Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

Abstract

This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities −Δ𝑢=𝜆𝑎(𝑥)𝑢^𝑝+𝑏(𝑥)𝑢^𝑞 in Ω, 𝑢>0 in Ω, 𝑢=0 on 𝜕Ω, where 𝜆>0 is a real parameter, 0<𝑝<1<𝑞. Ω is a bounded domain in ℝ^𝑁(𝑁≥3), and 𝑎(𝑥) and 𝑏(𝑥) are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.

1. Introduction

In this paper, we consider the elliptic problems with sublinear and superlinear nonlinearities −Δ𝑢=𝜆𝑎(𝑥)𝑢^𝑝+𝑏(𝑥)𝑢^𝑞(1)inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω,^𝜆 where 𝜆>0  is a real parameter,  0<𝑝<1<𝑞.  Ω is a bounded domain in ℝ^𝑁(𝑁≥3), and 𝑎(𝑥) and 𝑏(𝑥) are some given functions which satisfies the following assumptions:(𝐻₁)𝑎(𝑥),𝑏(𝑥)∈𝐿^∞(Ω),  𝑎(𝑥)≥𝑐₀,  𝑏(𝑥)≤−𝑐₁,  where 𝑐₀,  𝑐₁ are positive constants,

or(𝐻₂)𝑎(𝑥), 𝑏(𝑥)∈𝐿^∞(Ω),  𝑎(𝑥),𝑏(𝑥)≥𝑐₀,  where 𝑐₀ is a positive constant.

For convenience, we denote (1)_𝜆 with hypothesis (𝐻₁) or (𝐻₂) by (1)⁻_𝜆 and (1)⁺_𝜆, respectively.

Such problems occur in various branches of mathematical physics and population dynamics, and sublinear analogues or superlinear analogues of (1)_𝜆 have been considered by many authors in recent years (see [1–9] and their references). But most of such studies have been concerned with equations of the type involving sublinear nonlinearity (see [3–6, 8, 9]), with only few references dealing with the elliptic problems with sublinear and superlinear nonlinearities. In [1], Ambrosetti et al. deal with the analogue of (1)_𝜆 with 𝑎(𝑥)=𝑏(𝑥)≡1. It is known from [2] that there exist 𝜆^∗∈(0,∞), such that problem (1)_𝜆 has a solution if 𝜆≤𝜆^∗  and has no solution if 𝜆>𝜆^∗,  provided 𝑏(𝑥)≡1 on Ω.

Our goal in this paper is to show how variational method and super-subsolution method can be used to establish some existence results of problem (1)_𝜆. We work on the Sobolev space 𝐻¹₀(Ω)  equipped with the norm  ‖𝑥‖=(∫_Ω|∇𝑢|²𝑑𝑥)^1/2.   For 𝑢∈𝐻¹₀(Ω)  we define 𝐼_𝜆∶𝐻¹₀(Ω)→ℝ  by 𝐼_𝜆(𝑢)=12_Ω|∇𝑢|²𝜆𝑑𝑥−𝑝+1_Ω𝑎(𝑥)|𝑢|^𝑝+11𝑑𝑥−𝑞+1_Ω𝑏(𝑥)|𝑢|^𝑞+1𝑑𝑥. (1.1)

Let 𝜆₁ be the first eigenvalue of −Δ𝑢=𝜆𝑢,𝑥∈Ω,𝑢=0,𝑥∈𝜕Ω. (1.2)

𝜑₁  denotes the corresponding eigenfunction satisfying  0≤𝜑₁(𝑥)≤1.  𝐿^𝑝(Ω),  (1≤𝑝≤∞), denotes Lebesgue spaces, and the norm in 𝐿^𝑝 is denoted by ‖⋅‖_𝑝.

2. The Existence of Positive Solution of (1)⁻_𝜆

It is well known that ∇𝜑₁(𝑥)≠0,∀𝑥∈𝜕Ω. (2.1) Define  𝑎=min_𝜕Ω|∇𝜑₁|²;   from (2.1) we know  𝑎>0,  so we can split the domain Ω into two parts: Ω_𝜀 and Ω⧵Ω_𝜀, where Ω_𝜀={𝑥∈Ω∶|∇𝜑₁|²≥𝑎/2}⋂{𝑥∈Ω∶𝜑₁(𝑥)≤𝜀,𝜀issmallenough}. Let  𝑏=inf_Ω⧵Ω𝜀𝜑₁(𝑥);  we obtain that  𝑏≥𝜀  by the positivity of 𝜑₁  in  Ω,  and  Ω⧵Ω_𝜀  is nonempty when  𝜀  is small enough.

Theorem 2.1. Let  𝑎(𝑥),  𝑏(𝑥) satisfy assumption (𝐻₁),  and  0<𝑝<1<𝑞<2^∗−1, where  2^∗=2𝑁/(𝑁−2) is the limiting exponent in the Sobolev embedding. Then there exists a constant 𝜆>0  such that (1)⁻_𝜆 possesses at least a weak positive solution 𝑢^∗(𝑥)∈𝐻¹₀(Ω)  for  𝜆≥𝜆.

Proof. Let  𝑒(𝑥)  denote the positive solution of the following equation: −Δ𝑒=1,𝑥∈Ω,𝑒=0,𝑥∈𝜕Ω. (2.2) Here and hereafter we use the following notations: 𝐴=‖𝑎‖_∞,   𝐵=‖𝑏‖_∞,   𝐸=‖𝑒‖_∞. Since 0<𝑝<1, for all 𝜆∈ℝ⁺,  there exists  𝑇=𝑇(𝜆)>0  satisfying 𝑇≥𝜆𝐴𝑇^𝑝𝐸^𝑝. (2.3) Observing that  𝑏(𝑥)≤−𝑐₁<0,  as a consequence, the function  𝑇𝑒  verifies 𝑇=−Δ(𝑇𝑒)≥𝜆𝐴(𝑇𝑒)^𝑝≥𝜆𝑎(𝑥)(𝑇𝑒)^𝑝+𝑏(𝑥)(𝑇𝑒)^𝑞, (2.4) and hence it is a supersolution of (1)⁺_𝜆.  Let  𝑣(𝑥)=𝜑^𝑙₁,  𝑥∈Ω,  𝑙>1. For 𝑥∈Ω, we have 𝑥∈Ω_𝜀 or 𝑥∈Ω⧵Ω_𝜀. We will discuss it from two conditions.
(I)  For all 𝑥∈Ω_𝜀, observing that 𝑙>1 and when 𝜀 is small enough, we have ()𝑠𝑎𝑙𝑙−1⁻²2−𝐵𝑠^{𝑙(𝑞−1)}>𝜆₁𝑙,∀𝑠∈(0,𝜀). (2.5) Since 𝑥∈Ω_𝜀, then it follows that 𝜑₁(𝑥)≤𝜀,  |∇𝜑₁|²≥𝑎/2.  From (2.5) we infer 𝜆₁𝑙≤𝑙(𝑙−1)𝜑₁⁻²∇𝜑₁²−𝐵𝜑₁(𝑥)^{𝑙(𝑞−1)},∀𝑥∈Ω_𝜀. (2.6) Multiplying (2.6) with 𝜑^𝑙₁,  we get 𝑙(1−𝑙)𝜑₁^𝑙−2∇𝜑₁²+𝜆₁𝑙𝜑^𝑙₁≤−𝐵𝜑₁^𝑙𝑞. (2.7) It follows that −Δ𝜑^𝑙₁≤𝜆𝑎(𝑥)𝜑^𝑙₁^𝑝−𝑏(𝑥)𝜑^𝑙₁^𝑞. (2.8)
(II)  For all  𝑥∈Ω⧵Ω_𝜀, there exists  𝜆>0,  such that  for all  𝜆≥𝜆, and we have 𝜆𝑐₀𝑠^𝑝𝑙−𝐵𝑠^𝑞𝑙≥𝜆₁𝑙𝑠^𝑙,∀𝑠∈𝑅,𝑏≤𝑠≤1. (2.9) Since 𝑥∈Ω⧵Ω_𝜀,  then we have 𝜑₁(𝑥)≥𝑏  (and  𝜑₁(𝑥)≤1).  From (2.9), it follows that −Δ𝜑^𝑙₁≤𝜆₁𝑙𝜑^𝑙₁≤𝜆𝑐₀𝜑₁^𝑙𝑝−𝐵𝑠^𝑞𝑙≤𝜆𝑎(𝑥)𝜑^𝑙₁^𝑝+𝑏(𝑥)𝜑^𝑙₁^𝑞. (2.10) From (2.8) and (2.10), we derive that there exists  𝜆>0  such that   for all  𝑥∈Ω,   for all  𝜆≥𝜆, −Δ𝜑^𝑙₁≤𝜆𝑎(𝑥)𝜑^𝑙₁^𝑝+𝑏(𝑥)𝜑^𝑙₁^𝑞, (2.11) that is,  𝑣(𝑥)=𝜑^𝑙₁(𝑥) is a subsolution of (1)⁻_𝜆. Taking 𝑇 as sufficiently large, we also have 𝑇𝑒>𝜑^𝑙₁  by minimal principle. Define  𝑤(𝑥)=𝑇𝑒(𝑥),  and let 𝐾={𝑢∈𝐻¹₀(Ω):𝑣(𝑥)≤𝑢(𝑥)≤𝑤(𝑥),forall𝑥∈Ω},   then 𝐾 is closed and convex (and weakly closed). Let 𝑓(𝑠)=𝜆𝑎(𝑥)𝑠^𝑝+𝑏(𝑥)𝑠^𝑞,  for all  𝑠∈ℝ,𝑠>0.  We consider the function 𝐼_𝜆(𝑢)=12_Ω|∇𝑢|²𝑑𝑥−_Ω^𝑢₀𝑓(𝑠)𝑑𝑠𝑑𝑥. (2.12) Observe that  𝑏(𝑥)<0,  0<𝑝<1<𝑞<2^∗−1;  we infer that  𝐼_𝜆 is coercive, bounded, since it is blow and weakly lower semicontinuous. Using this fact, we conclude that there exists 𝑢^∗∈𝐾, such that 𝐼_𝜆(𝑢^∗)=inf_𝐾𝐼_𝜆 (see [10]). In the following, we will prove that 𝑢^∗ is a solution of problem (1)⁻_𝜆.
For 𝜙∈𝐾, define ℎ∶[0,1]→ℝ, such that ℎ(𝑡)=𝐼𝑡𝜙+(1−𝑡)𝑢^∗. (2.13) Clearly, ℎ(𝑡) achieves its minimum at  𝑡=0, and ℎ^′(𝑡)_𝑡=0=_Ω∇𝑢^∗∇𝜙−𝑢^∗𝑑𝑥−_Ω𝑓𝑢^∗𝜙−𝑢^∗𝑑𝑥≥0. (2.14) For all  𝜑∈𝐻¹₀(Ω),  𝜂>0,  define Ψ(𝑥)=𝑣,when𝑢^∗𝑢+𝜂𝜑<𝑣,_∗+𝜂𝜑,when𝑣≤𝑢^∗+𝜂𝜑≤𝑤,𝑤,when𝑢^∗+𝜂𝜑>𝑤. (2.15) Obviously,  Ψ∈𝐾,  and inserting (2.15) into (2.14),  we find 0≤_{𝑣≤𝑢∗+𝜂𝜑≤𝑤}∇𝑢^∗⋅∇𝜂𝜙−𝑓𝑢^∗𝜂𝜑𝑑𝑥+_{𝑢∗+𝜂𝜑>𝑤}∇𝑢^∗∇𝑤−𝑢^∗−𝑓𝑢^∗𝑤−𝑢^∗+𝑑𝑥_{𝑢∗+𝜂𝜑<𝑣}∇𝑢^∗∇𝑣−𝑢^∗−𝑓𝑢^∗𝑣−𝑢^∗𝑑𝑥=𝜂_{𝑣≤𝑢∗+𝜂𝜑≤𝑤}∇𝑢^∗⋅∇𝜑−𝑓𝑢^∗𝜑+𝑑𝑥_{𝑢∗+𝜂𝜑>𝑤}∇𝑤⋅∇𝑤−𝑢^∗−𝑓(𝑤)𝑤−𝑢^∗+𝑑𝑥_{𝑢∗+𝜂𝜑<𝑣}∇𝑣⋅∇𝑣−𝑢^∗−𝑓(𝑣)𝑣−𝑢^∗𝑑𝑥−_{𝑢∗+𝜂𝜑>𝑤}∇𝑤−∇𝑢^∗2𝑑𝑥−_{𝑢∗+𝜂𝜑<𝑣}∇𝑣−∇𝑢^∗2+𝑑𝑥_{𝑢∗+𝜂𝜑>𝑤}𝑓(𝑤)−𝑓𝑢^∗𝑤−𝑢^∗+𝑑𝑥_{𝑢∗+𝜂𝜑<𝑣}𝑓(𝑣)−𝑓𝑢^∗𝑣−𝑢^∗𝑑𝑥. (2.16) Since 𝑤(𝑥) and 𝑣(𝑥) are supersolution and subsolution, respectively, then _{𝑢∗+𝜂𝜑>𝑤}∇𝑤⋅∇𝑤−𝑢^∗−𝑓(𝑤)𝑤−𝑢^∗𝑑𝑥≤𝜂_{𝑢∗+𝜂𝜑>𝑤}∇𝑤⋅∇𝜑−𝑓(𝑤)𝜑𝑑𝑥,_{𝑢∗+𝜂𝜑<𝑣}∇𝑣⋅∇𝑣−𝑢^∗−𝑓(𝑣)𝑣−𝑢^∗𝑑𝑥≤𝜂_{𝑢∗+𝜂𝜑<𝑣}∇𝑣⋅∇𝜑−𝑓(𝑣)𝜑𝑑𝑥. (2.17) Observe that meas[𝑢^∗+𝜂𝜑>𝑤]→0, meas[𝑢^∗+𝜂𝜑<𝑣]→0, as 𝜂→0, ∇𝑤⋅∇𝜑−𝑓(_{𝑢∗+𝜂𝜑>𝑤}𝑤)𝜑𝑑𝑥⟶0,_{𝑢∗+𝜂𝜑<𝑣}∇𝑣⋅∇𝜑−𝑓(𝑣)𝜑𝑑𝑥⟶0. (2.18) Since  𝑢^∗∈𝐾,𝑏(𝑥)<0, it follows that _{𝑢∗+𝜂𝜑>𝑤}𝑓(𝑤)−𝑓𝑢^∗𝑤−𝑢^∗𝑑𝑥=_{𝑢∗+𝜂𝜑>𝑤}𝜆𝑎(𝑥)𝑤^𝑝−𝑢^∗𝑝𝑤−𝑢^∗𝑑𝑥+_{𝑢∗+𝜂𝜑>𝑤}𝑏(𝑥)𝑤^𝑞−𝑢^∗𝑞𝑤−𝑢^∗≤𝑑𝑥_{𝑢∗+𝜂𝜑>𝑤}𝜆𝑎(𝑥)𝑤^𝑝−𝑢^∗𝑝𝑤−𝑢^∗𝑑𝑥. (2.19) Similar to (2.19), we have _{𝑢∗+𝜂𝜑<𝑣}𝑓(𝑣)−𝑓𝑢^∗𝑣−𝑢^∗𝑑𝑥=_{𝑢∗+𝜂𝜑<𝑣}𝜆𝑎(𝑥)𝑣^𝑝−𝑢^∗𝑝𝑣−𝑢^∗𝑑𝑥+_{𝑢∗+𝜂𝜑<𝑣}𝑏(𝑥)𝑣^𝑞−𝑢^∗𝑞𝑣−𝑢^∗≤𝑑𝑥_{𝑢∗+𝜂𝜑<𝑣}𝜆𝑎(𝑥)𝑣^𝑝−𝑢^∗𝑝𝑣−𝑢^∗𝑑𝑥. (2.20) Similar to (2.18), as  𝜂→0, it follows that 𝜆𝑎(_{𝑢∗+𝜂𝜑>𝑤}𝑥)𝑤^𝑝−𝑢^∗𝑝𝑤−𝑢^∗𝑑𝑥⟶0,_{𝑢∗+𝜂𝜑<𝑣}𝜆𝑎(𝑥)𝑣^𝑝−𝑢^∗𝑝𝑣−𝑢^∗𝑑𝑥⟶0. (2.21) As 𝜂→0, we also have _{𝑣≤𝑢∗+𝜂𝜑≤𝑤}∇𝑢^∗⋅∇𝜑−𝑓𝑢^∗𝜑𝑑𝑥⟶_Ω∇𝑢^∗⋅∇𝜑−𝑓𝑢^∗𝜑𝑑𝑥. (2.22) Inserting (2.17), (2.19), and (2.20) into (2.16), we find 0≤𝜂_{𝑣≤𝑢∗+𝜂𝜑≤𝑤}∇𝑢^∗⋅∇𝜑−𝑓𝑢^∗𝜑𝑑𝑥+_{𝑢∗+𝜂𝜑>𝑤}(𝑤)𝜑+∇𝑤⋅∇𝜑−𝑓𝑑𝑥_{𝑢∗+𝜂𝜑<𝑣}∇𝑣⋅∇𝜑−𝑓(𝑣)𝜑𝑑𝑥+_{𝑢∗+𝜂𝜑>𝑤}𝜆𝑎(𝑥)𝑤^𝑝−𝑢^∗𝑝𝑤−𝑢^∗+𝑑𝑥_{𝑢∗+𝜂𝜑<𝑣}𝜆𝑎(𝑥)𝑣^𝑝−𝑢^∗𝑝𝑣−𝑢^∗𝑑𝑥. (2.23) Dividing by 𝜂 and letting  𝜂→0, using (2.18), (2.21), and (2.22), we derive _Ω∇𝑢^∗⋅∇𝜑−𝑓𝑢^∗𝜑𝑑𝑥≥0. (2.24) Noting that 𝜑 is arbitrary, this holds equally for  −𝜑, and it follows that 𝑢^∗ is indeed a weak solution of (1)⁻_𝜆,  and the strong maximum principle yields  𝑢^∗>𝜑^𝑙₁,  in Ω. Therefore it is a weak positive solution of (1)⁻_𝜆.

3. The Existence of Positive Solution of (1)⁺_𝜆

Theorem 3.1. Let 𝑎(𝑥),  𝑏(𝑥) satisfy assumption (𝐻₂),  and  0<𝑝<1<𝑞<+∞. Then there exists Λ∈ℝ,  Λ>0,  such that(i)for all 𝜆∈(0,Λ) problem (1)⁺_𝜆 has a minimal solution 𝑢_𝜆 such that 𝐼_𝜆(𝑢_𝜆)<0. Moreover 𝑢_𝜆 is increasing with respect to 𝜆;(ii)for  𝜆=Λ problem (1)⁺_𝜆 has at least one weak solution 𝑢∈𝐻∩𝐿^𝑝+1;(iii)for all  𝜆>Λ  problem  (1)⁺_𝜆 has no solution.

To prove Theorem 3.1, let us define Λ=sup𝜆>0∶(1)⁺_𝜆hasasolution. (3.1)First of all we prove a useful lemma.

Lemma 3.2. One has 0<Λ<+∞.

Proof. Let 𝑒(𝑥) denote the solution of the following equation: −Δ𝑒=1,𝑥∈Ω,𝑒=0,𝑥∈𝜕Ω. (3.2) Since  0<𝑝<1<𝑞,  we can find  𝜆₀>0  such that for all  0<𝜆≤𝜆₀  there exists  𝑇=𝑇(𝜆)>0  satisfying 𝑇≥𝜆𝐴𝑇^𝑝𝐸^𝑝+𝐵𝑇^𝑞𝐸^𝑞. (3.3) As a consequence,  the function 𝑇𝑒 verifies 𝑇=−Δ(𝑇𝑒)≥𝜆𝐴(𝑇𝑒)^𝑝+𝐵(𝑇𝑒)^𝑞≥𝜆𝑎(𝑥)(𝑇𝑒)^𝑝+𝑏(𝑥)(𝑇𝑒)^𝑞, (3.4) and hence it is a supersolution of (1)⁺_𝜆.  Moreover, let 𝑢₀ denote the solution of the following problem: −Δ𝑢=𝜆𝑎(𝑥)𝑢^𝑝₀𝑢,𝑥∈Ω,₀=0,𝑥∈𝜕Ω. (3.5) (From [3] we know that 𝑢₀ exists.)  Then 𝜀𝑢₀ is a subsolution of (1)⁺_𝜆, provided −Δ𝜀𝑢₀=𝜆𝜀𝑎(𝑥)𝑢^𝑝₀≤𝜆𝑎(𝑥)𝜀𝑢₀^𝑝+𝑏(𝑥)𝜀𝑢₀^𝑞, (3.6) which is satisfied for all  𝜀>0  small enough and all  𝜆. Taking  𝜀  as possibly smaller,  we also have 𝜀𝑢₀<𝑇𝑒. (3.7) It follows that  (1)⁺_𝜆 has a solution 𝑢,  𝜀𝑢₀≤𝑢≤𝑇𝑒 whenever 𝜆≤𝜆₀, and thus Λ≥𝜆₀.  
Next, let 𝜆^∗ be such that 𝑐₀𝜆^∗𝑡^𝑝+𝑡^𝑞>𝜆₁𝑡,∀𝑡>0. (3.8) If 𝜆 is such that  (1)⁺_𝜆  has a solution 𝑢, multiplying (1)⁺_𝜆 by 𝜑₁ and integrating over Ω we find 𝜆_1Ω𝑢𝜑₁𝑑𝑥=𝜆_Ω𝑎(𝑥)𝑢^𝑝𝜑₁𝑑𝑥+_Ω𝑏(𝑥)𝑢^𝑞𝜑₁𝑑𝑥≥𝑐_0Ω𝜆𝑢^𝑝𝜑₁+𝑢^𝑞𝜑₁𝑑𝑥. (3.9) This and (3.5)  immediately imply that  𝜆<𝜆^∗  and show that  Λ≤𝜆^∗,  hence  0<Λ<+∞.

We are now ready to give the proof of Theorem 3.1.

Proof. (i) From the proof of lemma, it follows that, for all  𝜆∈(0,Λ),   problem  (1)⁺_𝜆  has a solution 𝑢_𝜆.  Let 𝑢₀ satisfy (3.5); the iteration −Δ𝑢_𝑛+1=𝜆𝑎(𝑥)𝑢^𝑝_𝑛+𝑏(𝑥)𝑢^𝑞_𝑛 (3.10) satisfies  𝑢_𝑛↑𝑢_𝜆  by making use of Lemma  3.3 of [1] and maximum principle. It is easy to check that 𝑢_𝜆 is a minimal solution of  (1)⁺_𝜆. Indeed, if 𝑢 is any solution of (1)⁺_𝜆,  then  𝑢≥𝑢₀  and 𝑢 is a supersolution of  (1)⁺_𝜆. Thus  𝑢_𝑛≤𝑢,  for all  𝑛, by induction, and 𝑢_𝜆≤𝑢. Next, we will prove that 𝐼_𝜆(𝑢_𝜆)<0. Indeed, 𝐼_𝜆(𝑢)=12_Ω|∇𝑢|²𝜆𝑑𝑥−𝑝+1_Ω𝑎(𝑥)|𝑢|^𝑝+11𝑑𝑥−𝑞+1_Ω𝑏(𝑥)|𝑢|^𝑞+1𝑑𝑥. (3.11) Since 𝑢_𝜆 is a solution of  (1)⁺_𝜆 we have _Ω∇𝑢_𝜆²𝑑𝑥=_Ω𝜆𝑎(𝑥)𝑢_𝜆^𝑝+1𝑑𝑥+_Ω𝑏(𝑥)𝑢_𝜆^𝑞+1𝑑𝑥. (3.12) From Lemma  3.5 of [1], we know _Ω∇𝜑²−𝜆𝑝𝑎(𝑥)𝑢_𝜆^𝑝−1+𝑞𝑏(𝑥)𝑢_𝜆^𝑞−1𝜑²𝑑𝑥≥0,∀𝜑∈𝐻¹₀. (3.13) In particular with 𝜑=𝑢_𝜆, we infer _Ω∇𝑢_𝜆²𝑑𝑥−𝜆𝑝_Ω𝑎(𝑥)𝑢_𝜆^𝑝+1𝑑𝑥−𝑞_Ω𝑏(𝑥)𝑢_𝜆^𝑞+1𝑑𝑥≥0. (3.14) Combining (3.12) and (3.14), we obtain 𝐼_𝜆𝑢_𝜆1=𝜆2−1𝑝+1_Ω𝑎(𝑥)𝑢_𝜆^𝑝+11𝑑𝑥+2−1𝑞+1_Ω𝑏(𝑥)𝑢_𝜆^𝑞+1≤𝑑𝑥1−𝑝2−1+1𝑝+1𝑞+1_Ω𝑎(𝑥)𝑢_𝜆^𝑝+1𝑑𝑥<0. (3.15) To complete the proof of (i), it remains to show that 𝑢_𝜆<𝑢_𝜆1whenever𝜆<𝜆₁. (3.16)Indeed, if  𝜆<𝜆₁ then 𝑢_𝜆1 is a supersolution of  (1)⁺_𝜆. Since, for  𝜀>0  small,  𝜀𝑢₀  is a subsolution of (1)⁺_𝜆 and  𝜀𝑢₀<𝑢_𝜆1,  then (1)⁺_𝜆 possesses a solution 𝑣, with 𝜀𝑢₀≤𝑣≤𝑢_𝜆1. (3.17) Since 𝑢_𝜆 is the minimal solution of (1)⁺_𝜆, we infer that  𝑢_𝜆≤𝑣≤𝑢_𝜆1.   Moreover −Δ𝑢_𝜆1−𝑢_𝜆=𝜆₁𝑎(𝑥)𝑢^𝑝_𝜆1+𝑏(𝑥)𝑢^𝑞_𝜆1−𝜆𝑎(𝑥)𝑢^𝑝_𝜆+𝑏(𝑥)𝑢^𝑞_𝜆≥𝜆𝑎(𝑥)𝑢^𝑝_𝜆1+𝑏(𝑥)𝑢^𝑞_𝜆1−𝑎(𝑥)𝑢^𝑝_𝜆−𝑏(𝑥)𝑢^𝑞_𝜆≥0. (3.18) Since  𝑢_𝜆1≠𝑢_𝜆  (because 𝜆<𝜆₁), then the Hopf Maximum principle yields  𝑢_𝜆<𝑢_𝜆1.
(ii) Let 𝜆_𝑛 be a sequence such that  𝜆_𝑛↑Λ; then from  𝐼_𝜆𝑛(𝑢_𝜆𝑛)<0  we deduce that there exists 𝐶>0 such that ∇𝑢_𝑛²𝑢≤𝐶,_{𝑛𝑝+1𝑝+1}≤𝐶. (3.19) Then there exists 𝑢^∗∈𝐻¹₀ such that 𝑢_𝑛→𝑢^∗>0 a.e. in Ω, strongly in 𝐿^𝑝+1 and weakly in 𝐻¹₀. Such a 𝑢^∗ is thus a weak solution of (1)⁺_𝜆  for 𝜆=Λ.
(iii) This follows from the definition of Λ.

Acknowledgment

This work supported by the Physics and Mathematics Foundation of Changzhou University (ZMF10020065).