SEMIPOSITONE EQUATIONS

SEMIPOSITONE EQUATIONS

SHOBHA ORUGANTI AND R. SHIVAJI

Received 22 September 2005; Accepted 10 November 2005

We study positiveC¹( ¯Ω) solutions to classes of boundary value problems of the form

−Δpu=g(x,u,c) inΩ,u=0 on∂Ω, whereΔpdenotes thep-Laplacian operator defined byΔpz:=div(|∇z|^p−2∇z); p >1,c >0 is a parameter,Ωis a bounded domain inR^N; N≥2 with∂Ωof classC²and connected (ifN=1, we assume thatΩis a bounded open interval), andg(x, 0,c)<0 for somex∈Ω(semipositone problems). In particular, we first study the case wheng(x,u,c)=λ f(u)−cwhereλ >0 is a parameter andf is aC¹([0,∞)) function such that f(0)=0, f(u)>0 for 0< u < rand f(u)≤0 foru≥r. We establish positive constants c0(Ω,r) and λ^∗(Ω,r,c) such that the above equation has a positive solution whenc≤c0 andλ≥λ^∗. Next we study the case wheng(x,u,c)=a(x)u^p−1− u^γ−1−ch(x) (logistic equation with constant yield harvesting) where γ > pand ais a C¹( ¯Ω) function that is allowed to be negative near the boundary ofΩ. Herehis aC¹( ¯Ω) function satisfyingh(x)≥0 forx∈Ω,h(x)≡0, and max_x∈Ω¯h(x)=1. We establish a positive constantc1(Ω,a) such that the above equation has a positive solution whenc < c1. Our proofs are based on subsuper solution techniques.

Copyright © 2006 S. Oruganti and R. Shivaji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

We consider weak solutions to classes of boundary value problems of the form

−Δpu=g(x,u,c) inΩ,

u=0 on∂Ω, (1.1)

whereΔpdenotes thep-Laplacian operator defined byΔpz:=div(|∇z|^p−2∇z);p >1,c >

0 is a parameter,Ωis a bounded domain inR^N;N≥2 with∂Ωof classC²and connected (ifN=1, we assume thatΩis a bounded open interval) andg(x, 0,c)<0 for somex∈Ω (semipositone problems). By a weak solution to (1.1), we mean a functionu∈W₀^1,p(Ω)

Hindawi Publishing Corporation Boundary Value Problems

Volume 2006, Article ID 87483, Pages1–7 DOI10.1155/BVP/2006/87483

that satisfies

Ω|∇u|^p−2∇u· ∇w dx=

Ωg(x,u,c)w dx, ∀w∈C^∞₀(Ω). (1.2) However in this paper, we in fact study the existence ofC¹( ¯Ω) solutions that are strictly positive inΩ.

We first study the case wheng(x,u,c)=λ f(u)−cwhereλ >0 is a parameter and f satisfies:

(A1) f ∈C¹([0,∞)), f(0)=0, f(u)>0 for 0< u < rand f(u)≤0 foru≥rfor some r >0.

Whenc=0 it is easy to establish the existence of a positive solution for largeλ >0. Here we consider the challenging semipositone casec >0. Semipositone problems have been of great interest during the past two decades, and continue to pose mathematically difficult problems in the study of positive solutions (see [1–3,10–12]). Also most of the results established to date are for the case when p=2. Here we establish an existence result for p >1 for a class of nonlinearities satisfying (A1). Namely, we prove the following theorem.

Theorem 1.1. There exist positive constantsc0=c0(Ω,r) andλ^∗=λ^∗(Ω,r,c) such that (1.1) has a positive solution forc≤c0andλ≥λ^∗.

Remark 1.2. Refer to [2] where the authors study such a problem in the case whenp=2.

In particular, whencis very small they establish an existence of a positive solution for λnear the first eigenvalueλ1and then extend the existence forλ≥λ. In this paper, we establish the existence of a positive solution directly forλlarge. Our proof is new even in the casep=2.

Remark 1.3. The case wheng(x,u,c)=λ[f(u)−c] withh(u)=f(u)−cof the form

h(u)

u

has been studied for the case when p=2 in [6]. For p=2 this remains a challenging semipositone problem for existence of positive solutions for largeλ.

We next study the case wheng(x,u,c)=a(x)u^p−1−u^γ−1−ch(x) (Logistic equation with constant yield harvesting) whereγ > p,ais aC¹( ¯Ω) function that is allowed to be negative near the boundary ofΩ, andhis aC¹( ¯Ω) function satisfyingh(x)≥0 forx∈Ω, h(x)≡0 and max_x∈Ω¯h(x)=1. Again forc >0 this is a semipositone problem. In order to precisely state our result for this problem we introduce the region where we allowa(x) to be negative. Letλ1be the first eigenvalue of the−Δpwith Dirichlet boundary conditions

andφ1∈C¹( ¯Ω) be a corresponding eigenfunction such thatφ1>0 inΩ,∂φ/∂n <0 on

∂Ωandφ1∞=1. Letm >0,δ >0, andσ >0 be such that ∇φ1^p−λ1φ1^p≥m on ¯Ωδ,

φ1≥σ onΩ\Ω¯δ, (1.3)

where ¯Ωδ:= {x∈Ω|d(x,∂Ω)≤δ}. Further assume that there exists a constanta0>0 such that

a(x)≥a0 inΩ\Ω¯δ (1.4)

and letμ >0 be such that

a(x)≥ −μ in ¯Ωδ. (1.5)

Then we prove the following theorem.

Theorem 1.4. Let μ < m(p/(p−1))^p−1 anda0>(p/(p−1))^p−1λ1. Then there exists a positive constantc1=c1(Ω,μ,a0) such that (1.1) has a positive solution forc≤c1.

Remark 1.5. Refer to [7] where they studied the case whenc=0 anda(x) is a positive function throughout ¯Ω.

We establish Theorems1.1and1.4by the method of sub- and super-solutions. By a super-solutionφof (1.1) we mean a function inW^1,p(Ω)∩C( ¯Ω) such thatφ=0 on∂Ω and

Ω|∇φ|^p−2∇φ· ∇w dx≥

Ωg(x,φ,c)w dx, ∀w∈W, (1.6) whereW= {v∈C^∞₀(Ω)|v≥0 inΩ}. And by a subsolutionψof (1.1) we mean a func- tion inW^1,p(Ω)∩C( ¯Ω) such thatψ=0 on∂Ωand

Ω|∇ψ|^p−2∇ψ· ∇w dx≤

Ωg(x,ψ,c)w dx, ∀w∈W, (1.7) whereW is as defined before. Then if there exist sub- and super-solutionsψ andφre- spectively such thatψ≤φinΩthen (1.1) has aC¹( ¯Ω) solutionusuch thatψ≤u≤φ(see [7,8]).

In semipositone problems it is well documented that finding a nonnegative subsolu- tion is nontrivial. Recently in [4] an anti-maximum principle by [5,8,9] was used to create a crucial subsolution in the study of the problem wheng(x,u,c)=λf(u)−cwhere fsatisfiesf(0)=0, f(u)≥0 and lim_u→∞(f(u)/u)=0. Namely, the authors exploited the C¹( ¯Ω) solution of

−Δpzα−αzα^p−1= −1 inΩ,

z_α=0 on∂Ω, (1.8)

which is positive inΩby the anti-maximum principle forα∈(λ1,λ1+ν) for someν>0 whereλ1 is the first eigenvalue of the−Δp with Dirichlet boundary conditions. How- ever this requires a further restriction on fnamely: there existsm >0 such that f(v)>

v^p−1−m^p−1α^p−2+ (c/α),∀v∈[0,mαzα∞]. Moreover they obtain a positive a solution forλnear the first eigenvalueλ1. In provingTheorem 1.1we avoid the use of the anti- maximum principle in creating a crucial subsolution. Thus we avoid this above restriction on f for smalluwhich seems unnatural when we look for positive solutions for largeλ.

InTheorem 1.1we establish a subsolution by analyzing an appropriate power of the first eigenfunction of the−Δpwith Dirichlet boundary conditions.

Also recently in [13] the Logistic equation with constant yield harvesting was studied via an anti-maximum principle in the case whena(x) is a positive constant equal toA0

(> λ1) throughout ¯Ω. But in the case ofTheorem 1.4, since we allowa(x) to be negative near the boundary, the idea in [13] fails. Again we use an appropriate power of the eigen- function to create the crucial subsolution needed to establishTheorem 1.4. We will prove Theorem 1.1inSection 2andTheorem 1.4inSection 3.

2. Proof ofTheorem 1.1

Here note thatg(x,u,c)=λ f(u)−cwhere f satisfies (A1). Letλ1,φ1,δ,m,σ, andΩδbe as described inSection 1.

We now construct our positive subsolution. Letψ:=((p−1)/ p)rφ₁^p/(p−1). (Note that ψ∞< r.) Then∇ψ=rφ^1/(p1 ⁻¹⁾∇φ1andψwill be a subsolution if

Ω|∇ψ|^p−2∇ψ· ∇w dx≤

Ω

λ f(ψ)−cw dx, ∀w∈W. (2.1) But

Ω|∇ψ|^p−2∇ψ· ∇w dx=r^p−1

Ω

∇φ1^p−2φ1∇φ1· ∇w dx

=r^p−1

Ω

∇φ1^p−2∇φ1· ∇

φ1w dx−

Ω

∇φ1^pw dx

=r^p−1

Ω

λ1φ₁^p−∇φ1^p w dx.

(2.2) Nowr^p−1[λ1φ₁^p− |∇φ1|^p]≤ −mr^p−1 in ¯Ωδ. Hence ifc≤c0=mr^p−1thenr^p−1[λ1φ₁^p−

|∇φ1|^p]≤[λ f(ψ)−c] in ¯Ωδ, sincef(ψ)≥0.

Next inΩ−Ω¯δ,r^p−1[λ1φ₁^p− |∇φ1|^p]≤λ1r^p−1while

λ f(ψ)−c≥λα−c, (2.3)

whereα=inf{f(s)|((p−1)/ p)rσ^p/(p−1)≤s≤((p−1)/ p)r}. Hence ifλ≥λ^∗=(λ1r^p−1+ c)/αthen inΩ−Ω¯δ,

r^p−1λ1φ1^p−∇φ1^p

≤λ f(ψ)−c. (2.4)

Hence ifc≤c0andλ≥λ^∗then (2.1) is satisfied andψis a subsolution.

We next construct a super-solutionφsuch thatφ≥ψ. Letφ:=Mφ0whereφ0∈C¹(Ω) is the solution of

−Δpφ0=1 inΩ,

φ0=0 on∂Ω. (2.5)

Nowφwill be a super-solution if

Ω|∇φ|^p−2∇φ· ∇w dx≥

Ω

λ f(φ)−cw dx, ∀w∈W. (2.6)

But _Ω|∇φ|^p−2∇φ· ∇w dx=M^p−1_Ωw dx≥

Ω[λ f(φ)−c]w dx, provided M^p−1≥λ sup_[0,r]f(s) :=M(λ) (say). That is, ifM≥(M(λ))^1/(p−1) then (2.6) is satisfied andφis a super-solution. Sinceφ0>0 inΩand∂φ0/∂n <0 on∂Ω, we can chooseMlarge enough so thatφ≥ψis also satisfied. HenceTheorem 1.1is proven.

Remark 2.1. We have, in the proof ofTheorem 1.1, an explicit expression for bothc0(Ω,r) andλ^∗(Ω,r,c).

3. Proof ofTheorem 1.4

Here note thatg(x,u,c)=a(x)u^p−1−u^γ−1−ch(x). Letλ1,φ1,m,σ,δ,a0,μ, andΩδbe as described inSection 1.

Letψ=εφ1^p/(p−1) whereε will be chosen small enough later. (Note thatψ∞≤ε.) Thenψwill be a subsolution if

Ω|∇ψ|^p−2∇ψ· ∇w dx≤

Ω

a(x)ψ^p−1−ψ^γ−1−ch(x)w dx, ∀w∈W. (3.1)

Using a calculation similar to the one in the proof ofTheorem 1.1, we have

Ω|∇ψ|^p−2∇ψ· ∇w dx=ε^p−1 p

p−1 p−1

Ω

λ1φ₁^p−∇φ1^p

w dx. (3.2) Hence inequality (3.1) will be satisfied if both

ε^p−1 p

p−1 p−1

(−m)≤ −με^p−1−ε^γ−1−c considering ¯Ωδ , (3.3) ε^p−1

p p−1

p−1

λ1φ₁^p≤a0ε^p−1φ₁^p−ε^γ−1−c consideringΩ\Ω¯δ (3.4) are satisfied. Note that sinceμ < m(p/(p−1))^p−1inequality (3.3) will be satisfied if

ε < α1=

m p

p−1 p−1

−μ 1/(γ−p)

, c≤c1(ε)=ε^p−1

m

p p−1

p−1

−μ−ε^γ−p

.

(3.5)

Note thatc1(ε)>0. Similarly, sincea0>(p/(p−1))^p−1λ1, inequality (3.4) will be satisfied if

ε≤α2

a0−

p p−1

p−1

λ1

σ^p

1/(γ−p)

, c≤c2(ε)=ε^p−1

a0−

p p−1

p−1

λ1

σ^p−ε^γ−p

.

(3.6)

Note thatc2(ε)>0. Chooseα=min{α1,α2}andε=α/2. Then simplifying, bothc1(ε) andc2(ε) are greater than (α/2)^γ−1[2^γ−p−1]. Hence ifc≤(α/2)^γ−1[2^γ−p−1]=c1(Ω,a0,μ) thenψis a subsolution.

We next construct a super-solutionφsuch thatφ≥ψ. Letφ:=Mφ0whereφ0∈C¹( ¯Ω) is the solution of (2.5). Nowφwill be a super-solution if

Ω|∇φ|^p−2∇φ· ∇w dx≥

Ω

a(x)φ^p−1−φ^γ−1−ch(x)w dx, ∀w∈W. (3.7)

But_Ω|∇φ|^p−2∇φ· ∇w dx=M^p−1_Ωw dx≥

Ω[a(x)φ^p−1−φ^γ−1−ch(x)]w dx, provided M^p−1≥sup_[0,k][a∞s^p−1−s^γ−1] :=M1(say) wherek=a^1/(γ∞ ^−p). That is, ifM≥M^1/(p1 ⁻¹⁾

then (3.7) is satisfied andφ is a super-solution. Sinceφ0>0 in Ωand∂φ0/∂n <0 on

∂Ω, we can chooseMlarge enough so thatφ≥ψis also satisfied. HenceTheorem 1.4is proven.

Remark 3.1. We have, in the proof ofTheorem 1.4, an explicit expression forc1(Ω,a0,μ).

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Shobha Oruganti: Department of Mathematics, School of Science, The Behrend College, Penn State Erie, Erie, PA 16563, USA

E-mail address:[email protected]

R. Shivaji: Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA

E-mail address:[email protected]