Localization of solutions for semilinear problems with poly-Laplace type operators

Abstract

The paper presents an abstract theory regarding the problems with semilinear operator equations involving iterates of a strongly monotone symmetric linear operator. We obtain existence and localization results of positive solutions for such problems using Krasnosel’skiĭ’s technique and abstract Harnack inequality. In particular, we obtain results for problems with semilinear poly-Laplace operators.

Authors

Radu Precup
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Nataliia Kolun
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

Keywords

Krasnosel’skiĭ’s technique; fixed point; Harnack inequality; iterates of a symmetric linear operator. poly-Laplace type operator

Paper coordinates

N. Kolun, R. Precup, Localization of solutions for semilinear problems with poly-Laplace type operators, Applicable Analysis, published online 2023, https://doi.org/10.1080/00036811.2023.2218869

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Localization of solutions for semilinear problems with poly-Laplace type operators

Localization of solutions for semilinear problems with poly-Laplace type operators

Nataliia Kolun Military Academy, Department of Fundamental Sciences, 65009 Odessa, Ukraine & Babeş-Bolyai University, Faculty of Mathematics and Computer Science, 400084 Cluj-Napoca, Romania nataliiakolun@ukr.net  and  Radu Precup Babeş-Bolyai University, Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania r.precup@math.ubbcluj.ro
Abstract.

The paper presents an abstract theory regarding the problems with semilinear operator equations involving iterates of a strongly monotone symmetric linear operator. We obtain existence and localization results of positive solutions for such problems using Krasnosel’skiǐ’s technique and abstract Harnack inequality. In particular, we obtain results for semilinear problems involving poly-Laplace operators and Navier boundary conditions.


Key words: Krasnosel’skiǐ’s technique, fixed point, Harnack inequality, iterates of a symmetric linear operator, poly-Laplace type operator.

Mathematics Subject Classification: 34B15, 34K10, 47J05, 47H10.

1. Introduction

The goal of this paper is to investigate the existence and the localization of weak solutions in a cone, to semilinear operator equations involving iterates of a strongly monotone symmetric linear operator. For the first time, the idea of considering boundary value problems with equations of this type arose in [11] as an extension of the theory of poly-Laplace equations. The general theory about semilinear equations involving iterates of a symmetric linear operator constructed in [11] made it possible, in particular, to obtain results on the existence and uniquennes of solutions of problems involving poly-Laplace operators. The logical continuation of work [11] is the study of the localization of solutions of this type of equations.

More exactly we consider the problem

(1.1) {Apu=F(u)u,Au,,Ap1uXA.

Here A:D(A)X is a strongly monotone symmetric linear operator, X is a Hilbert space, D(A) is a linear subspace of X, Ap is the p-th iterate of A defined recursively by Ap=AAp1, F is any mapping and XA is the energetic space of A.

In particular, we consider equations involving the poly-Laplace operator, with Navier boundary conditions, more exactly to the problem

(1.2) {(Δ)pu=f(u(x))in Ω,u=Δu==Δp1u=0on Ω.

Here Ωn is bounded open and f:. In this case, X=L2(Ω), A=Δ and XA=H01(Ω).

For the classical theory of equations with poly-Laplace operators we refer the reader to the volume [6] which brings together the entire contribution of Miron Nicolescu to this field, and for a modern approach based on the notion of weak solution, to the works [1], [2], [3], [9] and the monograph [4]. See also works [16, 17, 18, 19, 20] which describe different methods of working with problems containing bi- and poly-Laplace operators.

In this paper we deal with the localization of weak solutions in a conical ”annulus” jointly defined by the norm and a semi-norm. The technique was first introduced in [12] (see also [10] and [8],  for its early form) and used after in [13] and [14]. The use of a semi-norm arises from the necessity to have a Harnack inequality for the estimation from below of the solutions. In many cases, particulary for ordinary differential equations, the semi-norm can be taken the norm itself. However, in case of partial differential equations, Moser-Harnack inequalities give us lower estimates only with respect to a semi-norm.

2. Preliminaries

2.1. The energetic space of the an iterate of a strong monotone symmetric linear operator

We shortly present the basic notions and results necessary for the investigation of problem (1.1). For details we refer the reader to paper [11].

Let X be a real Hilbert space with the inner product (,)X and the norm ||X. Let A:D(A)X be a strongly monotone symmetric linear operator, that is, a linear operator satisfying

(Au,v)X=(u,Av)Xfor all u,vD(A),
(2.1) (Au,u)Xc2|u|X2for all uD(A)

and some constant c>0. On D(A) one considers the energetic inner product

(u,v)A:=(Au,v)X(u,vD(A))

and the energetic norm

|u|A=(Au,u)X1/2(uD(A)).

The completion of the space (D(A),|.|A) is denoted by XA and is called the energetic space of A. By a standard technique, the inner product (.,.)A and norm |.|A are extended from D(A) to XA and denoted by (.,.)XA and |.|XA. Let XA be the dual space of XA. If the Hilbert space X is identified to its dual, then one has

XAXXA.

If the embedding XAX is compact, then the following Poincaré’s inequalities hold:

(2.2) |u|X1λ|u|XA(uXA),
(2.3) |u|XA1λ|u|X(uX),

where

(2.4) λ=inf{|u|XA2:uXA,|u|X=1}

and the last inf is reached, i.e., |ϕ|XA2=λ for some ϕXA with |ϕ|X=1, and

(2.5) Aϕ=λϕ.

The invers of A is the operator A1:XAXA defined by

(2.6) (A1h,v)XA=h,vfor allvXA,hXA,

where by h,v we mean the value of the linear functional h at the element v. Note that A1 is an isometry between XA and XA, i.e.,

(2.7) |A1h|XA=|h|XAfor allhXA.

As in [11], we consider the energetic space of the iterate Ap to be the space

H:=A(p1)(XA).

Here Ak=A1(A(k1)) for k=2,,p1. Since A1:XAXA and XAXA one has

H=A(p1)(XA)A(p2)(XA)A1(XA)XA.

Next the space H is endowed with the inner product (,)H and the norm ||H given by

(u,v)H=(Ap1u,Ap1v)XA,|u|H=|Ap1u|XA.

Note that for the embedding HXA, the following inequality holds

(2.8) |u|XA1λp1|u|Hfor alluH.

Indeed, if uH, then u=A(p1)v for some vXA and using successively (2.7) and Poincaré’s inequalities (2.2) and (2.3), we have

|u|XA = |A(p1)v|XA=|A1A(p2)v|XA=|A(p2)v|XA1λ|A(p2)v|X
1λ|A(p2)v|XA=1λ|A1A(p3)v|XA=1λ|A(p3)v|XA1λ3/2|A(p3)v|X
1λ2|A(p3)v|XA1λp1|v|XA=1λp1|u|H.

Moreover using (2.2) we obtain

(2.9) |u|X1λ|u|XA1λ2p12|u|Hfor all uH.

We also note that problem (1.1) is equivalent to the fixed point equation

u=ApF(u)

in the energetic space H, for the operator ApF.

2.2. A Krasnosel’skiǐ type theorem in a set defined by the norm and of a semi-norm

Here we state the abstract results that we use for the localization of solutions to problem (1.1).

Let Z be a normed linear space with norm ||Z and let . be a semi-norm on Z for which there is a constant σ>0 such that uσ|u|Z for all uZ.

Let K be a wedge in Z, i.e., a closed convex set with λKK for every λ+, and let ϕK be any fixed element with ϕ>0 and |ϕ|Z=1. Then for any positive numbers R0, R1 with R0<ϕR1, there exists a μ>0 such that μϕ>R0 and |μϕ|Z<R1. Hence the set {uK:R0<u,|u|Z<R1} is nonempty. Denote

KR0R1:={uK:R0u,|u|ZR1}.

Note that, in particular, when =||Z, KR0R1 is the conical shell {uK:R0|u|ZR1}.

The first theorem is a fixed point result in the set KR0R1.

Theorem 2.1.

Let N:KK be completely continuous and let hK with h>R0 and |h|ZR1. Assume that the following conditions are satisfied:

(2.10) |N(u)|ZR1for|u|ZR1;
(2.11) (1μ)N(u)+μhufor 0μ<1,u=R0,|u|ZR1.

Then N has a fixed point uK such that R0<u and |u|ZR1.

We also have a three solutions existence result.

Theorem 2.2.

Under the assumptions of Theorem 2.1, if in addition there exists a number R1 with 0<R1<σ1R0 and

N(u)μufor|u|Z=R1,μ1,

then N has three fixed points u1,u2,u3 with

R0<u1,|u1|ZR1;R1<|u2|ZR1,u2<R0;|u3|Z<R1.

The proofs of these theorems rely on the fixed point index theory and can be found in paper [13].

3. Main results

3.1. Existence and localization results for semilinear problems involving iterates of a symmetric linear operator

Let (Y,|.|Y) be a Banach space continuously embbeded in X and let K0 be a cone of Y. Denote by K0 the partial order relation on Y associated with K0 given by uK0v if and only if vuK0 and let u:Y+ be a semi-norm on Y for which there exists C>0 such that for every uY

(3.1) uC|u|Y.

Assume that the norm ||Y and the seminorm are monotone, i.e., if 0K0uK0v (u,vY), then |u|Y|v|Y and uv.

Our hypotheses are as follow:

(h1) (compactness):

The linear operator A1 is compact from Y to Y.

(h2) (abstract Harnack inequality):

There exists φK0{0} such that for every uK0 one has

A1uφK0A1u;
(h3) (norm estimate):

There exists ψK0{0} such that for every uK0 one has

uK0|u|Yψ;
(h4) (positivity and monotonicity):

The operator F:YY is continuous, bounded (maps bounded sets into bounded sets), positive and increasing with respect to the ordering induced by K0, i.e.,

0K0uK0vimplies 0K0F(u)K0F(v).
Remark 1.

In particular, if Y=X, the (h1) holds provided that the embedding XAX is compact. Indeed, the operator A1:XX can be descompose as JA1J0 with J0:XXA,J0u=u;A1:XAXA; J:XAX,Ju=u, where J is compact.

Now we consider a cone in Y,

K:={uK0:uφK0u}.

Clearly in view of (h2), A1(K0)K.

Denote N:YY the operator

N(u)=ApF(u).

The operator N is well-defined base on the assumption A1(Y)Y given by (h1). In addition, since

F:YYXXA,Ap:XAH,

one has N(Y)H. Thus, any fixed point from Y of N belongs to H.

Lemma 3.1.

Assume that the conditions (h2) and (h4) hold. Then

N(K)K.
Proof.

Let uK and denote v:=N(u)=ApF(u). We have to prove that vK, i.e., vK0 and vφK0v. Obviously, based on the above remark, vY. From uK one has uK0, whence in virtue of (h4), F(u)K0 and next, from (h2), vK0 and vφK0v. Hence vK.

We note that φ1. Indeed, if u is any nonzero element of K (take for example u=A1φ), then from uφK0u we have

uφu,

whence φ1.

Theorem 3.2.

Assume (h1)–(h4) hold and let ϕ0K with |ϕ0|Y=1. Assume that there exist numbers R0, R1 with

(3.2) 0<R0<ϕ0φR1

such that

(3.3) |ApF(R1ψ)|YR1,
(3.4) A1F(R0φ)R0λp1φϕϕ,

where φ,λ,ϕ have been identified in hypothesis (h2), in (2.4) and (2.5). Then (1.1) has at least one solution uKH with

R0<u,|u|YR1.
Proof.

We shall apply Theorem 2.1 on the space Y with norm ||Y and semi-norm . First, the continuity and boundedness of F together with the compactness of A1imply that the operator N=ApF is completely continuous from Y to Y.

From Lemma 3.1, N(K)K, hence N is completely continuous from K to K as required by Theorem 2.1.

Next we show that (2.10) holds. Let uK be any element with |u|YR1. From (h3) one has uK0|u|YψK0R1ψ. Next F(u)K0F(R1ψ) and

N(u)=ApF(u)K0ApF(R1ψ).

As a result

|N(u)|Y=|ApF(u)|Y|ApF(R1ψ)|Y.

Now (3.3) gives |N(u)|YR1.

Next we show that (2.11) holds for h:=R1ϕ0. From (3.2) one has

(3.5) h=R1ϕ0>R0.

Assume that (2.11) does not hold. Then

(3.6) (1μ)N(u)+μh=u

for some u, μ with u=R0, |u|YR1, 0<μ<1.

Since the abstract Harnack inequality holds (see (h2)) we have

uK0R0φ,hK0hφ,N(u)K0N(u)φ.

Then, since A1 is order-preserving and operator F is positive and increasing with respect to the ordening induced by K0 (see (h4))

N(u)K0N(R0φ)K00,

and taking into account the monotonicity on the semi-norm in K0, we have

N(u)N(R0φ).

Then (3.6) implies

uK0(1μ)N(u)φ+μhφ(1μ)N(R0φ)φ+μhφ=((1μ)N(R0φ)+μR1ϕ)φ,

whence applying the semi-norm, we obtain

R0((1μ)N(R0φ)+μR1ϕ)φ.

Then using (3.2) we get

R0>(1μ)N(R0φ)φ+μR0.

Consequently

N(R0φ)<R0φ,

i.e.

(3.7) ApF(R0φ)<R0φ.

On the other hand, from the condition (3.4) and (2.5) we have

A2F(R0φ)R0λp1φϕA1ϕ=R0λp2φϕϕ,
ApF(R0φ)R0φϕϕ.

Then

(3.8) ApF(R0φ)R0φ.

This contradicts of (3.7). Consequently (2.11) is true. Now the conclusion follows from Theorem 2.1. ∎

Theorem 2.2 yields the following three solutions existence result.

Theorem 3.3.

Under the assumptions of Theorem 3.2, if in addition there exists a number R1 with 0<CR1<R0<ϕ0φR1 and

|A1|p|F(R1ψ)|YR1,

then problem (1.1) has three solutions u1, u2,u3K with

R0<u1,|u1|YR1;R1<|u2|YR1,u2<R0;|u3|Y<R1.

If F(0)0, then all the three solutions are nontrivial.

Proof.

We only need to check the condition R1<σ1R0. In our case, σ=C. Indeed, from uC|u|Y the required relationship between R1 and R0 is satisfied if CR1<R0.

Theorem 3.2 immediately yields multiple solution results by a simple multiplication of the pair (R0,R1), aimed to produce disjoint sets of type KR0R1 for which the assumptions of the theorem are fulfilled. Thus we may state

Theorem 3.4.

Assume (h1)–(h4) hold and let ϕ0K with |ϕ0|Y=1.Assume that there exist increasing sequences of positive numbers (R0i)1ik, (R1i)1ik such that the following conditions are satisfied:

R0i<ϕ0φR1ifor i=1,,k;
(3.9) CR1i<R0i+1for i=1,,k1;
|A1|p|F(R1iψ)|YR1i,for i=1,,k;
A1F(R0iφ)R0iλp1φϕϕfor i=1,,k.

Then (1.1) has at least k solutions uiK with R0i<ui, |ui|YR1i for i=1,,k.

Proof.

It is enough to see that condition (3.9) guarantees that KR0iR1iKR0jR1j= for ij.

3.2. Existence and localization results for semilinear problems involving the poly-Laplace operator

In this section, we consider problem (1.2) as a particular case of problem (1.1). In order to apply the abstract results from Section 3.1, we let A=Δ, XA=H01(Ω), X=L2(Ω), H=(Δ)(p1)(H01(Ω)) and D(A)=C2(Ω¯)C0(Ω¯). Also

F(u)(x)=f(u(x))for xΩ.

Next we will use a theorem for superharmonic functions, where by a superharmonic function in a domain Ωn one means any function uH1(Ω) satisfying

Δu0in𝒟(Ω),

that is,

Ωuw0for every wC0(Ω)withw0inΩ.

Let Ω0 be any bounded subdomain of Ω with Ω¯0Ω (i.e. Ω0Ω). As shall need the following theorem:

Theorem 3.5 ([13]).

Let n3 and 1q<n/(n2), or n=2 and 1q<, and let Ω0Ω. Then there exists a constant M=M(n,q,Ω,Ω0)>0 such that for every nonnegative superharmonic function u in Ω, the following inequality holds:

(3.10) u(x)M|u|Lq(Ω0)forxΩ0.

This result was obtained in [13] as a consequence of the Moser-Harnack inequality (see [15, p. 305]).

In the following, we fix any Ω0Ω and we let Y=L(Ω). Clearly, one has

(3.11) Y=L(Ω)Lq(Ω)Lq(Ω0).

Next we consider the seminorm on Y given by

u=|u|Lq(Ω0)=(Ω0|u|q𝑑x)1q(uL(Ω)).

From (3.11) it follows that there exists a constant C>0 such that for every uL(Ω),

|u|Lq(Ω0)C|u|L(Ω),

which is the analogue of the formula (3.1). It easy to see that C=(mes(Ω0))1/q.

As cone K0 we have

K0={uL(Ω):u0onΩ}.

Note that the norm ||L(Ω) and the semi-norm are monotone, i.e., if 0uv (u,vL(Ω)), then |u|L(Ω)|v|L(Ω) and uv.

Now let us discuss the fulfillment of hypotheses (h1)–(h4) for this specific case:

Assumption (h1) (compactness): We recall (see [21], Lemma 1.1 and [22], p. 317) that if Ω is a bounded regular domain of class C1,β for some β(0,1) and gL(Ω), then the weak solution in H01(Ω) of the problem

(3.12) {Δu=ginΩ,u=0onΩ

belongs to C1(Ω¯). Also the linear solution operator (Δ)1:L(Ω)C1(Ω¯) assigning to each gL(Ω), the corresponding solution of (3.12), is continuous, compact and order-preserving. Therefore, all the more it is compact as an operator from L(Ω) to L(Ω). Thus (h1) is fulfilled.

Assumption (h2) (abstract Harnack inequality): We need to show that there exists φK0{0} such that for every fK0 one has

(Δ)1f|(Δ)1f|Lq(Ω0)φon Ω.

From inequality (3.10) we have

(Δ)1fM|(Δ)1f|Lq(Ω0)on Ω0.

Then clearly

(Δ)1fM|(Δ)1f|Lq(Ω0)χΩ0on Ω,

where χΩ0 is the characteristic function of Ω0, i.e. χΩ0(x)=1 if xΩ0 and χΩ0(x)=0 otherwise. So, (h2) is fulfilled with φ=MχΩ0.

Assumption (h3) (norm estimate): There exists ψK0{0} such that for every uK0 (uL(Ω),u0) one has

u|u|L(Ω)ψ.

Obviously, this condition is fulfilled with ψ1 (constant function 1).

Assumption (h4) (positivity and monotonicity): This hypothesis is clearly fulfilled if we assume that f: is continuous, nondecreasing and with f(+)+.

Notice that in virtue of (h2), in this case, the cone K is given by

K={uL(Ω):u|u|Lq(Ω0)MχΩ0onΩ},

where constant M>0 comes from inequality (3.10). Also, the operator N is well-defined from L(Ω) to L(Ω), by

N(u)=(Δ)pf(u()),

and N(K)K.

Now in order to apply Theorem 3.2 we take ϕ01. Then

ϕ0=|ϕ0|Lq(Ω0)=(Ω01q𝑑x)1q=(mes(Ω0))1q.

Note in addition that for this case, λ=λ1 and ϕ=ϕ1, where λ1 is the first eigenvalue of the Dirichlet problem for Δ and ϕ1 is the corresponding positive eigenfunction with |ϕ|L2(Ω)=1, i.e.,

{Δϕ=λ1ϕinΩ,ϕ=0onΩ.

Then ϕ=|ϕ1||Lq(Ω0).

With the above specifications, Theorem 3.2 gives the following result.

Theorem 3.6.

Let Ω is a bounded regular domain of class C1,β for some β(0,1) and f:  continuous, nondecreasing and with f(+)+. Assume that there exist numbers R0, R1 with

0<R0<M(mes(Ω0))2qR1

such that

f(R1)R11|(Δ)p1|L(Ω),
(3.13) (Δ)1f(R0MχΩ0)R0λ1p1M(mes(Ω0))1qϕ1ϕ1.

Then (1.2) has at least one solution uKH with

R0<u,|u|L(Ω)R1.
Remark 2.

A sufficient condition not involving A, for (3.13) to hold is

f(R0MχΩ0)R0λ1pM(mes(Ω0))1qϕ1ϕ1.

Since ϕ1(x)>0 for all xΩ and χΩ0(x)=0 on ΩΩ0, this inequality excludes the posibility to have f(0)=0. However, this is possible under condition (3.13).

4. Conclusions

Acknowledgments

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