# Existence and Nonexistence Results for Classes of Singular Elliptic Problem

Authors: Peng Zhang; Jia-Feng Liao
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

The singular semilinear elliptic problem-Δu+k(x)u-γ=λup in Ω, u>0 in Ω, u=0 on ∂Ω, is considered, where Ω is a bounded domain with smooth boundary in RN, k∈Clocα(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.


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## Body

## 1. Introduction and Main Results

In this paper, we study the existence or the nonexistence of solutions to the following singular semilinear elliptic problem(1.1)-Δu+k(x)u-γ=λup,inΩ,u>0,inΩ,u=0,on∂Ω,
where Ω⊂RN(N≥1) is a bounded domain with C2+α boundary for some α∈(0,1), k∈Clocα(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, whenk(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 C2+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∈C2,α(Ω¯) and k(x)>0 in Ω¯. Recently, Ghergu and Rădulescu in [12] considered more general sublinear singular elliptic problem with k∈Cα(Ω¯).In this paper, we consider the case thatk∈Clocα(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 thatk∈Clocα(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 uλ∈C2+α(Ω)∩C(Ω¯) and uλ-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 vλ which is increasing with respect to λ for all λ>λ¯.Remark 1.2.
Theorem1.1 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
(1.2)k(x)={-1ln(|x-x0|/(2d)),x∈Ω¯∖{x0},0,x=x0,
where x0∈∂Ω, and
(1.3)d=diam(Ω)=Δmax{|x-y|∣x,y∈Ω¯}.
Certainly, this example does not satisfy Theorem 1.2 in [12] yet.Theorem 1.3.
Suppose thatk∈Clocα(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, Theorem1.3 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
(1.4)k(x)={-1ln(|x-x0|/(2d))+ɛ,x∈Ω¯∖{x0},ɛ,x=x0,
where x0∈∂Ω, ɛ is any positive constant and d=diam(Ω) is the diameter of Ω.


## 2. Proof of Theorems

Consider the more general semilinear elliptic problem(2.1)-Δu=f(x,u),inΩ,u>0,inΩ,u=0,on∂Ω,
where the function f(x,s) is locally Hölder continuous in Ω×(0,∞) and continuously differentiable with respect to the variable s. A function u̲ is called to be a subsolution of problem (2.1) if u̲∈C2(Ω)∩C(Ω¯), and (2.2)-Δu̲≤f(x,u̲),inΩ,u̲>0,inΩ,u̲=0,on∂Ω.
A function u¯ is called to be a supersolution of problem (2.1) if u¯∈C2(Ω)∩C(Ω¯), and (2.3)-Δu¯≥f(x,u¯),inΩ,u¯>0,inΩ,u¯=0,on∂Ω.According to Lemma3 in the study of Cui [15], we can easily have the following basic existence of classical solution to problem (2.1).Lemma 2.1.
Letf∈Clocα(Ω×(0,∞)) be continuously differentiable with respect to the variable s.  Suppose that problem (2.1) has a supersolution u¯ and a subsolution u̲ such that
(2.4)u̲(x)≤u¯(x),inΩ,
then problem (2.1) has at least one solution u∈C2+α(Ω)∩C(Ω¯) satisfying
(2.5)u̲(x)≤u(x)≤u¯(x),inΩ¯.Letλ1 be the first eigenvalue of the eigenvalue problem (2.6)-Δu=λu,inΩ,u=0,on∂Ω,
and φ1>0 in Ω the corresponding eigenfunction. Then φ1∈C2+α(Ω¯). Moreover one has the following lemma.Lemma 2.2 (see [10]).
One has(2.7)∫Ωφ1rdx<∞
if and only if r>-1.Now we give the proof of our theorems.Proof of Theorem1.1.
Letp∈(0,1), and let u* denote the unique solution of
(2.8)-Δu=up,inΩ,u>0,inΩ,u=0,on∂Ω,
where u* belongs to C2(Ω¯) (see [16]). Then u=λ1/(1-p)u* is a solution of
(2.9)-Δu=λup,inΩ,u>0,inΩ,u=0,on∂Ω,
where 0<p<1 and λ>0. Then fix λ>0 and set
(2.10)u¯=λ1/(1-p)u*,
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). Let(2.11)u̲=Mφ12/(1+γ),
where M is a positive constant; now we will prove that u̲ is a subsolution of problem (1.1). By Hopf's maximum principle in [17], there exist δ>0 and ɛ0>0 such that
(2.12)|∇φ1|≥δ,onΩ∖Ω',φ1≥δ,onΩ',
where Ω'={x∈Ω∣dist(x,∂Ω)>ɛ0}.  On Ω', we choose M≥M1=Δ((∥k∥∞(1+γ))/λ1δ2)1/(1+γ), then we have
(2.13)k(x)Mγφ12γ/(1+γ)≤λ1M1+γφ12/(1+γ),
where ∥k∥∞=max{|k(x)|∣x∈Ω¯} for k∈C(Ω¯). On Ω∖Ω', we choose M≥M2=Δ(∥k∥∞(1+γ)2/2(1-γ)δ2)1/(1+γ), then one obtains
(2.14)k(x)Mγφ12γ/(1+γ)≤2(1-γ)M|∇φ1|2(1+γ)2φ12γ/(1+γ).
Thus, we choose M≥max{M1,M2}, then fixing M, let λ>λ'=Δ(3λ1M1-p)/(1+γ)∥φ1∥∞2(1-p)/(1+γ), it follows from (2.13) and (2.14) that
(2.15)-Δu̲+k(x)u̲λ-γ=-MΔφ12/(1+γ)+k(x)Mγφ12γ/(1+γ)=-M(2(1-γ)(1+γ)2|∇φ1|2φ1-2γ/(1+γ)+21+γφ1(1-γ)/(1+γ)Δφ1)+k(x)Mγφ12γ/(1+γ)=2λ1M1+γφ12/(1+γ)+k(x)Mγφ12γ/(1+γ)-2(1-γ)M|∇φ1|2(1+γ)2φ12γ/(1+γ)≤3λ1M1+γφ12/(1+γ)≤λ(Mφ12/(1+γ))p=λu̲λp.
Thus we proved thatu̲=Mφ12/(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 that(2.16)φ1(x)≤Cu*(x),inΩ¯.
Set λ≥λ′′=Δ(MC∥φ1∥∞(1-γ)/(1+γ))1-p, then we have
(2.17)u¯=λ1/(1-p)u*≥u̲=Mφ12/(1+γ),inΩ.
Thus we choose λ*=max{λ′,λ′′}; via Lemma 2.1, problem (1.1) has at least one solution uλ∈C2+α(Ω)∩C(Ω¯) and satisfying
(2.18)u̲(x)≤uλ(x)≤u¯(x),inΩ¯,
for all λ≥λ*.
Sinceuλ≥Mφ12/(1+γ) in Ω¯ for all λ≥λ* and -2γ/(1+γ)>-1, according to Lemma 2.2 one has(2.19)∫Ωuλ-γ(x)dx≤1Mγ∫Ωφ1-2γ/(1+γ)(x)dx<+∞.
So we obtain uλ-γ∈L1(Ω).
LetΩj={x∈Ω∣dist(x,∂Ω)>r/2j}, j=1,2,3,…, and let uj be the unique solution of(2.20)-Δu+k(x)uj-1-γ=λuj-1p,inΩj,u=uj-1,onΩ¯∖Ωj,
for j=1,2,3,…, and with u0=u¯=λ1/(1-p)u*, where
(2.21)r=maxx∈Ωminy∈∂Ω|x-y|.
We claim that uj 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
(2.22)-Δ(u0-u1)=-Δu0+Δu1=-Δu0+k(x)u0-γ-λu0p=-Δu¯+k(x)u¯λ-γ-λu¯λp>0,
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, un≤un-1 in Ω¯. Then we obtain
(2.23)-Δ(un-un+1)=-Δun+Δun+1=λ(un-1p-unp)+k(x)(un-γ-un-1-γ)>0,
for all x∈Ωn+1. Since un=un+1 in Ω¯∖Ωn+1, so by the maximum principle, one has un≥un+1 in Ω¯. Thus by the induction, one obtains
(2.24)uj+1≤uj,inΩ¯,
for all j∈N. Then by the monotonicity of uj, we have
(2.25)-Δuj=λuj-1p-k(x)uj-1-γ≥λujp-k(x)uj-γ,
for all x∈Ωj and j∈N+. According to the definitions of uj and u0, we obtain that uj is a supersolution of problem (1.1) for all j∈N+. Let uλ be a classical solution of problem (1.1), thus one has
(2.26)uλ(x)≤uj+1(x)≤uj(x)≤u0(x),inΩ¯.
Assume that vλ(x)=limj→∞uj(x) for all x∈Ω¯, then by standard elliptic arguments (see [17]) it follows that vλ is a solution of problem (1.1), and vλ≥uλ in Ω for any uλ. Therefore, vλ 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 Theorem1.1, setting(2.27)σ={λ>0∣problem(1.1)hasatleastonesolutionuλ},λ¯=infσ,
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 uλ0 be a solution of problem (1.1) corresponding to λ0,  then uλ0 is a subsolution of problem (1.1) with every fixed λ>λ0. Since u¯=λ1/(1-p)u* is a supersolution of problem (1.1) for any λ>0, then one has
(2.28)λ1/(1-p)u*≥λ01/(1-p)u*≥uλ0,inΩ,
for all λ>λ0. According to Lemma 2.1, problem (1.1) has at least one solution uλ∈C2+α(Ω)∩C(Ω¯) for all λ>λ0. Moreover,
(2.29)uλ0(x)≤uλ(x)≤u¯(x),inΩ.
Consequently, the maximal solution vλ 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 Theorem1.3.
Suppose to the contrary that there existsλ>0 such that problem (1.1) has one solution uλ∈C2(Ω)∩C(Ω¯).  Let e be the unique solution of
(2.30)-Δu=1,inΩ,u=0,on∂Ω,e∈C2+α(Ω¯). By the maximum principle, e>0 in Ω. We claim that for any solution uλ of problem (1.1), there exists a constant M=M(λ)>0 such that
(2.31)Me(x)>uλ(x),inΩ.
Indeed, let M=λ∥uλ∥∞p+1,  then one obtains
(2.32)-Δ(Me-uλ)=-MΔe+Δuλ=λ∥uλ∥∞p+1-λuλp(x)+k(x)uλ-γ>0,
for all x∈Ω.  Since (Me-uλ)∣∂Ω=0, by the maximum principle we have
(2.33)Me(x)>uλ(x),inΩ.
According to Lemma 4 in [14], there exists a positive constant C such that
(2.34)e(x)≤Cφ1(x),inΩ.
Since γ≥1, from Lemma 2.2, it follows that
(2.35)∫Ωuλ-γ(x)dx≥1(CM)γ∫Ωφ1-γ(x)dx=+∞.
Thus we obtain
(2.36)∫Ωuλ-γdx=+∞.
Set(2.37)Ωi={x∈Ω∣dist(x,∂Ω)>r2i,i∈N+},
and Ω=⋃i=1∞Ωi,  then Ωi⊂Ω and uλ∈C2(Ω¯i), satisfying
(2.38)-Δuλ+k(x)uλ-γ=λuλp,
for all x∈Ω¯i and i∈N+. Consequently, integrating (2.38) we have
(2.39)-∫ΩiΔuλdx+∫Ωik(x)uλ-γdx=λ∫Ωiuλpdx≤λ∫Ωuλpdx,
noting that
(2.40)∫ΩiΔuλdx=∫∂Ωi∂uλ∂nds,
where n denotes the outward normal to ∂Ωi.  From (2.39) and (2.40), letting i→∞, one has
(2.41)∫Ωk(x)uλ-γdx-∫∂Ω∂uλ∂nds≤λ∥uλ∥∞p|Ω|,
where |Ω| denotes the Lebesgue measure of Ω. According to (2.36) and k(x)>0 in Ω¯, one obtains
(2.42)∫∂Ω∂uλ∂nds=+∞.
But this is impossible, by Hopf's maximum principle, we have
(2.43)∂uλ∂n<0,
for all x∈∂Ω, where n denotes the outward normal to ∂Ω at x. Therefore Theorem 1.3 is true.



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