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

3 years ago

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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, 103 (2024) no. 5, pp. 985-997, https://doi.org/10.1080/00036811.2023.2218869

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https://www.tandfonline.com/doi/epdf/10.1080/00036811.2023.2218869?needAccess=true&role=button

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Journal

Applicable Analysis\Taylor & Francis Online

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DOI

https://doi.org/10.1080/00036811.2023.2218869

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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.

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 ${(\cdot, \cdot)}_{X}$ and the norm $| \cdot |_{X} .$ Let $A : D (A) \to X$ be a strongly monotone symmetric linear operator, that is, a linear operator satisfying

{(A u, v)}_{X} = {(u, A v)}_{X} for all u, v \in D (A),

(2.1)

{(A u, u)}_{X} \geq c^{2} {| u |}_{X}^{2} for all u \in D (A)

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

{(u, v)}_{A} := {(A u, v)}_{X} (u, v \in D (A))

and the energetic norm

{| u |}_{A} = {(A u, u)}_{X}^{1 / 2} (u \in D (A)) .

The completion of the space $(D (A), | . |_{A})$ is denoted by $X_{A}$ 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 $X_{A}$ and denoted by ${(., .)}_{X_{A}}$ and $| . |_{X_{A}} .$ Let $X_{A}^{'}$ be the dual space of $X_{A} .$ If the Hilbert space $X$ is identified to its dual, then one has

X_{A} \subset X \subset X_{A}^{'} .

If the embedding $X_{A} \subset X$ is compact, then the following Poincaré’s inequalities hold:

(2.2)

{| u |}_{X} \leq \frac{1}{\sqrt{λ}} {| u |}_{X_{A}} (u \in X_{A}),

(2.3)

{| u |}_{X_{A}^{'}} \leq \frac{1}{\sqrt{λ}} {| u |}_{X} (u \in X),

where

(2.4)

λ = inf {| u |_{X_{A}}^{2} : u \in X_{A}, | u |_{X} = 1}

and the last $inf$ is reached, i.e., ${| ϕ |}_{X_{A}}^{2} = λ$ for some $ϕ \in X_{A}$ with ${| ϕ |}_{X} = 1,$ and

(2.5)

A ϕ = λ ϕ .

The invers of $A$ is the operator $A^{- 1} : X_{A}^{'} \to X_{A}$ defined by

(2.6)

{(A^{- 1} h, v)}_{X_{A}} = ⟨ h, v ⟩ for all v \in X_{A}, h \in X_{A}^{'},

where by $⟨ h, v ⟩$ we mean the value of the linear functional $h$ at the element $v .$ Note that $A^{- 1}$ is an isometry between $X_{A}^{'}$ and $X_{A}$ , i.e.,

(2.7)

{| A^{- 1} h |}_{X_{A}} = {| h |}_{X_{A}^{'}} for all h \in X_{A}^{'} .

As in [11], we consider the energetic space of the iterate $A^{p}$ to be the space

H := A^{- (p - 1)} (X_{A}) .

Here $A^{- k} = A^{- 1} (A^{- (k - 1)})$ for $k = 2, \dots, p - 1 .$ Since $A^{- 1} : X_{A}^{'} \to X_{A}$ and $X_{A} \subset X_{A}^{'}$ one has

H = A^{- (p - 1)} (X_{A}) \subset A^{- (p - 2)} (X_{A}) \subset \dots \subset A^{- 1} (X_{A}) \subset X_{A} .

Next the space $H$ is endowed with the inner product ${(\cdot, \cdot)}_{H}$ and the norm $| \cdot |_{H}$ given by

{(u, v)}_{H} = {(A^{p - 1} u, A^{p - 1} v)}_{X_{A}}, {| u |}_{H} = {| A^{p - 1} u |}_{X_{A}} .

Note that for the embedding $H \subset X_{A},$ the following inequality holds

(2.8)

{| u |}_{X_{A}} \leq \frac{1}{λ^{p - 1}} {| u |}_{H} for all u \in H .

Indeed, if $u \in H$ , then $u = A^{- (p - 1)} v$ for some $v \in X_{A}$ and using successively (2.7) and Poincaré’s inequalities (2.2) and (2.3), we have

${\| u \|}_{X_{A}}$	$=$	${\| A^{- (p - 1)} v \|}_{X_{A}} = {\| A^{- 1} A^{- (p - 2)} v \|}_{X_{A}} = {\| A^{- (p - 2)} v \|}_{X_{A}^{'}} \leq \frac{1}{\sqrt{λ}} {\| A^{- (p - 2)} v \|}_{X}$
	$\leq$	$\frac{1}{λ} {\| A^{- (p - 2)} v \|}_{X_{A}} = \frac{1}{λ} {\| A^{- 1} A^{- (p - 3)} v \|}_{X_{A}} = \frac{1}{λ} {\| A^{- (p - 3)} v \|}_{X_{A}^{'}} \leq \frac{1}{λ^{3 / 2}} {\| A^{- (p - 3)} v \|}_{X}$
	$\leq$	$\frac{1}{λ^{2}} {\| A^{- (p - 3)} v \|}_{X_{A}} \leq \dots \leq \frac{1}{λ^{p - 1}} {\| v \|}_{X_{A}} = \frac{1}{λ^{p - 1}} {\| u \|}_{H} .$

Moreover using (2.2) we obtain

(2.9)

{| u |}_{X} \leq \frac{1}{\sqrt{λ}} {| u |}_{X_{A}} \leq \frac{1}{λ^{\frac{2 p - 1}{2}}} {| u |}_{H} for all u \in H .

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

u = A^{- p} F (u)

in the energetic space $H,$ for the operator $A^{- p} F .$

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 $| \cdot |_{Z}$ and let $∥ . ∥$ be a semi-norm on $Z$ for which there is a constant $σ > 0$ such that $‖ u ‖ \leq σ {| u |}_{Z}$ for all $u \in Z .$

Let $K$ be a wedge in $Z,$ i.e., a closed convex set with $λ K \subset K$ for every $λ \in ℝ_{+},$ and let $ϕ \in K$ be any fixed element with $‖ ϕ ‖ > 0$ and ${| ϕ |}_{Z} = 1$ . Then for any positive numbers $R_{0},$ $R_{1}$ with $R_{0} < ‖ ϕ ‖ R_{1},$ there exists a $μ > 0$ such that $‖ μ ϕ ‖ > R_{0}$ and ${| μ ϕ |}_{Z} < R_{1}$ . Hence the set ${u \in K : R_{0} < ‖ u ‖, {| u |}_{Z} < R_{1}}$ is nonempty. Denote

K_{R_{0} R_{1}} := {u \in K : R_{0} \leq ‖ u ‖, {| u |}_{Z} \leq R_{1}} .

Note that, in particular, when $∥ \cdot ∥ = | \cdot |_{Z},$ $K_{R_{0} R_{1}}$ is the conical shell ${u \in K : R_{0} \leq {| u |}_{Z} \leq R_{1}}$ .

The first theorem is a fixed point result in the set $K_{R_{0} R_{1}} .$

Theorem 2.1.

Let $N : K \to K$ be completely continuous and let $h \in K$ with $‖ h ‖ > R_{0}$ and ${| h |}_{Z} \leq R_{1} .$ Assume that the following conditions are satisfied:

(2.10)

{| N (u) |}_{Z} \leq R_{1} for {| u |}_{Z} \leq R_{1};

(2.11)

(1 - μ) N (u) + μ h \neq u for 0 \leq μ < 1, ‖ u ‖ = R_{0}, {| u |}_{Z} \leq R_{1} .

Then $N$ has a fixed point $u \in K$ such that $R_{0} < ‖ u ‖$ and ${| u |}_{Z} \leq R_{1} .$

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 $R_{- 1}$ with $0 < R_{- 1} < σ^{- 1} R_{0}$ and

N (u) \neq μ u for {| u |}_{Z} = R_{- 1}, μ \geq 1,

then $N$ has three fixed points $u_{1}, u_{2}, u_{3}$ with

R_{0} < ‖ u_{1} ‖, {| u_{1} |}_{Z} \leq R_{1}; R_{- 1} < {| u_{2} |}_{Z} \leq R_{1}, ‖ u_{2} ‖ < R_{0}; {| u_{3} |}_{Z} < R_{- 1} .

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 $K_{0}$ be a cone of $Y .$ Denote by $\leq_{K_{0}}$ the partial order relation on $Y$ associated with $K_{0}$ given by $u \leq_{K_{0}} v$ if and only if $v - u \in K_{0}$ and let $‖ u ‖ : Y \to ℝ_{+}$ be a semi-norm on $Y$ for which there exists $C > 0$ such that for every $u \in Y$

(3.1)

‖ u ‖ \leq C {| u |}_{Y} .

Assume that the norm $| \cdot |_{Y}$ and the seminorm $∥ \cdot ∥$ are monotone, i.e., if $0 \leq_{K_{0}} u \leq_{K_{0}} v$ $(u, v \in Y),$ then ${| u |}_{Y} \leq {| v |}_{Y}$ and $‖ u ‖ \leq ‖ v ‖ .$

Our hypotheses are as follow:

(h1) (compactness):

The linear operator $A^{- 1}$ is compact from $Y$ to $Y .$

(h2) (abstract Harnack inequality):

There exists $φ \in K_{0} ∖ {0}$ such that for every $u \in K_{0}$ one has

‖ A^{- 1} u ‖ φ \leq_{K_{0}} A^{- 1} u;

(h3) (norm estimate):

There exists $ψ \in K_{0} ∖ {0}$ such that for every $u \in K_{0}$ one has

u \leq_{K_{0}} {| u |}_{Y} ψ;

(h4) (positivity and monotonicity):

The operator $F : Y \to Y$ is continuous, bounded (maps bounded sets into bounded sets), positive and increasing with respect to the ordering induced by $K_{0},$ i.e.,

0 \leq_{K_{0}} u \leq_{K_{0}} v implies 0 \leq_{K_{0}} F (u) \leq_{K_{0}} F (v) .

Remark 1.

In particular, if $Y = X,$ the (h1) holds provided that the embedding $X_{A} \subset X$ is compact. Indeed, the operator $A^{- 1} : X \to X$ can be descompose as $J A^{- 1} J_{0}$ with $J_{0} : X \to X_{A}^{^{'}}, J_{0} u = u; A^{- 1} : X_{A}^{'} \to X_{A};$ $J : X_{A} \to X, J u = u,$ where $J$ is compact.

Now we consider a cone in $Y,$

K := {u \in K_{0} : ‖ u ‖ φ \leq_{K_{0}} u} .

Clearly in view of (h2), $A^{- 1} (K_{0}) \subset K .$

Denote $N : Y \to Y$ the operator

N (u) = A^{- p} F (u) .

The operator $N$ is well-defined base on the assumption $A^{- 1} (Y) \subset Y$ given by (h1). In addition, since

F : Y \to Y \subset X \subset X_{A}^{'}, A^{- p} : X_{A}^{'} \to H,

one has $N (Y) \subset H .$ 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) \subset K .

Proof.

Let $u \in K$ and denote $v := N (u) = A^{- p} F (u) .$ We have to prove that $v \in K,$ i.e., $v \in K_{0}$ and $‖ v ‖ φ \leq_{K_{0}} v .$ Obviously, based on the above remark, $v \in Y$ . From $u \in K$ one has $u \in K_{0},$ whence in virtue of (h4), $F (u) \in K_{0}$ and next, from (h2), $v \in K_{0}$ and $‖ v ‖ φ \leq_{K_{0}} v .$ Hence $v \in K .$ ∎

We note that $‖ φ ‖ \leq 1 .$ Indeed, if $u$ is any nonzero element of $K$ (take for example $u = A^{- 1} φ),$ then from $‖ u ‖ φ \leq_{K_{0}} u$ we have

‖ u ‖ ‖ φ ‖ \leq ‖ u ‖,

whence $‖ φ ‖ \leq 1 .$

Theorem 3.2.

Assume (h1)–(h4) hold and let $ϕ_{0} \in K$ with ${| ϕ_{0} |}_{Y} = 1 .$ Assume that there exist numbers $R_{0},$ $R_{1}$ with

(3.2)

0 < R_{0} < ‖ ϕ_{0} ‖ ‖ φ ‖ R_{1}

such that

(3.3)

{| A^{- p} F (R_{1} ψ) |}_{Y} \leq R_{1},

(3.4)

A^{- 1} F (R_{0} φ) \geq \frac{R_{0} λ^{p - 1}}{‖ φ ‖ ‖ ϕ ‖} ϕ,

where $φ, λ, ϕ$ have been identified in hypothesis (h2), in (2.4) and (2.5). Then $(1.1)$ has at least one solution $u \in K \cap H$ with

R_{0} < ‖ u ‖, {| u |}_{Y} \leq R_{1} .

Proof.

We shall apply Theorem 2.1 on the space $Y$ with norm $| \cdot |_{Y}$ and semi-norm $∥ \cdot ∥ .$ First, the continuity and boundedness of $F$ together with the compactness of $A^{- 1}$ imply that the operator $N = A^{- p} F$ is completely continuous from $Y$ to $Y .$

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

Next we show that $(2.10)$ holds. Let $u \in K$ be any element with ${| u |}_{Y} \leq R_{1} .$ From (h3) one has $u \leq_{K_{0}} {| u |}_{Y} ψ \leq_{K_{0}} R_{1} ψ .$ Next $F (u) \leq_{K_{0}} F (R_{1} ψ)$ and

N (u) = A^{- p} F (u) \leq_{K_{0}} A^{- p} F (R_{1} ψ) .

As a result

{| N (u) |}_{Y} = {| A^{- p} F (u) |}_{Y} \leq {| A^{- p} F (R_{1} ψ) |}_{Y} .

Now (3.3) gives ${| N (u) |}_{Y} \leq R_{1} .$

Next we show that $(2.11)$ holds for $h := R_{1} ϕ_{0} .$ From $(3.2)$ one has

(3.5)

‖ h ‖ = R_{1} ‖ ϕ_{0} ‖ > R_{0} .

Assume that (2.11) does not hold. Then

(3.6)

(1 - μ) N (u) + μ h = u

for some $u,$ $μ$ with $‖ u ‖ = R_{0},$ ${| u |}_{Y} \leq R_{1},$ $0 < μ < 1 .$

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

u \geq_{K_{0}} R_{0} φ, h \geq_{K_{0}} ‖ h ‖ φ, N (u) \geq_{K_{0}} ‖ N (u) ‖ φ .

Then, since $A^{- 1}$ is order-preserving and operator $F$ is positive and increasing with respect to the ordening induced by $K_{0}$ (see (h4))

N (u) \geq_{K_{0}} N (R_{0} φ) \geq_{K_{0}} 0,

and taking into account the monotonicity on the semi-norm in $K_{0},$ we have

‖ N (u) ‖ \geq ‖ N (R_{0} φ) ‖ .

Then $(3.6)$ implies

u \geq_{K_{0}} (1 - μ) ‖ N (u) ‖ φ + μ ‖ h ‖ φ \geq (1 - μ) ‖ N (R_{0} φ) ‖ φ + μ ‖ h ‖ φ = ((1 - μ) ‖ N (R_{0} φ) ‖ + μ R_{1} ‖ ϕ ‖) φ,

whence applying the semi-norm, we obtain

R_{0} \geq ((1 - μ) ‖ N (R_{0} φ) ‖ + μ R_{1} ‖ ϕ ‖) ‖ φ ‖ .

Then using $(3.2)$ we get

R_{0} > (1 - μ) ‖ N (R_{0} φ) ‖ ‖ φ ‖ + μ R_{0} .

Consequently

‖ N (R_{0} φ) ‖ < \frac{R_{0}}{‖ φ ‖},

i.e.

(3.7)

‖ A^{- p} F (R_{0} φ) ‖ < \frac{R_{0}}{‖ φ ‖} .

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

A^{- 2} F (R_{0} φ) \geq \frac{R_{0} λ^{p - 1}}{‖ φ ‖ ‖ ϕ ‖} A^{- 1} ϕ = \frac{R_{0} λ^{p - 2}}{‖ φ ‖ ‖ ϕ ‖} ϕ,

⋮

A^{- p} F (R_{0} φ) \geq \frac{R_{0}}{‖ φ ‖ ‖ ϕ ‖} ϕ .

Then

(3.8)

‖ A^{- p} F (R_{0} φ) ‖ \geq \frac{R_{0}}{‖ φ ‖} .

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 $R_{- 1}$ with $0 < C R_{- 1} < R_{0} < ‖ ϕ_{0} ‖ ‖ φ ‖ R_{1}$ and

{| A^{- 1} |}^{p} {| F (R_{- 1} ψ) |}_{Y} \leq R_{- 1},

then problem $(1.1)$ has three solutions $u_{1},$ $u_{2}, u_{3} \in K$ with

R_{0} < ‖ u_{1} ‖, {| u_{1} |}_{Y} \leq R_{1}; R_{- 1} < {| u_{2} |}_{Y} \leq R_{1}, ‖ u_{2} ‖ < R_{0}; {| u_{3} |}_{Y} < R_{- 1} .

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

Proof.

We only need to check the condition $R_{- 1} < σ^{- 1} R_{0} .$ In our case, $σ = C .$ Indeed, from $‖ u ‖ \leq C {| u |}_{Y}$ the required relationship between $R_{- 1}$ and $R_{0}$ is satisfied if $C R_{- 1} < R_{0} .$ ∎

Theorem 3.2 immediately yields multiple solution results by a simple multiplication of the pair $(R_{0}, R_{1}),$ aimed to produce disjoint sets of type $K_{R_{0} R_{1}}$ for which the assumptions of the theorem are fulfilled. Thus we may state

Theorem 3.4.

Assume (h1)–(h4) hold and let $ϕ_{0} \in K$ with ${| ϕ_{0} |}_{Y} = 1 .$ Assume that there exist increasing sequences of positive numbers ${(R_{0}^{i})}_{1 \leq i \leq k},$ ${(R_{1}^{i})}_{1 \leq i \leq k}$ such that the following conditions are satisfied:

R_{0}^{i} < ‖ ϕ_{0} ‖ ‖ φ ‖ R_{1}^{i} for i = 1, \dots, k;

(3.9)

C R_{1}^{i} < R_{0}^{i + 1} for i = 1, \dots, k - 1;

{| A^{- 1} |}^{p} {| F (R_{1}^{i} ψ) |}_{Y} \leq R_{1}^{i}, for i = 1, \dots, k;

A^{- 1} F (R_{0}^{i} φ) \geq \frac{R_{0}^{i} λ^{p - 1}}{‖ φ ‖ ‖ ϕ ‖} ϕ for i = 1, \dots, k .

Then $(1.1)$ has at least $k$ solutions $u_{i} \in K$ with $R_{0}^{i} < ‖ u_{i} ‖,$ ${| u_{i} |}_{Y} \leq R_{1}^{i}$ for $i = 1, \dots, k .$

Proof.

It is enough to see that condition (3.9) guarantees that $K_{R_{0}^{i} R_{1}^{i}} \cap K_{R_{0}^{j} R_{1}^{j}} = \emptyset$ for $i \neq j .$ ∎

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 $A = - Δ$ , $X_{A} = H_{0}^{1} (Ω)$ , $X = L^{2} (Ω)$ , $H = {(- Δ)}^{- (p - 1)} (H_{0}^{1} (Ω))$ and $D (A) = C^{2} (\bar{Ω}) \cap C_{0} (\bar{Ω})$ . Also

F (u) (x) = f (u (x)) for x \in Ω .

Next we will use a theorem for superharmonic functions, where by a superharmonic function in a domain $Ω \subset ℝ^{n}$ one means any function $u \in H^{1} (Ω)$ satisfying

Δ u \leq 0 in 𝒟^{'} (Ω),

that is,

\int_{Ω} \nabla u \cdot \nabla w \geq 0 for every w \in C_{0}^{\infty} (Ω) with w \geq 0 in Ω .

Let $Ω_{0}$ be any bounded subdomain of $Ω$ with ${\bar{Ω}}_{0} \subset Ω$ (i.e. $Ω_{0} ⋐ Ω$ ). As shall need the following theorem:

Theorem 3.5 ([13]).

Let $n \geq 3$ and $1 \leq q < n / (n - 2)$ , or $n = 2$ and $1 \leq q < \infty$ , and let $Ω_{0} ⋐ Ω$ . Then there exists a constant $M = M (n, q, Ω, Ω_{0}) > 0$ such that for every nonnegative superharmonic function $u$ in $Ω$ , the following inequality holds:

(3.10)

u (x) \geq M {| u |}_{L^{q} (Ω_{0})} for x \in Ω_{0} .

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

In the following, we fix any $Ω_{0} ⋐ Ω$ and we let $Y = L^{\infty} (Ω) .$ Clearly, one has

(3.11)

Y = L^{\infty} (Ω) \subset L^{q} (Ω) \subset L^{q} (Ω_{0}) .

Next we consider the seminorm on $Y$ given by

‖ u ‖ = {| u |}_{L^{q} (Ω_{0})} = {(\int_{Ω_{0}} {| u |}^{q} 𝑑 x)}^{\frac{1}{q}} (u \in L^{\infty} (Ω)) .

From (3.11) it follows that there exists a constant $C > 0$ such that for every $u \in L^{\infty} (Ω),$

{| u |}_{L^{q} (Ω_{0})} \leq C {| u |}_{L^{\infty} (Ω)},

which is the analogue of the formula (3.1). It easy to see that $C = {(mes (Ω_{0}))}^{1 / q} .$

As cone $K_{0}$ we have

K_{0} = {u \in L^{\infty} (Ω) : u \geq 0 on Ω} .

Note that the norm $| \cdot |_{L^{\infty} (Ω)}$ and the semi-norm $∥ \cdot ∥$ are monotone, i.e., if $0 \leq u \leq v$ $(u, v \in L^{\infty} (Ω))$ , then ${| u |}_{L^{\infty} (Ω)} \leq {| v |}_{L^{\infty} (Ω)}$ and $‖ u ‖ \leq ‖ v ‖$ .

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 $C^{1, β}$ for some $β \in (0, 1)$ and $g \in L^{\infty} (Ω)$ , then the weak solution in $H_{0}^{1} (Ω)$ of the problem

(3.12)

{\begin{matrix} - Δ u = g & in Ω, \\ u = 0 & on \partial Ω \end{matrix}

belongs to $C^{1} (\bar{Ω})$ . Also the linear solution operator ${(- Δ)}^{- 1} : L^{\infty} (Ω) \to C^{1} (\bar{Ω})$ assigning to each $g \in L^{\infty} (Ω)$ , the corresponding solution of (3.12), is continuous, compact and order-preserving. Therefore, all the more it is compact as an operator from $L^{\infty} (Ω)$ to $L^{\infty} (Ω) .$ Thus (h1) is fulfilled.

Assumption (h2) (abstract Harnack inequality): We need to show that there exists $φ \in K_{0} ∖ {0}$ such that for every $f \in K_{0}$ one has

{(- Δ)}^{- 1} f \geq {| {(- Δ)}^{- 1} f |}_{L^{q} (Ω_{0})} φ on Ω .

From inequality (3.10) we have

{(- Δ)}^{- 1} f \geq M {| {(- Δ)}^{- 1} f |}_{L^{q} (Ω_{0})} on Ω_{0} .

Then clearly

{(- Δ)}^{- 1} f \geq M {| {(- Δ)}^{- 1} f |}_{L^{q} (Ω_{0})} χ_{Ω_{0}} on Ω,

where $χ_{Ω_{0}}$ is the characteristic function of $Ω_{0}$ , i.e. $χ_{Ω_{0}} (x) = 1$ if $x \in Ω_{0}$ and $χ_{Ω_{0}} (x) = 0$ otherwise. So, (h2) is fulfilled with $φ = M χ_{Ω_{0}} .$

Assumption (h3) (norm estimate): There exists $ψ \in K_{0} ∖ {0}$ such that for every $u \in K_{0}$ $(u \in L^{\infty} (Ω), u \geq 0)$ one has

u \leq {| u |}_{L^{\infty} (Ω)} ψ .

Obviously, this condition is fulfilled with $ψ \equiv 1$ (constant function $1$ ).

Assumption (h4) (positivity and monotonicity): This hypothesis is clearly fulfilled if we assume that $f : ℝ \to ℝ$ is continuous, nondecreasing and with $f (ℝ_{+}) \subset ℝ_{+} .$

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

K = {u \in L^{\infty} (Ω) : u \geq | u |_{L^{q} (Ω_{0})} M χ_{Ω_{0}} on Ω},

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

N (u) = {(- Δ)}^{- p} f (u (\cdot)),

and $N (K) \subset K$ .

Now in order to apply Theorem 3.2 we take $ϕ_{0} \equiv 1$ . Then

‖ ϕ_{0} ‖ = {| ϕ_{0} |}_{L^{q} (Ω_{0})} = {(\int_{Ω_{0}} 1^{q} 𝑑 x)}^{\frac{1}{q}} = {(mes (Ω_{0}))}^{\frac{1}{q}} .

Note in addition that for this case, $λ = λ_{1}$ and $ϕ = ϕ_{1},$ where $λ_{1}$ is the first eigenvalue of the Dirichlet problem for $- Δ$ and $ϕ_{1}$ is the corresponding positive eigenfunction with ${| ϕ |}_{L^{2} (Ω)} = 1,$ i.e.,

{\begin{matrix} - Δ ϕ = λ_{1} ϕ & in Ω, \\ ϕ = 0 & on \partial Ω . \end{matrix}

Then $‖ ϕ ‖ = {| ϕ_{1} | |}_{L^{q} (Ω_{0})} .$

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

Theorem 3.6.

Let $Ω$ is a bounded regular domain of class $C^{1, β}$ for some $β \in (0, 1)$ and $f : ℝ \to ℝ$ continuous, nondecreasing and with $f (ℝ_{+}) \subset ℝ_{+} .$ Assume that there exist numbers $R_{0}$ , $R_{1}$ with

0 < R_{0} < M {(mes (Ω_{0}))}^{\frac{2}{q}} R_{1}

such that

\frac{f (R_{1})}{R_{1}} \leq \frac{1}{{| {(- Δ)}^{- p} 1 |}_{L^{\infty} (Ω)}},

(3.13)

{(- Δ)}^{- 1} f (R_{0} M χ_{Ω_{0}}) \geq \frac{R_{0} λ_{1}^{p - 1}}{M {(mes (Ω_{0}))}^{\frac{1}{q}} ‖ ϕ_{1} ‖} ϕ_{1} .

Then (1.2) has at least one solution $u \in K \cap H$ with

R_{0} < ‖ u ‖, {| u |}_{L^{\infty} (Ω)} \leq R_{1} .

Remark 2.

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

f (R_{0} M χ_{Ω_{0}}) \geq \frac{R_{0} λ_{1}^{p}}{M {(mes (Ω_{0}))}^{\frac{1}{q}} ‖ ϕ_{1} ‖} ϕ_{1} .

Since $ϕ_{1} (x) > 0$ for all $x \in Ω$ and $χ_{Ω_{0}} (x) = 0$ on $Ω ∖ Ω_{0},$ this inequality excludes the posibility to have $f (0) = 0 .$ However, this is possible under condition (3.13).

4. Conclusions

Acknowledgments

References

[1] F. BERNIS, J. GARCIA-AZOREBO, I. PERAL, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1, pp. 210–240, 1996.
[2] M. BHAKTA, Solutions to semilinear elliptic PDE’s with biharmonic operator and singular potential, Electronic J. Differential Equations, 2016, 261, pp. 1–17, 2016.
[3] X. CHENG, Z. FENG, L. WEI, Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with weight functions, Discrete Cont. Dyn. Syst. Ser. S, 14, pp. 3067–3083, 2021.
[4] F. GAZZOLA, H.-C. GRUNAU, G. SWEERS, Polyharmonic boundary value problems, Springer, 2009.
[5] L.V. KANTOROVICH, G.P. AKILOV, Functional analysis, Pergamon Press, 1982.
[6] M. NICOLESCU, Opera matematică. Funcţii poliarmonice, Ed. Academiei, Bucureşti, 1980.
[7] M. NICOLESCU, Les fonctions polyharmoniques, Actualité Sci. 331, Herman, 1936.
[8] D. O’Regan and R. PrecupCompression-expansion fixed point theorem in two norms and applications, J. Math. Anal. Appl. 309 (2005), 383–391.
[9] M. PÉREZ-LLANOS, A. PRIMO, Semilinear biharmonic problems with a singular term, J. Differential Equations, 257, pp. 3200–3225, 2014.
[10] R. Precup Compression-expansion fixed point theorems in two norms, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 3 (2005), 157–163.
[11] R. Precup, Semilinear problems with poly-Laplace type operators, Proceedings of the Romanian Academy - Series A: Mathematics, Physics, Technical Sciences, Information Science ????
[12] R. Precup Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations, Journal of Fixed Point Theory and Applications 12 (2012), 193–206.
[13] R. Precup Moser-Harnack inequality, Krasnosel’skiǐ type fixed point theorems in cones and elliptic problems, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder University Centre 40 (2012), 301–313.
[14] R. Precup Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal. 75 (2012), 834–851.
[15] J. Jost, Partial Differential Equations, Springer, New York, 2007.
[16] Yajing Zhang, Yinmei Lü, and Ningning Wang Existence of Positive Solutions of Semilinear Biharmonic Equations, Abstract and Applied Analysis (2014), 1–11.
[17] Tacksun Jungy and Q-Heung Choi Applications of topological methods to the semilinear biharmonic problem with different powers, Korean J. Math. 25 (2017), No. 3, 455–468.
[18] Guoxin Wei and Lingzhong Zeng Estimates for Eigenvalues of Poly-Harmonic Operators, Adv. Nonlinear Stud. 16 (2016), No. 1, 31–44.
[19] Selma Yildrim Yolcu and Tëurkay Yolcu Eigenvalue bounds for the poly-harmonic operators, Illinois Journal of Mathematics 58 (2014), No. 3, 847–865.
[20] QING-MING CHENG, XUERONG QI AND GUOXIN WEI A lower bound for eigenvalues of the poly-Laplace with arbitrary order, Pacific Journal of Mathematics, 262 (2013), No. 1, 35–47.
[21] C. Azizieh and P. Clement, A priori estimates and continuation methods for positive solutions of p-Laplace equations, J. Differential Equations 179 (2002), 213–245.
[22] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[1] Precup R. Semilinear problems with poly-Laplace type operators. Proc Rom Acad Ser A.2022;23:319–328.
[2] Nicolescu M. Opera matematică: funcţii poliarmonice. Bucureşti: Ed. Academiei;1980.
[3] Bernis F, Garcia-Azorebo J, Peral I. Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv Differ Equ.1996;1:210–240.[4] Bhakta M. Solutions to semilinear elliptic PDE’s with biharmonic operator and singular potential. Electron J DifferEqu.2016;261:1–17.
[5] Cheng X, Feng Z, Wei L. Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equationwith weight functions. Discrete Cont Dyn Syst Ser S.2021;14:3067–3083.
[6] Pérez-Llanos M, Primo A. Semilinear biharmonic problems with a singular term. J Differ Equ.2014;257:3200–3225.
[7] GazzolaF,GrunauHC,SweersG.Polyharmonicboundaryvalueproblems.Berlin:Springer;2009.
[8] Zhang Y, Lü Y, Wang N. Existence of positive solutions of semilinear biharmonic equations. Abstr Appl Anal.2014;2014:11 p. Article ID 624328.
[9] Jungy T, Choi QH. Applications of topological methods to the semilinear biharmonic problem with differentpowers. Korean J Math.2017;25:455–468.
[10] Wei G, Zeng L. Estimates for eigenvalues of Poly-Harmonic operators. Adv Nonlinear Stud.2016;16:31–44.
[11] Yolcu SY, Yolcu T. Eigenvalue bounds for the poly-harmonic operators. Illinois J Math.2014;58:847–865.
[12] Cheng QM, Qi X, Wei G. A lower bound for eigenvalues of the poly-Laplace with arbitrary order. Pacific J Math.2013;262:35–47.
[13] Precup R. Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations. J Fixed Point TheoryAppl.2012;12:193–206.
[14] Precup R. Compression-expansion fixed point theorems in two norms. Ann Tiberiu Popoviciu Semin Funct EquApprox Convexity.2005;3:157–163.
[15] O’Regan D, Precup R. Compression-expansion fixed point theorem in two norms and applications. J Math AnalAppl.2005;309:383–391.
[16] Precup R. Moser-Harnack inequality, Krasnoselskiı type fixed point theorems in cones and elliptic problems.Topol Methods Nonlinear Anal.2012;40:301–313.
[17] Precup R. Critical point theorems in cones and multiple positive solutions of elliptic problems. Nonlinear Anal.2012;75:834–851.
[18] Jost J. Partial differential equations. New York: Springer;2007.
[19] Azizieh C, Clement P. A priori estimates and continuation methods for positive solutions of p-Laplace equations.J Differ Equ.2002;179:213–245.
[20] Brezis H. Functional analysis, Sobolev spaces and partial differential equations. New York: Springer;2011.
[21] Precup R. Linear and semilinear partial differential equations. Berlin: De Gruyter;2013.
[22] Herlea DR. Positive solutions for second-order boundary-value problems with phi-Laplacian. Addendum Elec-tron J Differ Equ.2016;51:1–12.

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${\| u \|}_{X_{A}}$	$=$	${\| A^{- (p - 1)} v \|}_{X_{A}} = {\| A^{- 1} A^{- (p - 2)} v \|}_{X_{A}} = {\| A^{- (p - 2)} v \|}_{X_{A}^{'}} \leq \frac{1}{\sqrt{λ}} {\| A^{- (p - 2)} v \|}_{X}$
	$\leq$	$\frac{1}{λ} {\| A^{- (p - 2)} v \|}_{X_{A}} = \frac{1}{λ} {\| A^{- 1} A^{- (p - 3)} v \|}_{X_{A}} = \frac{1}{λ} {\| A^{- (p - 3)} v \|}_{X_{A}^{'}} \leq \frac{1}{λ^{3 / 2}} {\| A^{- (p - 3)} v \|}_{X}$
	$\leq$	$\frac{1}{λ^{2}} {\| A^{- (p - 3)} v \|}_{X_{A}} \leq \dots \leq \frac{1}{λ^{p - 1}} {\| v \|}_{X_{A}} = \frac{1}{λ^{p - 1}} {\| u \|}_{H} .$

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

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

Abstract.

1. Introduction

2. Preliminaries

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

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

Theorem 2.1.

Theorem 2.2.

3. Main results

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

Remark 1.

Lemma 3.1.

Proof.

Theorem 3.2.

Proof.

Theorem 3.3.

Proof.

Theorem 3.4.

Proof.

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

Theorem 3.5 ([13]).

Theorem 3.6.

Remark 2.

4. Conclusions

Acknowledgments

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