑝/𝑝∗=𝑆||𝑔⁺||^{∞−(𝑁−𝑝)/𝑁}𝑙^𝑝/𝑝∗,(4.11) which implies that 𝑙≥||𝑔⁺||^{∞−(𝑁−𝑝)/𝑝}𝑆^𝑁/𝑝.(4.12) Hence, from (4.6) to (4.12) we get 𝑐==1lim_𝑛→∞𝐽_𝜆𝑢_𝑛𝑝lim^𝑛→∞‖‖𝑢_𝑛‖‖^𝑝−𝜆𝑞lim^𝑛→∞_Ω𝑓||𝑢_𝑛||^𝑞𝑑𝑥−1𝑝^∗lim_𝑛→∞_Ω=1𝑔||𝑢_𝑛||^𝑝∗𝑑𝑥≥1𝑝−𝑝1^∗𝑙𝑁||𝑔⁺||^{∞−(𝑁−𝑝)/𝑝}𝑆^𝑁/𝑝.(4.13) This is a contradiction to 𝑐<(1/𝑁)|𝑔⁺|^{∞−(𝑁−𝑝)/𝑝}𝑆^𝑁/𝑝. Therefore 𝑢₀ is a nontrivial solution of (𝐸_{𝜆𝑓,𝑔}).

Lemma 4.3. Assume that (𝑓1)-(𝑓2) and (𝑔1)-(𝑔4) hold. Then for any 𝜆>0, there exists 𝑣_𝜆∈𝑊 such that sup_𝑡≥0𝐽_𝜆𝑡𝑣_𝜆<1𝑁||𝑔⁺||^{∞−(𝑁−𝑝)/𝑝}𝑆^𝑁/𝑝.(4.14)
In particular, 𝛼^−𝜆<(1/𝑁)|𝑔⁺|^{∞−(𝑁−𝑝)/𝑝}𝑆^𝑁/𝑝 for all 𝜆∈(0,Λ₁) where Λ₁ is as in (1.5).

Proof. For convenience, we introduce the following notations: 𝐼(𝑢)=_Ω1𝑝||∇𝑢||^𝑝−1𝑝^∗𝑔|𝑢|^𝑝∗𝜒𝑑𝑥,_{𝐵(0,2𝜌0)}=10if𝑥∈𝐵0,2𝜌₀,𝑄(𝑢)=if||∇𝑥𝑢∉||_𝑝𝑝𝐵0,2𝜌₀,|||𝑔𝜒_{𝐵(0,2𝜌0)}^1/𝑝∗𝑢|||^𝑝𝑝_∗.(4.15) From (𝑔3) to (𝑔4), we know that there exists 𝛿₀∈(0,𝜌₀) such that for all 𝑥∈𝐵(0,2𝛿₀), 𝑔(𝑥)=𝑔(0)+𝑜|𝑥|^𝛽forsome𝛽>𝑁𝑝−1.(4.16) Motivated by some ideas of selecting cut-off functions in [20, Lemma 4