Existence and Nonexistence of Positive Solutions for Quasilinear Elliptic Problem

Abstract

Using variational arguments we prove some existence and nonexistence results for positive solutions of a class of elliptic boundary-value problems involving the 𝑝-Laplacian.

1. Introduction

In a recent paper, Rădulescu and Repovš [1] studied the existence and nonexistence of positive solutions of the nonlinear elliptic problem −Δ𝑢=𝜆𝑘(𝑥)𝑢^𝑞±ℎ(𝑥)𝑢^𝑝𝑢|inΩ,_𝜕Ω=0,𝑢>0inΩ, (1.1) where Ω is a smooth bounded domain in ℝ^𝑛, 𝜆>0 is a parameter, 0<𝑞<1<𝑝, and ℎ, 𝑘 in 𝐿^∞(Ω) such that essinf_𝑥∈Ω𝑘(𝑥)>0,essinf_𝑥∈Ωℎ(𝑥)>0. (1.2) They showed using sub-supersolutions arguments and monotonicity methods that the problem (1.1)₊ has a minimal solution, provided that 𝜆>0 is small enough. The next result is concerned with problem (1.1)₋ and asserts that there is some 𝜆^∗>0 such that (1.1)₋ has a nontrivial solution if 𝜆>𝜆^∗ and no solution exists provided that 𝜆<𝜆^∗.

In the present paper we consider that the corresponding quasilinear problem −Δ_𝑝𝑢=𝜆𝑘(𝑥)𝑢^𝑞±ℎ(𝑥)𝑢^𝑟𝑢|inΩ,_𝜕Ω=0,𝑢>0inΩ, (1.3) where Δ_𝑝𝑢=div(|∇𝑢|^𝑝−2∇𝑢), denotes the 𝑝-Laplacian operator, 1<𝑝<∞, 𝜆>0, 0≤𝑞<𝑝−1<𝑟<𝑝^∗−1, with 𝑝^∗=𝑁𝑝/(𝑁−𝑝) if 𝑝<𝑁, and 𝑝^∗=+∞ otherwise, and ℎ, 𝑘 in 𝐿^∞(Ω) such that essinf_𝑥∈Ω𝑘(𝑥)>0,essinf_𝑥∈Ωℎ(𝑥)>0. (1.4) We are concerned with the existence of weak solutions of problems (1.3)₊ and (1.3)₋, that is, for functions 𝑢∈𝑊₀^1,𝑝(Ω) satisfying essinf_𝐾𝑢>0 over every compact set 𝐾⊂Ω and _Ω|∇𝑢|^𝑝−2∇𝑢⋅∇𝜙d𝑥=𝜆_Ω𝑘(𝑥)𝑢^𝑞𝜙d𝑥±_Ωℎ(𝑥)𝑢^𝑟𝜙d𝑥 (1.5) for all 𝜙∈𝐶^∞_c(Ω). As usual, 𝐶^∞_c(Ω) denotes the space of all 𝐶^∞ functions 𝜙∶Ω→ℝ with compact support. Using variational methods, we will prove the following theorems.

Theorem 1.1. Assume 0≤𝑞<𝑝−1<𝑟<𝑝^∗−1. Then there exists a positive number Λ such that the following properties hold: (1)for all 𝜆∈(0,Λ) problem (1.3)₊ has a minimal solution 𝑢_𝜆;(2)Problem (1.3)₊ has a solution if 𝜆=Λ;(3)Problem (1.3)₊ does not have any solution if 𝜆>Λ.

Theorem 1.2. Assume 0≤𝑞<𝑝−1<𝑟<𝑝^∗−1. Then there exists a positive number Λ such that the following properties hold: (1)If 𝜆>Λ, then problem (1.3)₋ has at least one solution; (2)If 𝜆<Λ, then problem (1.3)₋ does not have any solution.

2. Proof of Theorem 1.1

At first, we give the definition of weak supersolution and subsolution of (1.3)₊. By definition 𝑢∈𝑊₀^1,𝑝(Ω) is a weak subsolution to (1.3)₊ if 𝑢>0 in Ω and _Ω|∇𝑢|^𝑝−2∇𝑢⋅∇𝜙d𝑥≤𝜆_Ω𝑘(𝑥)𝑢^𝑞𝜙d𝑥±_Ωℎ(𝑥)𝑢^𝑟𝜙d𝑥 (2.1) for all 𝜙∈𝐶^∞_c(Ω). Similarly 𝑢∈𝑊₀^1,𝑝(Ω) is a weak supersolution to (1.3)₊ if in the above the reverse inequalities hold.

Let us define Λ=^defsup𝜆>0∶(1.3)₊hasaweaksolution (2.2) and the energy functional 𝐸_𝜆∶𝑊₀^1,𝑝(Ω)→ℝ defined by 𝐸_𝜆(𝑢)=1𝑝_Ω|∇𝑢|^𝑝−𝜆d𝑥𝑞+1_Ω𝑘(𝑥)𝑢^𝑞+11d𝑥−𝑟+1_Ωℎ(𝑥)𝑢^𝑞+1d𝑥 (2.3) in the Sobolev space 𝑊₀^1,𝑝(Ω).

The proof of the theorem is organized in several steps.

Step 1 (existence of minimal solution for 0<𝜆<Λ). To show the existence of a solution to (1.3)₊, we construct a subsolution 𝑢_𝜆, and a supersolution 𝑢_𝜆, such that 𝑢_𝜆≤𝑢_𝜆.
We introduce the following Dirichlet problem: −Δ_𝑝𝑢=𝜆𝑘(𝑥)𝑢^𝑞|inΩ,𝑢_𝜕Ω=0,𝑢>0inΩ. (2.4) From [2] we know there exists a unique solution, say 𝑢, satisfying the problem (2.4). Define 𝑢_𝜆=𝜖𝑢. Then −Δ_𝑝(𝑢_𝜆)=𝜆𝑘(𝑥)𝜖^𝑝−1𝑢^𝑞 and 𝑢_𝜆 is a subsolution of the problem (1.3)₊ if 𝜆𝑘(𝑥)𝜖^𝑝−1𝑢^𝑞≤𝜆𝑘(𝑥)𝜖^𝑞𝑢^𝑞+ℎ(𝑥)𝜖^𝑟𝑢^𝑟. (2.5) Indeed, for 𝜖 small enough we get 𝜆𝑘(𝑥)𝜖^𝑝−1𝑢^𝑞≤𝜆𝑘(𝑥)𝜖^𝑞𝑢^𝑞≤𝜆𝑘(𝑥)𝜖^𝑞𝑢^𝑞+ℎ(𝑥)𝜖^𝑟𝑢^𝑟. (2.6) (Since 𝑞<𝑝−1 and for 𝜖∈(0,1)). Then 𝜖𝑢 is a subsolution of the problem (1.3)₊.
On the other hand, let 𝑣 the solution to the following problem be: −Δ_𝑝𝑣=𝜆+1inΩ,𝑣|_𝜕Ω=0,𝑣>0inΩ. (2.7)
Then 0<𝑣<𝐾 in Ω. By simplicity of writing we call 𝐹(𝑢)=𝜆𝑘(𝑥)𝑢^𝑞+ℎ(𝑥)𝑢^𝑟. (2.8)
Define 𝑢_𝜆(𝑥)=𝑇𝑣(𝑥) where 𝑇 is a constant that will be chosen in such a way that −Δ_𝑝𝑢_𝜆≥𝐹(𝑇𝑀)≥𝐹𝑢_𝜆, (2.9) where 𝑀=max{1,‖𝑣‖_∞}. Now −Δ_𝑝𝑢_𝜆=𝑇^𝑝−1(𝜆+1) and 𝐹𝑢_𝜆≡𝜆𝑘(𝑥)𝑇^𝑞𝑣^𝑞+𝑇^𝑟𝑣^𝑟≤𝜆𝑐₁𝑇^𝑞𝑀^𝑞+𝑐₂𝑇^𝑟𝑀^𝑟, (2.10) where 𝑐₁=‖𝑘‖_𝐿∞ et 𝑐₂=‖ℎ‖_𝐿∞. Then, it is sufficient to find 𝑇 such that (𝜆+1)≥𝜆𝑐₁𝑇^{𝑞+1−𝑝}𝑀^𝑞+𝑐₂𝑇^{𝑟+1−𝑝}𝑀^𝑟. (2.11)
We call 𝜑(𝑇)=𝜆𝐴𝑇^{𝑞+1−𝑝}+𝐵𝑇^{𝑟+1−𝑝}, (2.12) with 𝐴=𝑐₁𝑀^𝑞,𝐵=𝑐₂𝑀^𝑟. Then lim_𝑇→0+𝜑(𝑇)=lim_𝑇→∞𝜑(𝑇)=∞, (2.13) because 𝑞+1−𝑝<0<𝑟+1−𝑝; then 𝜑 attains a minimum in [0,∞). Elementary computations shows that this function attains its minimum for 𝑇₀=𝐶𝜆^{1/(𝑟−𝑞)} where 𝐶=[𝐴𝐵⁻¹(𝑟−𝑝+1)(𝑝−𝑞−1)⁻¹]^{1/(𝑟−𝑞)}. For the validity of (2.11) it suffices that 𝜑𝑇₀≤𝜆+1, (2.14) that is, 𝐷𝜆^{(𝑟+1−𝑝)/(𝑟−𝑞)}<𝜆+1, (2.15) where 𝐷 is a constant, depends on 𝑝,𝑞, and 𝑀. Then there exists 𝜆₀ such that for 0<𝜆<𝜆₀,𝑢(𝑥)=𝑇₀𝑣 is a supersolution of problem (1.3)₊. It remains to show that 𝜖𝑢≤𝑇₀𝑣. In turn, fix the supersolution, that is, 𝑇, for 𝜖 small enough, we get −Δ_𝑝𝑢_𝜆=𝜆𝑘(𝑥)𝜖^𝑝−1𝑢^𝑞≤𝜆𝜖^𝑝−1≤−Δ_𝑝𝑢_𝜆. (2.16) Consequently, we may apply the weak comparison principle (see Proposition 2.3 in [3]) in order to conclude that 𝑢_𝜆≤𝑢_𝜆. Thus, By the classical iteration method (1.3)₊ has a solution between the subsolution and supersolution.
Let us now prove that 𝑢_𝜆 is a minimal solution of (1.3)₊. We use here the weak comparison principle (see Proposition 2.3 in Cuesta and Takác̆ [3]) and the following monotone iterative scheme: −Δ_𝑝𝑢_𝑛=𝜆𝑘(𝑥)𝑢^𝑞_𝑛−1+ℎ(𝑥)𝑢^𝑟_𝑛−1𝑢inΩ;_𝑛𝜕Ω=0, (2.17) where 𝑢₀=𝑢_𝜆, the unique solution to (2.4). Note that 𝑢₀ is a weak subsolution to (1.3)₊. and 𝑢₀≤𝑈 where 𝑈 is any weak solution to (1.3)₊. Then, from the weak comparison principle, we get easily that 𝑢₀≤𝑢₁ and {𝑢_𝑛}^∞_𝑛=1 is a nondecreasing sequence. Furthermore, 𝑢_𝑛≤𝑈 and {𝑢_𝑛}^∞_𝑛=1 is uniformly bounded in 𝑊₀^1,𝑝(Ω). Hence, it is easy to prove that {𝑢_𝑛} converges weakly in 𝑊₀^1,𝑝(Ω) and pointwise to 𝑢_𝜆, a weak solution to (1.3)₊. Let us show that 𝑢_𝜆 is the minimal solution to (1.3)₊ for any 0<𝜆<Λ. Let 𝑣_𝜆 a weak solution to (1.3)₊ for any 0<𝜆<Λ. Then, 𝑢₀=𝑢_𝜆≤𝑣_𝜆. From the weak comparison principle, 𝑢_𝑛≤𝑣_𝜆 for any 𝑛≥0. Letting 𝑛→∞, we get 𝑢_𝜆≤𝑣_𝜆. This completes the proof of the Step 1.

Step 2 (there exists Λ>0 such that (1.3)₊ has no positive solution for 𝜆>Λ). From the definition of Λ, problem (1.3)₊ does not have any solution if 𝜆>Λ. In what follows we claim that Λ<∞. We argue by contradiction: suppose there exists a sequence 𝜆_𝑛→∞ such that (1.3)₊ admits a solution 𝑢_𝑛. Denote 𝑚∶=minessinf_𝑥∈Ω𝑘(𝑥),essinf_𝑥∈Ωℎ(𝑥)>0. (2.18) There exists 𝜆_∗>0 such that 𝑚𝜆𝑡^𝑞+𝑡^𝑟≥𝜆₁+𝜖𝑡^𝑝−1∀𝑡>0,𝜖∈(0,1),𝜆>𝜆_∗, (2.19) where 𝜆₁ is the first Dirichlet eigenvalue of −Δ_𝑝 is positive and is given by 𝜆₁=min_𝑢≠0∫_Ω|∇𝑢|^𝑝∫_Ω|𝑢|^𝑝 (2.20) (see Lindqvist [4]). Choose 𝜆_𝑛>𝜆_∗. Clearly 𝑢_𝑛 is a supersolution of the problem −Δ_𝑝𝑢=𝜆₁+𝜖𝑢^𝑝−1|inΩ,𝑢>0,𝑢_𝜕Ω=0 (2.21) for all 𝜖∈(0,1). We now use the result in [2] to choose 𝜇<𝜆₁+𝜖 small enough so that 𝜇𝜙₁(𝑥)<𝑢_𝑛(𝑥) and 𝜇𝜙₁ is a subsolution to problem (2.8). By a monotone interation procedure we obtain a solution to (2.8) for any 𝜖∈(0,1), contradicting the fact that 𝜆₁ is an isolated point in the spectrum of −Δ_𝑝 in 𝑊₀^1,𝑝(Ω) (see Anane [5]). This proves the claim and completes the proof of the Step 2.

Step 3 (there exists at least one positive-weak solution for 𝜆=Λ to (1.3)₊). Let {𝜆_𝑘}_𝑘∈ℕ be such that 𝜆_𝑘↑Λ as 𝑘→∞. Then, from Step 1, there exists 𝑢_𝑘=𝑢_𝜆𝑘≥𝑢_𝜆𝑘 to a weak positive solution to (1.3)₊ for 𝜆=𝜆_𝑘. Therefore, for any 𝜙∈𝐶^∞_𝑐(Ω), we have _Ω∇𝑢_𝑘^𝑝−2∇𝑢_𝑘∇𝜙d𝑥=𝜆_𝑘𝑘(𝑥)_Ω𝑢_𝑘^𝑞𝜙d𝑥+ℎ(𝑥)_Ω𝑢^𝑟_𝑘𝜙d𝑥. (2.22) Since 𝑢_𝑘∈𝑊₀^1,𝑝(Ω) and 𝑢_𝑘≥𝑢_𝜆𝑘 it is easy to see that (2.22) holds also for 𝜙∈𝑊₀^1,𝑝(Ω). Moreover, from above 𝐸_𝜆𝑘𝑢_𝑘≤𝐸_𝜆𝑘𝑢_𝜆𝑘<1𝑝_Ω∇𝑢_𝜆𝑘^𝑝𝜆d𝑥−^𝑘𝑘(𝑥)𝑞+1_Ω𝑢_𝜆𝑘^𝑞+1d𝑥<0, (2.23) it follows that sup_𝑘𝑢_𝑘𝑝<∞. (2.24)
Hence, there exists 𝑢_Λ≥𝑢_𝜆𝑘 such that 𝑢_𝑘⇀𝑢_Λ in 𝑊₀^1,𝑝(Ω) as 𝑘→∞ and then by Sobolev imbedding and using the fact that 𝑘,ℎ∈𝐿^∞(Ω): 𝑢_𝑘⇀𝑢in𝐿^𝑞(Ω)andpointwisea.e.as𝑘⟶∞. (2.25) From (2.22), (2.24), and (2.25), we get for any 𝜙∈𝑊₀^1,𝑝(Ω)_Ω∇𝑢_Λ^𝑝−2∇𝑢_Λ∇𝜙d𝑥=𝜆_Ω𝑘(𝑥)𝑢^𝑞_Λ𝜙d𝑥+_Ωℎ(𝑥)𝑢^𝑟_Λ𝜙d𝑥 (2.26) which completes the proof of the Step 3 and gives the proof of Theorem 1.1.

3. Proof of Theorem 1.2

At first, we introduce some notation which will be used throughout the proof. The norm in 𝑊₀^1,𝑝(Ω) will be denoted by ‖𝑢‖=_𝑝^def_Ω|∇𝑢|^𝑝d𝑥^1/𝑝. (3.1) The norm in 𝐿^𝑞+1(Ω) will be denoted by ‖𝑢‖=_𝑞+1^def_Ω|𝑢|^𝑞+1d𝑥^1/𝑞+1. (3.2) The norm in 𝐿^𝑟+1(Ω) will be denoted by ‖𝑢‖=_𝑟+1^def_Ω|𝑢|^𝑟+1d𝑥^1/𝑟+1. (3.3) Let us define the energy functional 𝐽_𝜆∶𝑊₀^1,𝑝(Ω)→ℝ defined by 𝐽_𝜆(𝑢)=1𝑝_Ω|∇𝑢|^𝑝−𝜆d𝑥𝑞+1_Ω𝑘(𝑥)𝑢^𝑞+11d𝑥+𝑟+1_Ωℎ(𝑥)𝑢^𝑟+1d𝑥 (3.4) in the Sobolev space 𝑊₀^1,𝑝(Ω).

The proof of the theorem is organized in several steps.

Step 1 (coercivity of 𝐽_𝜆:). For any 𝑢∈𝑊₀^1,𝑝(Ω) and all 𝜆>0𝐽_𝜆(𝑢)≥1𝑝‖𝑢‖^𝑝−𝐶₁‖𝑢‖^𝑞+1_𝑞+1+𝐶₂‖𝑢‖^𝑟+1_𝑟+1, (3.5) where 𝐶₁=𝜆‖𝑘‖_𝐿∞/(𝑞+1) and 𝐶₂=(𝑟+1)⁻¹essinf_𝑥∈Ωℎ(𝑥) are positive constants. We call 𝜙(𝑇)=𝐴𝑇^{𝑞+1−𝑝}−𝐵𝑇^{𝑟+1−𝑝}. (3.6) Then lim_𝑇→0+𝜙(𝑇)=lim_𝑇→∞𝜙(𝑇)=∞, (3.7) because 𝑞+1−𝑝<0<𝑟+1−𝑝; then 𝜑 attains a minimum 𝑚<0 in [0,∞). By elementary computations shows that this function attains its minimum for 𝑇=[𝐴(𝑞+1−𝑝)/(𝐵𝑟+1−𝑝)]^{1/(𝑟−𝑞)}.
Returning to (3.5), we deduce that 𝐽_𝜆(𝑢)≥1𝑝‖𝑢‖^𝑝+𝑚. (3.8) Hence, from (3.8), we get that 𝐽_𝜆(𝑢)⟶+∞as‖𝑢‖⟶∞. (3.9)
Let 𝑛↦𝑢_𝑛 be a minimizing sequence of 𝐽_𝜆 in 𝑊₀^1,𝑝(Ω), which is bounded in 𝑊₀^1,𝑝(Ω) by Step 1. Without loss of generality, we may assume that (𝑢_𝑛)_𝑛 is nonnegative, converges weakly to some 𝑢 in 𝑊₀^1,𝑝(Ω), and converges also pointwise. Moreover, by the weak lower semicontinuity of the norm ‖⋅‖ and the boundedness of (𝑢_𝑛)_𝑛 in 𝑊₀^1,𝑝(Ω) we get 𝐽_𝜆(𝑢)≤lim_𝑛→∞inf𝐽_𝜆𝑢_𝑛. (3.10) Hence 𝑢 is a global minimizer of 𝐽_𝜆 in 𝑊₀^1,𝑝(Ω), which completes the proof of the Step 1.

Step 2 (the weak limit 𝑢 is a nonnegative weak solution of (1.3)_- if 𝜆>0 is sufficiently large). Firstly, observe that 𝐽_𝜆(0)=0. Thus, to prove that the nonnegative solution is nontrivial, it suffices to prove that there exists 𝜆^∗>0 such that inf_{𝑢∈𝑊01,𝑝(Ω)}𝐽_𝜆(𝑢)<0∀𝜆>𝜆^∗. (3.11) For this, we consider the constrained minimization problem 𝜆^∗def1=inf𝑝_Ω|∇𝑤|^𝑝1d𝑥+𝑟+1_Ωℎ(𝑥)|𝑤|^𝑟+1d𝑥∶𝑤∈𝑊₀^1,𝑝(Ω)1and×𝑞+1_Ω𝑘(𝑥)|𝑤|^𝑞+1d𝑥=1. (3.12)
Let 𝑛↦𝑣_𝑛 be a minimizing sequence of (3.12) in 𝑊₀^1,𝑝(Ω), which is bounded in 𝑊₀^1,𝑝(Ω), so that we can assume, without loss of generality, that it converges weakly to some 𝑣∈𝑊₀^1,𝑝(Ω), with 1𝑞+1_Ω𝑘(𝑥)|𝑣|^𝑞+1d𝑥=1,𝜆^∗=1𝑝_Ω|∇𝑣|^𝑝1d𝑥+𝑟+1_Ωℎ(𝑥)|𝑣|^𝑟+1d𝑥. (3.13) Thus, 𝐽_𝜆(𝑣)=𝜆^∗−𝜆<0 for any 𝜆>𝜆^∗.
Now put Λ=^definf𝜆>0∶(1.3)₋admitsanontrivialweaksolution. (3.14) From above 𝜆^∗≥Λ and that problem (1.3)₋ has a solution for all 𝜆>𝜆^∗. The proof of the Step 2 is now completed.

Step 3 (problem (1.3)₋ has a weak solution for any 𝜆>Λ). By the definition of Λ, there exists 𝜇∈(Λ,𝜆) such that 𝐽_𝜇 has a nontrivial critical point 𝑢_𝜇∈𝑊₀^1,𝑝(Ω). Since 𝜇<𝜆,𝑢_𝜇 is a subsolution of the problem (1.3)₋. In order to find a super-solution of the problem (1.3)₋ which dominates 𝑢_𝜇, we consider the constrained minimization problem inf𝐽_𝜆(𝑤);𝑤∈𝑊₀^1,𝑝(Ω)and𝑤≥𝑢_𝜇.. (3.15) Arguments similar to those used in Step 2 show that the above minimization problem has a solution 𝑢_𝜆≥𝑢_𝜇 which is also a weak solution of problem (1.3)₋, provided 𝜆>Λ.
Using similar arguments as in [6]. Thus, from Theorem 2.2 in Pucci and Servadei [7], based on the Moser iteration, it is clear that  𝑢∈𝐿^∞_loc. Next, again by bootstrap regularity [Corollary on p. 830] due to DiBenedetto, [8] shows that the weak solution 𝑢∈𝐶^1,𝛼(Ω) where 𝛼∈(0,1). Finally, the nonnegative follows immediately by the strong maximum principle since 𝑢 is a 𝐶¹ nonnegative weak solution of the differential inequality ∇(|∇𝑢|^𝑝−2∇𝑢)−ℎ(𝑥)𝑢^𝑟≤0 in Ω, with 𝑝−1<𝑟, see, for instance, Section 4.8 of Pucci and Serrin [9]. Thus, 𝑢>0 in Ω. The proof of the Step 3 is now completed.

Step 4 (nonexistence for 𝜆>0 is small). The same monotonicity arguments as in Step 3 show that (1.3)₋ does not have any solution if 𝜆<Λ, which completes the proof of the Theorem 1.2.