R E S E A R C H
Open Access
Existence and global behavior of positive
solutions for some eigenvalue problems
Noureddine Zeddini
*, Ramzi S Alsaedi and Habib Mâagli
*Correspondence: [email protected] Department of Mathematics, College of Sciences and Arts-Rabigh Campus, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh, 21911, Saudi Arabia
Abstract
In this paper, we study the existence and nonexistence of positive bounded solutions of the integral equationu=
λV
(af(u)), whereλ
is a positive parameter,ais a nontrivial nonnegative measurable function with bounded potential andVbelongs to a class of positive kernels that contains in particular the potential kernel of the classicalLaplacianV= (–
)–1orV= (∂
∂t–
)–1or the inverse of the polyharmonic Laplacian
(–
)m,m≥2.
1 Introduction
Letbe a smooth domain ofRn(n≥) andf: [,∞)→(,∞) be a nondecreasing
con-tinuous function. In this paper, we are interested in the existence of a positive solution of the following integral equation:
u=λVaf(u), (.)
whereλis a positive parameter,ais a nontrivial nonnegative measurable function satisfy-ing an appropriate condition andV belongs to a class of positive kernels that contains in particular the potential kernel of the classical LaplacianV= (–)–orV= (∂∂t –)–or the inverse of the polyharmonic Laplacian (–)m,m≥.
We note that ifV is the inverse of a differential operator –L(i.e. V = (–L)–) anduis a solution of (.), thenusatisfies
–Lu=λa(x)f(u) in (.)
under some appropriate Dirichlet conditions on the boundary of.
Forλ= , (.) has been extensively studied either for nonincreasing positive nonlin-earities (see [–]) or for sublinear positive nonlinnonlin-earities (see [, –]). In fact existence, uniqueness and global behavior of the solution are discussed in the above works.
Here we are interested in the general equation (.). In fact, we will show that there exists λ> such that (.) has a positive solution for <λ<λunder the following hypotheses
on the functionsaandf:
(H) f : [,∞)→(,∞)is a nondecreasing continuous function.
(H) ais a nontrivial nonnegative measurable function onsuch that the potential
func-tionVais bounded in.
More precisely, letVa∞=supx∈Va(x);g(t) =f(tt) fort> andβ=supt>g(t)∈(,∞]. So, we prove the following result.
Theorem . Assume(H)-(H).Then we have:
(i) Ifg(t) <βfor eacht> ,then(.)has a positive solutionufor each <λ<Vaβ∞. (ii) If there existst> such thatg(t) =β,then equation(.)has a positive solutionu
for each <λ≤Vaβ
∞.
Moreover,in the two cases there exists C> such that for each x∈,
CVa(x)≤u(x)≤CVa(x).
Remark .(see [] and []) Letbe a smooth bounded domain and letδ(x) denote the Euclidean distance fromx∈to the boundary∂. Then for eachλ< , the function
a(x) =
(δ(x))λ satisfies (H).
Next, we give a nonexistence result under some restrictions on the functiona. So we as-sume that the kernelVis self-adjoint, namely for each nonnegative measurable functions
h,kinwe haveVh(x)k(x)dx=h(x)Vk(x)dx. The functionsf andaare required to satisfy the following hypotheses:
(H) The functiong(t) =f(tt) is nonnegative and bounded on(,∞).
(H) There exist λ> and a positive measurable functionϕsuch thatλV(aϕ) =ϕand
a(x)ϕ(x)dx<∞.
Then we have the following nonexistence result.
Theorem . Assume(H)-(H).Then for eachλ>λg∞, (.)has no positive bounded solution in.
Remark . The problem
u= –λ(u+ ), <x< ,
u() =u() =
has the positive bounded solutionu(x) =
cos(
√ λ )
sin(
√
λ
x)sin(
√
λ
( –x)) for <λ<π =λ
.
Hence the nonexistence result in Theorem . is optimal.
Our paper is organized as follows. Section is devoted to the proof of Theorems . and .. The last section is devoted to the study of some examples and applications.
2 Proof of main results
Proof of Theorem. Assume thatasatisfies (H) and let <λ<Vaβ∞ =Va∞supt>f(tt). Consider <c≤λf() and <c such that <λ≤f(ccVa∞). Then, from hypothesis (H), we deduce that <c≤c. Consider the nonempty closed convex set
=u∈B+() such thatcVa≤u≤cVa
whereB+() denotes the set of nonnegative Borel measurable functions in. LetT
λbe
the operator defined onby
Tλu=λV
af(u).
Sincef is nondecreasing on [,∞),Tλis nondecreasing on. Moreover, for eachu∈,
we have
cVa≤λV
f()a≤λVaf(u)≤λfcVa∞
Va≤cVa.
So Tλ⊂. Consider the sequence (uk)k≥defined byu=cVaanduk+=Tλuk for k≥. SinceTλ⊂andTλis nondecreasing, we find by induction that the sequence
(uk)k≥is nondecreasing and satisfies for eachk≥
cVa≤uk≤cVa.
Hence, it follows from the monotone convergence theorem that the sequence (uk)k≥ con-verges to a functionu∈satisfying the integral equation
u=λVaf(u). (.)
This proves assertion (i) of the theorem.
Now, assume that there existst> such thatg(t) =β. Then forλ=Vaβ∞, we take
c=Vat∞ and <c<λf() =Vatf∞f()(t), to adapt the previous proof.
Proof of Theorem. Assume that (.) has a positive bounded solutionuin. Then we have
λ
ϕ(x)u(x)a(x)dx=λλ
Vaf(u)(x)ϕ(x)a(x)dx
=λλ
a(x)fu(x)V(aϕ)(x)dx
=λ
a(x)fu(x)ϕ(x)dx
≥λ inf
t>
f(t)
t
a(x)u(x)ϕ(x)dx.
This shows thatλg∞≥λ.
Example . As a consequence of Theorem ., we deduce that the problem
u= –λeu, inB(, )⊂R,
u/∂B=
has no positive bounded solution forλ> π
e . Indeed, in this case we have λ=π
and
3 Examples and applications 3.1 Examples
Next, we give some examples where (H) and (H) are satisfied.
Example . Letf(t) =tp+ forp≥ andt≥. Then we discuss three cases: Case : p∈[, ). In this caseβ= +∞and the existence result for (.) holds for
λ∈(,∞).
Case : p= . Thenβ= and the existence result for (.) holds forλ∈(,Va∞). Case : p> . In this caseg(t) =+ttp andβ=g(
(p–)p) =
(p–)– p
p . So the existence result
for (.) holds forλ∈(,(p–)
– p
pVa∞].
Example . Letf(t) =et. Thenβ=sup
t>g(t) =g() =e. In this case, the existence result
for (.) holds forλ∈(,eVa∞].
3.2 Applications
Throughout the following first three applications, we assume that the function f : [,∞)→(,∞) is continuous and nondecreasing.
Application . Letbe a smooth domain(bounded or unbounded)with compact bound-ary or=Rn+={x= (x,x, . . . ,xn)∈Rn:xn> }(n≥)be the upper half space and let G(x,y)be the Green function of the Laplacian(–)onwith Dirichlet boundary condi-tions.Assume that a is a nontrivial nonnegative Borel measurable function such that its Green potential Va(x) =G(x,y)a(y)dy is continuous and satisfieslimx→∂∞Va(x) = ,
where∂∞=∂ifis bounded and∂∞=∂∪ {∞}wheneveris unbounded.Then for each <λ<Va∞supt>f(tt),the problem
–u=λa(x)f(u), in,
u/∂∞=
has a positive continuous solution u satisfying for each x∈
CVa(x)≤u(x)≤CVa(x) for some C> .
Application . Let m be a positive integer, =B(, ) be the unit ball inRn (n≥) and Gm,n(x,y)be the Green function of the polyharmonic Laplacian(–)mon B(, )with Dirichlet boundary conditions (see[]). Let a be a nontrivial nonnegative function on B(, )such that its Green potential Va(x) =B(,)Gm,n(x,y)a(y)dy is continuous on B(, ) andlim|x|→ Va(x)
(–|x|)m– = .Then,for each <λ<Va∞supt>f(tt),the problem
(–)mu=λa(x)f(u), in, lim|x|→(–u|x(|x))m– =
has a positive continuous solution u satisfying for each x∈B(, )
Application . For t>s and x,y∈Rn
+,we denote by
(x,t,y,s) =π(t–s)–n
–exp – xnyn (t–s)
exp –|x–y|
(t–s)
the Green function of the heat operator ∂
∂t–onRn+×(,∞)with Dirichlet boundary
con-ditions.Put Pt(x) =P(x,t) =Rn
+(x,t,y, )dy=
√
πt xn
exp(–
ξ
t)dξ.Consider a nontriv-ial nonnegative measurable function a onRn
+×(,∞)such that the function(x,t)→Pat(x(,tx))
belongs to the parabolic Kato class P∞(Rn
+)introduced and studied in[].Then from[]
the function(x,t)→Va(x,t) =Rn
+×(,∞)(x,t,y,s)a(y,s)dy ds is continuous and bounded inRn
+×(,∞)and for each(x,t)∈Rn+×(,∞)we have lim
ξ→∂Rn+
Va(ξ,t) = lim
r→+Va(x,r) = .
Consequently,for <λ<Va∞supt>f(tt),the problem
⎧ ⎪ ⎨ ⎪ ⎩
∂u
∂t –u=λa(x,t)f(u), inRn+×(,∞),
u(x, ) = for x∈Rn
+,
u/Rn+×(,∞)=
has a positive continuous solution u satisfying for each(x,t)∈Rn
+×(,∞)
CVa(x,t)≤u(x,t)≤CVa(x,t) for some C> .
Application . Letbe a smooth bounded ofRn(n≥),f : [,∞)→(,∞)be a contin-uous function such that g(t) =f(tt) is bounded and let a(x) =(δ(x))λ withλ< .Letλ> be
the first positive eigenvalue of the problem–u=λa(x)u inwith Dirichlet boundary con-ditions.From[],λis a simple eigenvalue.Letϕbe the positive normalized(ϕ∞= )
eigenfunction associated withλ.Thenϕsatisfies the equationλV(aϕ) =ϕ.Moreover,it
is well known thatϕ(x)≈δ(x).Namely,there exists C> such thatCδ(x)≤ϕ(x)≤Cδ(x),
for each x∈.Soa(x)ϕ(x)≤Cω(δ(x))λ–dx<∞.Hence from Theorem.,the problem
–u=λa(x)f(u), in,
u/∂=
has no positive bounded solution infor eachλ>λg∞.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. 257/662/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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