Existence and Nonexistence Results for Classes of Singular Elliptic Problem

Abstract

The singular semilinear elliptic problem −Δ𝑢+𝑘(𝑥)𝑢^−𝛾=𝜆𝑢^𝑝 in Ω, 𝑢>0 in Ω, 𝑢=0 on 𝜕Ω, is considered, where Ω is a bounded domain with smooth boundary in 𝑅^𝑁, 𝑘∈𝐶^𝛼_loc(Ω)∩𝐶(Ω), and 𝛾,𝑝,𝜆 are three positive constants. Some existence or nonexistence results are obtained for solutions of this problem by the sub-supersolution method.

1. Introduction and Main Results

In this paper, we study the existence or the nonexistence of solutions to the following singular semilinear elliptic problem−Δ𝑢+𝑘(𝑥)𝑢^−𝛾=𝜆𝑢^𝑝,inΩ,𝑢>0,inΩ,𝑢=0,on𝜕Ω, (1.1) where Ω⊂𝑅^𝑁(𝑁≥1) is a bounded domain with 𝐶^2+𝛼 boundary for some 𝛼∈(0,1), 𝑘∈𝐶^𝛼_loc(Ω)∩𝐶(Ω), and 𝛾,𝑝,and𝜆 are three nonnegative constants. This problem arises in the study of non-Newtonian fluids, chemical heterogeneous catalysts, in the theory of heat conduction in electrically conducting materials (see [1–7] and their references).

Many authors have considered this problem. For examples, when 𝑘(𝑥)<0 in Ω, problem (1.1) was studied in [3, 8–11]; when 𝑘(𝑥)>0 in Ω, problem (1.1) was considered in [12–14]. Particularly, when 𝑘(𝑥)≡1, it has been established in Zhang [14] that there exists 𝜆>0 such that problem (1.1) has at least one solution in 𝐶^2+𝛼(Ω)∩𝐶(Ω) for all 𝜆>𝜆 and has no solution in 𝐶²(Ω)∩𝐶(Ω) if 𝜆<𝜆. After that Shi and Yao in [13] have also obtained the same results with 𝑘∈𝐶^2,𝛼(Ω) and 𝑘(𝑥)>0 in Ω. Recently, Ghergu and Rădulescu in [12] considered more general sublinear singular elliptic problem with 𝑘∈𝐶^𝛼(Ω).

In this paper, we consider the case that 𝑘∈𝐶^𝛼_loc(Ω)∩𝐶(Ω), and 𝑘 may have zeros in Ω. The following main results are obtained by the sub-supersolution method with restriction on the boundary in Cui [15].

Theorem 1.1. Suppose that 𝑘∈𝐶^𝛼_loc(Ω)∩𝐶(Ω),𝑘≥0, and 𝑘≠0. Assume that 0<𝛾<1 and 0<𝑝<1, then there exists 𝜆∈(0,∞) such that problem (1.1) has at least one solution 𝑢_𝜆∈𝐶^2+𝛼(Ω)∩𝐶(Ω) and 𝑢_𝜆^−𝛾∈𝐿¹(Ω) for all 𝜆>𝜆, and problem (1.1) has no solution in 𝐶²(Ω)∩𝐶(Ω) if 𝜆<𝜆. Moreover, problem (1.1) has a maximal solution 𝑣_𝜆 which is increasing with respect to 𝜆 for all 𝜆>𝜆.

Remark 1.2. Theorem 1.1 generalizes Theorem 1.2 in [13] in coefficient 𝑘(𝑥) of the singular term. Consequently, it also generalizes Theorem 1 in [14]. Moreover, there are functions 𝑘 satisfying our Theorem 1.1 and not satisfying Theorem 1.2 in [13]. For example, let −1𝑘(𝑥)=ln(|𝑥−𝑥₀|/(2𝑑)),𝑥∈Ω⧵{𝑥₀},0,𝑥=𝑥₀, (1.2) where 𝑥₀∈𝜕Ω, and 𝑑=diam(Ω)=^Δmax{|𝑥−𝑦|∣𝑥,𝑦∈Ω}. (1.3) Certainly, this example does not satisfy Theorem 1.2 in [12] yet.

Theorem 1.3. Suppose that 𝑘∈𝐶^𝛼_loc(Ω)∩𝐶(Ω) and 𝑘(𝑥)>0 in Ω. If 𝛾≥1, problem (1.1) has no solution in 𝐶²(Ω)∩𝐶(Ω) for all 𝜆>0 and 𝑝>0.

Remark 1.4. Obviously, Theorem 1.3 is a generalization of Theorem 2 in [14]. There are also functions 𝑘(𝑥) satisfying our Theorem 1.3 and not satisfying Theorem 2 in [14] and Theorem 1.1 in [12]. For example, let −1𝑘(𝑥)=ln(|𝑥−𝑥₀|/(2𝑑))+𝜀,𝑥∈Ω⧵{𝑥₀},𝜀,𝑥=𝑥₀, (1.4) where 𝑥₀∈𝜕Ω, 𝜀 is any positive constant and 𝑑=diam(Ω) is the diameter of Ω.

2. Proof of Theorems

Consider the more general semilinear elliptic problem−Δ𝑢=𝑓(𝑥,𝑢),inΩ,𝑢>0,inΩ,𝑢=0,on𝜕Ω, (2.1) where the function 𝑓(𝑥,𝑠) is locally Hölder continuous in Ω×(0,∞) and continuously differentiable with respect to the variable 𝑠. A function 𝑢 is called to be a subsolution of problem (2.1) if 𝑢∈𝐶²(Ω)∩𝐶(Ω), and −Δ𝑢≤𝑓(𝑥,𝑢𝑢),inΩ,𝑢>0,inΩ,=0,on𝜕Ω. (2.2) A function 𝑢 is called to be a supersolution of problem (2.1) if 𝑢∈𝐶²(Ω)∩𝐶(Ω), and −Δ𝑢≥𝑓(𝑥,𝑢),inΩ,𝑢>0,inΩ,𝑢=0,on𝜕Ω. (2.3)

According to Lemma 3 in the study of Cui [15], we can easily have the following basic existence of classical solution to problem (2.1).

Lemma 2.1. Let 𝑓∈𝐶^𝛼_loc(Ω×(0,∞)) be continuously differentiable with respect to the variable 𝑠. Suppose that problem (2.1) has a supersolution 𝑢 and a subsolution 𝑢 such that 𝑢(𝑥)≤𝑢(𝑥),inΩ, (2.4) then problem (2.1) has at least one solution 𝑢∈𝐶^2+𝛼(Ω)∩𝐶(Ω) satisfying 𝑢(𝑥)≤𝑢(𝑥)≤𝑢(𝑥),inΩ. (2.5)

Let 𝜆₁ be the first eigenvalue of the eigenvalue problem −Δ𝑢=𝜆𝑢,inΩ,𝑢=0,on𝜕Ω, (2.6) and 𝜑₁>0 in Ω the corresponding eigenfunction. Then 𝜑₁∈𝐶^2+𝛼(Ω). Moreover one has the following lemma.

Lemma 2.2 (see [10]). One has _Ω𝜑^𝑟₁𝑑𝑥<∞ (2.7) if and only if 𝑟>−1.

Now we give the proof of our theorems.

Proof of Theorem 1.1. Let 𝑝∈(0,1), and let 𝑢^∗ denote the unique solution of −Δ𝑢=𝑢^𝑝,inΩ,𝑢>0,inΩ,𝑢=0,on𝜕Ω, (2.8) where 𝑢^∗ belongs to 𝐶²(Ω) (see [16]). Then 𝑢=𝜆^1/(1−𝑝)𝑢^∗ is a solution of −Δ𝑢=𝜆𝑢^𝑝,inΩ,𝑢>0,inΩ,𝑢=0,on𝜕Ω, (2.9) where 0<𝑝<1 and 𝜆>0. Then fix 𝜆>0 and set 𝑢=𝜆^1/(1−𝑝)𝑢^∗, (2.10) thus we can easily obtain that 𝑢 is a supersolution of problem (1.1).
Now, we want to find a subsolution of problem (1.1). Let𝑢=𝑀𝜑₁^2/(1+𝛾), (2.11) where 𝑀 is a positive constant; now we will prove that 𝑢 is a subsolution of problem (1.1). By Hopf's maximum principle in [17], there exist 𝛿>0 and 𝜀₀>0 such that |∇𝜑₁|≥𝛿,onΩ⧵Ω^′,𝜑₁≥𝛿,onΩ^′, (2.12) where Ω^′={𝑥∈Ω∣dist(𝑥,𝜕Ω)>𝜀₀}. On Ω^′, we choose 𝑀≥𝑀=₁^Δ((‖𝑘‖_∞(1+𝛾))/𝜆₁𝛿²)^1/(1+𝛾), then we have 𝑘(𝑥)𝑀^𝛾𝜑₁^{2𝛾/(1+𝛾)}≤𝜆₁𝑀𝜑1+𝛾^12/(1+𝛾), (2.13) where ‖𝑘‖_∞=max{|𝑘(𝑥)|∣𝑥∈Ω} for 𝑘∈𝐶(Ω). On Ω⧵Ω^′, we choose 𝑀≥𝑀=₂^Δ(‖𝑘‖_∞(1+𝛾)²/2(1−𝛾)𝛿²)^1/(1+𝛾), then one obtains 𝑘(𝑥)𝑀^𝛾𝜑₁^{2𝛾/(1+𝛾)}≤2(1−𝛾)𝑀|∇𝜑₁|²(1+𝛾)²𝜑₁^{2𝛾/(1+𝛾)}. (2.14) Thus, we choose 𝑀≥max{𝑀₁,𝑀₂}, then fixing 𝑀, let 𝜆>𝜆=^′Δ(3𝜆₁𝑀^1−𝑝)/(1+𝛾)‖𝜑₁‖_∞^{2(1−𝑝)/(1+𝛾)}, it follows from (2.13) and (2.14) that −Δ𝑢+𝑘(𝑥)𝑢_𝜆^−𝛾=−𝑀Δ𝜑₁^2/(1+𝛾)+𝑘(𝑥)𝑀^𝛾𝜑₁^{2𝛾/(1+𝛾)}=−𝑀(2(1−𝛾)(1+𝛾)²|∇𝜑₁|²𝜑₁^{−2𝛾/(1+𝛾)}+2𝜑1+𝛾^{1(1−𝛾)/(1+𝛾)}Δ𝜑₁)+𝑘(𝑥)𝑀^𝛾𝜑₁^{2𝛾/(1+𝛾)}=2𝜆₁𝑀𝜑1+𝛾^12/(1+𝛾)+𝑘(𝑥)𝑀^𝛾𝜑₁^{2𝛾/(1+𝛾)}−2(1−𝛾)𝑀|∇𝜑₁|²(1+𝛾)²𝜑₁^{2𝛾/(1+𝛾)}≤3𝜆₁𝑀𝜑1+𝛾^12/(1+𝛾)≤𝜆(𝑀𝜑₁^2/(1+𝛾))^𝑝=𝜆𝑢^𝑝_𝜆. (2.15)
Thus we proved that 𝑢=𝑀𝜑₁^2/(1+𝛾) is a subsolution of problem (1.1) for all 𝜆>𝜆^′. According to Lemma 4 in [14], there exists a positive constant 𝐶 such that𝜑₁(𝑥)≤𝐶𝑢^∗(𝑥),inΩ. (2.16) Set 𝜆≥𝜆^Δ′′=(𝑀𝐶‖𝜑₁‖_∞^{(1−𝛾)/(1+𝛾)})^1−𝑝, then we have 𝑢=𝜆^1/(1−𝑝)𝑢^∗≥𝑢=𝑀𝜑₁^2/(1+𝛾),inΩ. (2.17) Thus we choose 𝜆^∗=max{𝜆^′,𝜆^′′}; via Lemma 2.1, problem (1.1) has at least one solution 𝑢_𝜆∈𝐶^2+𝛼(Ω)∩𝐶(Ω) and satisfying 𝑢(𝑥)≤𝑢_𝜆(𝑥)≤𝑢(𝑥),inΩ, (2.18) for all 𝜆≥𝜆^∗.
Since 𝑢_𝜆≥𝑀𝜑₁^2/(1+𝛾) in Ω for all 𝜆≥𝜆^∗ and −2𝛾/(1+𝛾)>−1, according to Lemma 2.2 one has_Ω𝑢_𝜆^−𝛾1(𝑥)𝑑𝑥≤𝑀^𝛾_Ω𝜑₁^{−2𝛾/(1+𝛾)}(𝑥)𝑑𝑥<+∞. (2.19) So we obtain 𝑢_𝜆^−𝛾∈𝐿¹(Ω).
Let Ω_𝑗={𝑥∈Ω∣dist(𝑥,𝜕Ω)>𝑟/2𝑗}, 𝑗=1,2,3,…, and let 𝑢_𝑗 be the unique solution of−Δ𝑢+𝑘(𝑥)𝑢^−𝛾_𝑗−1=𝜆𝑢^𝑝_𝑗−1,inΩ_𝑗,𝑢=𝑢_𝑗−1,onΩ⧵Ω_𝑗, (2.20) for 𝑗=1,2,3,…, and with 𝑢₀=𝑢=𝜆^1/(1−𝑝)𝑢^∗, where 𝑟=max_𝑥∈Ωmin_{𝑦∈𝜕Ω}|𝑥−𝑦|. (2.21) We claim that 𝑢_𝑗 is nonincreasing with respect to 𝑗 in Ω for all 𝑗∈𝑁. Indeed, since 𝑢 is a supersolution of problem (1.1) for all 𝜆>0, then we have −Δ(𝑢₀−𝑢₁)=−Δ𝑢₀+Δ𝑢₁=−Δ𝑢₀+𝑘(𝑥)𝑢₀^−𝛾−𝜆𝑢^𝑝₀=−Δ𝑢+𝑘(𝑥)𝑢_𝜆^−𝛾−𝜆𝑢^𝑝_𝜆>0, (2.22) for all 𝑥∈Ω₁. Since 𝑢₁=𝑢₀ in Ω⧵Ω₁, so by the maximum principle, one has 𝑢₀≥𝑢₁ in Ω. So when 𝑗=0 our claim is true. We assume that our claim is true when 𝑗=𝑛; that is, 𝑢_𝑛≤𝑢_𝑛−1 in Ω. Then we obtain −Δ(𝑢_𝑛−𝑢_𝑛+1)=−Δ𝑢_𝑛+Δ𝑢_𝑛+1=𝜆(𝑢^𝑝_𝑛−1−𝑢^𝑝_𝑛)+𝑘(𝑥)(𝑢_𝑛^−𝛾−𝑢^−𝛾_𝑛−1)>0, (2.23) for all 𝑥∈Ω_𝑛+1. Since 𝑢_𝑛=𝑢_𝑛+1 in Ω⧵Ω_𝑛+1, so by the maximum principle, one has 𝑢_𝑛≥𝑢_𝑛+1 in Ω. Thus by the induction, one obtains 𝑢_𝑗+1≤𝑢_𝑗,inΩ, (2.24) for all 𝑗∈𝑁. Then by the monotonicity of 𝑢_𝑗, we have −Δ𝑢_𝑗=𝜆𝑢^𝑝_𝑗−1−𝑘(𝑥)𝑢^−𝛾_𝑗−1≥𝜆𝑢^𝑝_𝑗−𝑘(𝑥)𝑢_𝑗^−𝛾, (2.25) for all 𝑥∈Ω_𝑗 and 𝑗∈𝑁⁺. According to the definitions of 𝑢_𝑗 and 𝑢₀, we obtain that 𝑢_𝑗 is a supersolution of problem (1.1) for all 𝑗∈𝑁⁺. Let 𝑢_𝜆 be a classical solution of problem (1.1), thus one has 𝑢_𝜆(𝑥)≤𝑢_𝑗+1(𝑥)≤𝑢_𝑗(𝑥)≤𝑢₀(𝑥),inΩ. (2.26) Assume that 𝑣_𝜆(𝑥)=lim_𝑗→∞𝑢_𝑗(𝑥) for all 𝑥∈Ω, then by standard elliptic arguments (see [17]) it follows that 𝑣_𝜆 is a solution of problem (1.1), and 𝑣_𝜆≥𝑢_𝜆 in Ω for any 𝑢_𝜆. Therefore, 𝑣_𝜆 is the maximal solution of problem (1.1). According to the above arguments, problem (1.1) has a maximal solution for 𝜆≥𝜆^∗.
To complete the proof of Theorem 1.1, setting𝜎={𝜆>0∣problem(1.1)hasatleastonesolution𝑢_𝜆},𝜆=inf𝜎, (2.27) then [𝜆^∗,+∞)⊂𝜎, 𝜆≤𝜆^∗. It suffices to prove that if 𝜆₀∈𝜎, then [𝜆₀,+∞)⊂𝜎; that is, assume that 𝜆>𝜆₀, then problem (1.1) has at least one solution. Let 𝑢_𝜆0 be a solution of problem (1.1) corresponding to 𝜆₀, then 𝑢_𝜆0 is a subsolution of problem (1.1) with every fixed 𝜆>𝜆₀. Since 𝑢=𝜆^1/(1−𝑝)𝑢^∗ is a supersolution of problem (1.1) for any 𝜆>0, then one has 𝜆^1/(1−𝑝)𝑢^∗≥𝜆₀^1/(1−𝑝)𝑢^∗≥𝑢_𝜆0,inΩ, (2.28) for all 𝜆>𝜆₀. According to Lemma 2.1, problem (1.1) has at least one solution 𝑢_𝜆∈𝐶^2+𝛼(Ω)∩𝐶(Ω) for all 𝜆>𝜆₀. Moreover, 𝑢_𝜆0(𝑥)≤𝑢_𝜆(𝑥)≤𝑢(𝑥),inΩ. (2.29) Consequently, the maximal solution 𝑣_𝜆 of problem (1.1) is increasing with respect to 𝜆 for all 𝜆>𝜆. So the proof of Theorem 1.1 is completed.

Proof of Theorem 1.3. Suppose to the contrary that there exists 𝜆>0 such that problem (1.1) has one solution 𝑢_𝜆∈𝐶²(Ω)∩𝐶(Ω). Let 𝑒 be the unique solution of −Δ𝑢=1,inΩ,𝑢=0,on𝜕Ω, (2.30)𝑒∈𝐶^2+𝛼(Ω). By the maximum principle, 𝑒>0 in Ω. We claim that for any solution 𝑢_𝜆 of problem (1.1), there exists a constant 𝑀=𝑀(𝜆)>0 such that 𝑀𝑒(𝑥)>𝑢_𝜆(𝑥),inΩ. (2.31) Indeed, let 𝑀=𝜆‖𝑢_𝜆‖^𝑝_∞+1, then one obtains −Δ(𝑀𝑒−𝑢_𝜆)=−𝑀Δ𝑒+Δ𝑢_𝜆=𝜆‖𝑢_𝜆‖^𝑝_∞+1−𝜆𝑢^𝑝_𝜆(𝑥)+𝑘(𝑥)𝑢_𝜆^−𝛾>0, (2.32) for all 𝑥∈Ω. Since (𝑀𝑒−𝑢_𝜆)∣_𝜕Ω=0, by the maximum principle we have 𝑀𝑒(𝑥)>𝑢_𝜆(𝑥),inΩ. (2.33) According to Lemma 4 in [14], there exists a positive constant 𝐶 such that 𝑒(𝑥)≤𝐶𝜑₁(𝑥),inΩ. (2.34) Since 𝛾≥1, from Lemma 2.2, it follows that _Ω𝑢_𝜆^−𝛾1(𝑥)𝑑𝑥≥(𝐶𝑀)^𝛾_Ω𝜑₁^−𝛾(𝑥)𝑑𝑥=+∞. (2.35) Thus we obtain _Ω𝑢_𝜆^−𝛾𝑑𝑥=+∞. (2.36)
SetΩ_𝑖𝑟={𝑥∈Ω∣dist(𝑥,𝜕Ω)>2𝑖,𝑖∈𝑁⁺}, (2.37) and Ω=⋃^∞_𝑖=1Ω_𝑖, then Ω_𝑖⊂Ω and 𝑢_𝜆∈𝐶²(Ω_𝑖), satisfying −Δ𝑢_𝜆+𝑘(𝑥)𝑢_𝜆^−𝛾=𝜆𝑢^𝑝_𝜆, (2.38) for all 𝑥∈Ω_𝑖 and 𝑖∈𝑁⁺. Consequently, integrating (2.38) we have −_Ω𝑖Δ𝑢_𝜆𝑑𝑥+_Ω𝑖𝑘(𝑥)𝑢_𝜆^−𝛾𝑑𝑥=𝜆_Ω𝑖𝑢^𝑝_𝜆𝑑𝑥≤𝜆_Ω𝑢^𝑝_𝜆𝑑𝑥, (2.39) noting that _Ω𝑖Δ𝑢_𝜆𝑑𝑥=_𝜕Ω𝑖𝜕𝑢_𝜆𝜕𝑛𝑑𝑠, (2.40) where 𝑛 denotes the outward normal to 𝜕Ω_𝑖. From (2.39) and (2.40), letting 𝑖→∞, one has _Ω𝑘(𝑥)𝑢_𝜆^−𝛾𝑑𝑥−_𝜕Ω𝜕𝑢_𝜆𝜕𝑛𝑑𝑠≤𝜆‖𝑢_𝜆‖^𝑝_∞|Ω|, (2.41) where |Ω| denotes the Lebesgue measure of Ω. According to (2.36) and 𝑘(𝑥)>0 in Ω, one obtains _𝜕Ω𝜕𝑢_𝜆𝜕𝑛𝑑𝑠=+∞. (2.42) But this is impossible, by Hopf's maximum principle, we have 𝜕𝑢_𝜆𝜕𝑛<0, (2.43) for all 𝑥∈𝜕Ω, where 𝑛 denotes the outward normal to 𝜕Ω at 𝑥. Therefore Theorem 1.3 is true.

Acknowledgment

This paper is supported by NNSF of China under Grant 10771173 and the Natural Science Foundation of Education of Guizhou Province under Grant 2008067.