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 polyLaplace operators.
Authors
Radu Precup
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, ClujNapoca, Romania
Department of Mathematics BabesBolyai University, ClujNapoca, Romania
Nataliia Kolun
Faculty of Mathematics and Computer Science, BabeşBolyai University, ClujNapoca, Romania
Keywords
Krasnosel’skiĭ’s technique; fixed point; Harnack inequality; iterates of a symmetric linear operator. polyLaplace type operator
Paper coordinates
N. Kolun, R. Precup, Localization of solutions for semilinear problems with polyLaplace type operators, Applicable Analysis, 103(5) (2024), pp. 985997, https://doi.org/10.1080/00036811.2023.2218869
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Localization of solutions for semilinear problems with polyLaplace 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’skiǐ’s technique and abstract Harnack inequality. In particular, we obtain results for semilinear problems involving polyLaplace operators and Navier boundary conditions.
Key words: Krasnosel’skiǐ’s technique, fixed point, Harnack inequality, iterates of a symmetric linear operator, polyLaplace 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 polyLaplace 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 polyLaplace 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)  $$\{\begin{array}{c}{A}^{p}u=F(u)\hfill \\ u,Au,\mathrm{\dots},{A}^{p1}u\in {X}_{A}.\hfill \end{array}$$ 
Here $A:D\left(A\right)\to X$ is a strongly monotone symmetric linear operator, $X$ is a Hilbert space, $D\left(A\right)$ is a linear subspace of $X,$ ${A}^{p}$ is the $p$th iterate of $A$ defined recursively by ${A}^{p}=A{A}^{p1}$, $F$ is any mapping and ${X}_{A}$ is the energetic space of $A.$
In particular, we consider equations involving the polyLaplace operator, with Navier boundary conditions, more exactly to the problem
(1.2)  $$\{\begin{array}{cc}{(\mathrm{\Delta})}^{p}u=f(u\left(x\right))\hfill & \text{in}\mathrm{\Omega},\hfill \\ u=\mathrm{\Delta}u=\mathrm{\dots}={\mathrm{\Delta}}^{p1}u=0\hfill & \text{on}\mathrm{\Omega}.\hfill \end{array}$$ 
Here $\mathrm{\Omega}\subset {\mathbb{R}}^{n}$ is bounded open and $f:\mathbb{R}\to \mathbb{R}$. In this case, $X={L}^{2}(\mathrm{\Omega}),$ $A=\mathrm{\Delta}$ and ${X}_{A}={H}_{0}^{1}(\mathrm{\Omega})$.
For the classical theory of equations with polyLaplace 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 polyLaplace operators.
In this paper we deal with the localization of weak solutions in a conical ”annulus” jointly defined by the norm and a seminorm. 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 seminorm 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 seminorm can be taken the norm itself. However, in case of partial differential equations, MoserHarnack inequalities give us lower estimates only with respect to a seminorm.
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 $A:D\left(A\right)\to X$ be a strongly monotone symmetric linear operator, that is, a linear operator satisfying
$${(Au,v)}_{X}={(u,Av)}_{X}\phantom{\rule{1.5em}{0ex}}\text{for all}u,v\in D\left(A\right),$$ 
(2.1)  $${(Au,u)}_{X}\ge {c}^{2}{\leftu\right}_{X}^{2}\phantom{\rule{1em}{0ex}}\text{for all}u\in D(A)$$ 
and some constant $c>0.$ On $D\left(A\right)$ one considers the energetic inner product
$${(u,v)}_{A}:={(Au,v)}_{X}\phantom{\rule{2em}{0ex}}\left(u,v\in D\left(A\right)\right)$$ 
and the energetic norm
$${\leftu\right}_{A}={(Au,u)}_{X}^{1/2}\phantom{\rule{1.5em}{0ex}}\left(u\in D\left(A\right)\right).$$ 
The completion of the space $(D\left(A\right),.{}_{A})$ 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 $D\left(A\right)$ to ${X}_{A}$ and denoted by ${(.,.)}_{{X}_{A}}$ and $.{}_{{X}_{A}}.$ Let ${X}_{A}^{\prime}$ be the dual space of ${X}_{A}.$ If the Hilbert space $X$ is identified to its dual, then one has
$${X}_{A}\subset X\subset {X}_{A}^{\prime}.$$ 
If the embedding ${X}_{A}\subset X$ is compact, then the following Poincaré’s inequalities hold:
(2.2)  $${u}_{X}\le \frac{1}{\sqrt{\lambda}}{u}_{{X}_{A}}\phantom{\rule{1em}{0ex}}(u\in {X}_{A}),$$ 
(2.3)  $${u}_{{X}_{A}^{\prime}}\le \frac{1}{\sqrt{\lambda}}{u}_{X}\phantom{\rule{1em}{0ex}}(u\in X),$$ 
where
(2.4)  $$\lambda =inf\{u{}_{{X}_{A}}^{2}:u\in {X}_{A},u{}_{X}=1\}$$ 
and the last $inf$ is reached, i.e., ${\varphi }_{{X}_{A}}^{2}=\lambda $ for some $\varphi \in {X}_{A}$ with ${\varphi }_{X}=1,$ and
(2.5)  $$A\varphi =\lambda \varphi .$$ 
The invers of $A$ is the operator ${A}^{1}:{X}_{A}^{\prime}\to {X}_{A}$ defined by
(2.6)  $${({A}^{1}h,v)}_{{X}_{A}}=\u27e8h,v\u27e9\phantom{\rule{1em}{0ex}}\text{for all}v\in {X}_{A},h\in {X}_{A}^{\prime},$$ 
where by $\u27e8h,v\u27e9$ we mean the value of the linear functional $h$ at the element $v.$ Note that ${A}^{1}$ is an isometry between ${X}_{A}^{\prime}$ and ${X}_{A}$, i.e.,
(2.7)  $${{A}^{1}h}_{{X}_{A}}={h}_{{X}_{A}^{\prime}}\text{for all}h\in {X}_{A}^{\prime}.$$ 
As in [11], we consider the energetic space of the iterate ${A}^{p}$ to be the space
$$H:={A}^{(p1)}({X}_{A}).$$ 
Here ${A}^{k}={A}^{1}({A}^{(k1)})$ for $k=2,\mathrm{\dots},p1.$ Since ${A}^{1}:{X}_{A}^{\prime}\to {X}_{A}$ and ${X}_{A}\subset {X}_{A}^{\prime}$ one has
$$H={A}^{(p1)}({X}_{A})\subset {A}^{(p2)}({X}_{A})\subset \mathrm{\dots}\subset {A}^{1}({X}_{A})\subset {X}_{A}.$$ 
Next the space $H$ is endowed with the inner product ${(\cdot ,\cdot )}_{H}$ and the norm $\cdot {}_{H}$ given by
$${(u,v)}_{H}={({A}^{p1}u,{A}^{p1}v)}_{{X}_{A}},{u}_{H}={{A}^{p1}u}_{{X}_{A}}.$$ 
Note that for the embedding $H\subset {X}_{A},$ the following inequality holds
(2.8)  $${u}_{{X}_{A}}\le \frac{1}{{\lambda}^{p1}}{u}_{H}\phantom{\rule{2em}{0ex}}\text{for all}u\in H.$$ 
Indeed, if $u\in H$, then $u={A}^{(p1)}v$ for some $v\in {X}_{A}$ and using successively (2.7) and Poincaré’s inequalities (2.2) and (2.3), we have
${\leftu\right}_{{X}_{A}}$  $=$  ${{A}^{(p1)}v}_{{X}_{A}}={{A}^{1}{A}^{(p2)}v}_{{X}_{A}}={{A}^{(p2)}v}_{{X}_{A}^{\prime}}\le {\displaystyle \frac{1}{\sqrt{\lambda}}}{{A}^{(p2)}v}_{X}$  
$\le $  $\frac{1}{\lambda}}{{A}^{(p2)}v}_{{X}_{A}}={\displaystyle \frac{1}{\lambda}}{{A}^{1}{A}^{(p3)}v}_{{X}_{A}}={\displaystyle \frac{1}{\lambda}}{{A}^{(p3)}v}_{{X}_{A}^{\prime}}\le {\displaystyle \frac{1}{{\lambda}^{3/2}}}{{A}^{(p3)}v}_{X$  
$\le $  $\frac{1}{{\lambda}^{2}}}{{A}^{(p3)}v}_{{X}_{A}}\le \mathrm{\dots}\le {\displaystyle \frac{1}{{\lambda}^{p1}}}{v}_{{X}_{A}}={\displaystyle \frac{1}{{\lambda}^{p1}}}{u}_{H}.$ 
Moreover using (2.2) we obtain
(2.9)  $${\leftu\right}_{X}\le \frac{1}{\sqrt{\lambda}}{\leftu\right}_{{X}_{A}}\le \frac{1}{{\lambda}^{\frac{2p1}{2}}}{\leftu\right}_{H}\phantom{\rule{1.5em}{0ex}}\text{for all}u\in H.$$ 
We also note that problem $(\text{1.1})$ 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 seminorm
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 $\parallel .\parallel $ be a seminorm on $Z$ for which there is a constant $\sigma >0$ such that $\Vert u\Vert \le \sigma {\leftu\right}_{Z}$ for all $u\in Z.$
Let $K$ be a wedge in $Z,$ i.e., a closed convex set with $\lambda K\subset K$ for every $\lambda \in {\mathbb{R}}_{+},$ and let $\varphi \in K$ be any fixed element with $\Vert \varphi \Vert >0$ and ${\varphi }_{Z}=1$. Then for any positive numbers ${R}_{0},$ ${R}_{1}$ with $$ there exists a $\mu >0$ such that $\Vert \mu \varphi \Vert >{R}_{0}$ and $$. Hence the set $$ is nonempty. Denote
$${K}_{{R}_{0}{R}_{1}}:=\{u\in K:{R}_{0}\le \Vert u\Vert ,{u}_{Z}\le {R}_{1}\}.$$ 
Note that, in particular, when $\parallel \cdot \parallel =\cdot {}_{Z},$ ${K}_{{R}_{0}{R}_{1}}$ is the conical shell $\{u\in K:{R}_{0}\le {u}_{Z}\le {R}_{1}\}$.
The first theorem is a fixed point result in the set ${K}_{{R}_{0}{R}_{1}}.$
Theorem 2.1.
Let $N:K\to K$ be completely continuous and let $h\in K$ with $\Vert h\Vert >{R}_{0}$ and ${\lefth\right}_{Z}\le {R}_{1}.$ Assume that the following conditions are satisfied:
(2.10)  $${\leftN\left(u\right)\right}_{Z}\le {R}_{1}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}{u}_{Z}\le {R}_{1};$$ 
(2.11)  $$ 
Then $N$ has a fixed point $u\in K$ such that $$ and ${\leftu\right}_{Z}\le {R}_{1}.$
We also have a three solutions existence result.
Theorem 2.2.
Under the assumptions of Theorem 2.1, if in addition there exists a number ${R}_{1}$ with $$ and
$$N\left(u\right)\ne \mu u\phantom{\rule{1.5em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}{u}_{Z}={R}_{1},\mu \ge 1,$$ 
then $N$ has three fixed points ${u}_{1},{u}_{2},{u}_{3}$ with
$$ 
The proofs of these theorems rely on the fixed point index theory and can be found in paper [13].
3. Main results
3.1. Existence and localization results for semilinear problems involving iterates of a symmetric linear operator
Let $(Y,.{}_{Y})$ be a Banach space continuously embbeded in $X$ and let ${K}_{0}$ be a cone of $Y.$ Denote by ${\le}_{{K}_{0}}$ the partial order relation on $Y$ associated with ${K}_{0}$ given by $u{\le}_{{K}_{0}}v$ if and only if $vu\in {K}_{0}$ and let $\Vert u\Vert :Y\to {\mathbb{R}}_{+}$ be a seminorm on $Y$ for which there exists $C>0$ such that for every $u\in Y$
(3.1)  $$\Vert u\Vert \le C{u}_{Y}.$$ 
Assume that the norm $\cdot {}_{Y}$ and the seminorm $\parallel \cdot \parallel $ are monotone, i.e., if $0{\le}_{{K}_{0}}u{\le}_{{K}_{0}}v$ $(u,v\in Y),$ then ${u}_{Y}\le {v}_{Y}$ and $\Vert u\Vert \le \Vert v\Vert .$
Our hypotheses are as follow:
 (h1) (compactness):

The linear operator ${A}^{1}$ is compact from $Y$ to $Y.$
 (h2) (abstract Harnack inequality):

There exists $\phi \in {K}_{0}\setminus \{0\}$ such that for every $u\in {K}_{0}$ one has
$$\Vert {A}^{1}u\Vert \phi {\le}_{{K}_{0}}{A}^{1}u;$$  (h3) (norm estimate):

There exists $\psi \in {K}_{0}\setminus \{0\}$ such that for every $u\in {K}_{0}$ one has
$$u{\le}_{{K}_{0}}{\leftu\right}_{Y}\psi ;$$  (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.,
$$0{\le}_{{K}_{0}}u{\le}_{{K}_{0}}v\phantom{\rule{1em}{0ex}}\text{implies}0{\le}_{{K}_{0}}F(u){\le}_{{K}_{0}}F(v).$$
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 ${J}_{0}:X\to {X}_{A}^{{}^{\prime}},{J}_{0}u=u;{A}^{1}:{X}_{A}^{\prime}\to {X}_{A};$ $J:{X}_{A}\to X,Ju=u,$ where $J$ is compact.
Now we consider a cone in $Y,$
$$K:=\{u\in {K}_{0}:\Vert u\Vert \phi {\le}_{{K}_{0}}u\}.$$ 
Clearly in view of (h2), ${A}^{1}\left({K}_{0}\right)\subset K.$
Denote $N:Y\to Y$ the operator
$$N(u)={A}^{p}F(u).$$ 
The operator $N$ is welldefined base on the assumption ${A}^{1}(Y)\subset Y$ given by (h1). In addition, since
$$F:Y\to Y\subset X\subset {X}_{A}^{\prime},{A}^{p}:{X}_{A}^{\prime}\to H,$$ 
one has $N\left(Y\right)\subset H.$ 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 $\Vert v\Vert \phi {\le}_{{K}_{0}}v.$ 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 $\Vert v\Vert \phi {\le}_{{K}_{0}}v.$ Hence $v\in K.$ ∎
We note that $\Vert \phi \Vert \le 1.$ Indeed, if $u$ is any nonzero element of $K$ (take for example $u={A}^{1}\phi ),$ then from $\Vert u\Vert \phi {\le}_{{K}_{0}}u$ we have
$$\Vert u\Vert \Vert \phi \Vert \le \Vert u\Vert ,$$ 
whence $\Vert \phi \Vert \le 1.$
Theorem 3.2.
Assume (h1)–(h4) hold and let ${\varphi}_{0}\in K$ with ${{\varphi}_{0}}_{Y}=1.$ Assume that there exist numbers ${R}_{0},$ ${R}_{1}$ with
(3.2)  $$ 
such that
(3.3)  $${\left{A}^{p}F\left({R}_{1}\psi \right)\right}_{Y}\le {R}_{1},$$ 
(3.4)  $${A}^{1}F({R}_{0}\phi )\ge \frac{{R}_{0}{\lambda}^{p1}}{\Vert \phi \Vert \Vert \varphi \Vert}\varphi ,$$ 
where $\phi ,\lambda ,\varphi $ have been identified in hypothesis (h2), in (2.4) and (2.5). Then $(\text{1.1})$ has at least one solution $u\in K\cap H$ with
$$ 
Proof.
We shall apply Theorem 2.1 on the space $Y$ with norm $\cdot {}_{Y}$ and seminorm $\parallel \cdot \parallel .$ 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 $(\text{2.10})$ holds. Let $u\in K$ be any element with ${\leftu\right}_{Y}\le {R}_{1}.$ From (h3) one has $u{\le}_{{K}_{0}}{\leftu\right}_{Y}\psi {\le}_{{K}_{0}}{R}_{1}\psi .$ Next $F\left(u\right){\le}_{{K}_{0}}F\left({R}_{1}\psi \right)$ and
$$N\left(u\right)={A}^{p}F\left(u\right){\le}_{{K}_{0}}{A}^{p}F\left({R}_{1}\psi \right).$$ 
As a result
$${\leftN\left(u\right)\right}_{Y}={\left{A}^{p}F\left(u\right)\right}_{Y}\le {\left{A}^{p}F\left({R}_{1}\psi \right)\right}_{Y}.$$ 
Now (3.3) gives ${\leftN\left(u\right)\right}_{Y}\le {R}_{1}.$
Next we show that $(\text{2.11})$ holds for $h:={R}_{1}{\varphi}_{0}.$ From $(\text{3.2})$ one has
(3.5)  $$\Vert h\Vert ={R}_{1}\Vert {\varphi}_{0}\Vert >{R}_{0}.$$ 
Assume that (2.11) does not hold. Then
(3.6)  $$(1\mu )N(u)+\mu h=u$$ 
for some $u,$ $\mu $ with $\Vert u\Vert ={R}_{0},$ ${u}_{Y}\le {R}_{1},$ $$
Since the abstract Harnack inequality holds (see (h2)) we have
$$u{\ge}_{{K}_{0}}{R}_{0}\phi ,h{\ge}_{{K}_{0}}\Vert h\Vert \phi ,N(u){\ge}_{{K}_{0}}\Vert N(u)\Vert \phi .$$ 
Then, since ${A}^{1}$ is orderpreserving and operator $F$ is positive and increasing with respect to the ordening induced by ${K}_{0}$ (see (h4))
$$N(u){\ge}_{{K}_{0}}N({R}_{0}\phi ){\ge}_{{K}_{0}}0,$$ 
and taking into account the monotonicity on the seminorm in ${K}_{0},$ we have
$$\Vert N(u)\Vert \ge \Vert N({R}_{0}\phi )\Vert .$$ 
Then $(\text{3.6})$ implies
$$u{\ge}_{{K}_{0}}(1\mu )\Vert N(u)\Vert \phi +\mu \Vert h\Vert \phi \ge (1\mu )\Vert N({R}_{0}\phi )\Vert \phi +\mu \Vert h\Vert \phi =\left((1\mu )\Vert N({R}_{0}\phi )\Vert +\mu {R}_{1}\Vert \varphi \Vert \right)\phi ,$$ 
whence applying the seminorm, we obtain
$${R}_{0}\ge \left((1\mu )\Vert N({R}_{0}\phi )\Vert +\mu {R}_{1}\Vert \varphi \Vert \right)\Vert \phi \Vert .$$ 
Then using $(\text{3.2})$ we get
$${R}_{0}>(1\mu )\Vert N({R}_{0}\phi )\Vert \Vert \phi \Vert +\mu {R}_{0}.$$ 
Consequently
$$ 
i.e.
(3.7)  $$ 
On the other hand, from the condition $(\text{3.4})$ and (2.5) we have
$${A}^{2}F({R}_{0}\phi )\ge \frac{{R}_{0}{\lambda}^{p1}}{\Vert \phi \Vert \Vert \varphi \Vert}{A}^{1}\varphi =\frac{{R}_{0}{\lambda}^{p2}}{\Vert \phi \Vert \Vert \varphi \Vert}\varphi ,$$ 
$$\mathrm{\vdots}$$ 
$${A}^{p}F({R}_{0}\phi )\ge \frac{{R}_{0}}{\Vert \phi \Vert \Vert \varphi \Vert}\varphi .$$ 
Then
(3.8)  $$\Vert {A}^{p}F({R}_{0}\phi )\Vert \ge \frac{{R}_{0}}{\Vert \phi \Vert}.$$ 
This contradicts of $(\text{3.7})$. Consequently (2.11) is true. Now the conclusion follows from Theorem 2.1. ∎
Theorem 2.2 yields the following three solutions existence result.
Theorem 3.3.
Under the assumptions of Theorem 3.2, if in addition there exists a number ${R}_{1}$ with $$ and
$${\left{A}^{1}\right}^{p}{\leftF\left({R}_{1}\psi \right)\right}_{Y}\le {R}_{1},$$ 
then problem $(\text{1.1})$ has three solutions ${u}_{1},$ ${u}_{2},{u}_{3}\in K$ with
$$ 
If $F\left(0\right)\ne 0,$ then all the three solutions are nontrivial.
Proof.
We only need to check the condition $$ In our case, $\sigma =C.$ Indeed, from $\Vert u\Vert \le C{\leftu\right}_{Y}$ the required relationship between ${R}_{1}$ and ${R}_{0}$ is satisfied if $$ ∎
Theorem 3.2 immediately yields multiple solution results by a simple multiplication of the pair $({R}_{0},{R}_{1}),$ aimed to produce disjoint sets of type ${K}_{{R}_{0}{R}_{1}}$ for which the assumptions of the theorem are fulfilled. Thus we may state
Theorem 3.4.
Assume (h1)–(h4) hold and let ${\varphi}_{0}\in K$ with ${{\varphi}_{0}}_{Y}=1.$Assume that there exist increasing sequences of positive numbers ${({R}_{0}^{i})}_{1\le i\le k},$ ${({R}_{1}^{i})}_{1\le i\le k}$ such that the following conditions are satisfied:
$$ 
(3.9)  $$ 
$${\left{A}^{1}\right}^{p}{\leftF\left({R}_{1}^{i}\psi \right)\right}_{Y}\le {R}_{1}^{i},\text{for}i=1,\mathrm{\dots},k;$$ 
$${A}^{1}F({R}_{0}^{i}\phi )\ge \frac{{R}_{0}^{i}{\lambda}^{p1}}{\Vert \phi \Vert \Vert \varphi \Vert}\varphi \phantom{\rule{1.5em}{0ex}}\text{for}i=1,\mathrm{\dots},k.$$ 
Then $(\text{1.1})$ has at least $k$ solutions ${u}_{i}\in K$ with $$ ${{u}_{i}}_{Y}\le {R}_{1}^{i}$ for $i=1,\mathrm{\dots},k.$
Proof.
It is enough to see that condition (3.9) guarantees that ${K}_{{R}_{0}^{i}{R}_{1}^{i}}\cap {K}_{{R}_{0}^{j}{R}_{1}^{j}}=\mathrm{\varnothing}$ for $i\ne j.$ ∎
3.2. Existence and localization results for semilinear problems involving the polyLaplace 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 $A=\mathrm{\Delta}$, ${X}_{A}={H}_{0}^{1}(\mathrm{\Omega})$, $X={L}^{2}(\mathrm{\Omega})$, $H={(\mathrm{\Delta})}^{(p1)}({H}_{0}^{1}\left(\mathrm{\Omega}\right))$ and $D(A)={C}^{2}(\overline{\mathrm{\Omega}})\cap {C}_{0}(\overline{\mathrm{\Omega}})$. Also
$$F\left(u\right)\left(x\right)=f\left(u\left(x\right)\right)\phantom{\rule{1.5em}{0ex}}\text{for}x\in \mathrm{\Omega}.$$ 
Next we will use a theorem for superharmonic functions, where by a superharmonic function in a domain $\mathrm{\Omega}\subset {\mathbb{R}}^{n}$ one means any function $u\in {H}^{1}(\mathrm{\Omega})$ satisfying
$$\mathrm{\Delta}u\le 0\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{1em}{0ex}}{\mathcal{D}}^{\prime}(\mathrm{\Omega}),$$ 
that is,
$${\int}_{\mathrm{\Omega}}\nabla u\cdot \nabla w\ge 0\phantom{\rule{1em}{0ex}}\text{for every}\phantom{\rule{1.5em}{0ex}}w\in {C}_{0}^{\mathrm{\infty}}(\mathrm{\Omega})\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}w\ge 0\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{1em}{0ex}}\mathrm{\Omega}.$$ 
Let ${\mathrm{\Omega}}_{0}$ be any bounded subdomain of $\mathrm{\Omega}$ with ${\overline{\mathrm{\Omega}}}_{0}\subset \mathrm{\Omega}$ (i.e. ${\mathrm{\Omega}}_{0}\u22d0\mathrm{\Omega}$). As shall need the following theorem:
Theorem 3.5 ([13]).
Let $n\ge 3$ and $$, or $n=2$ and $$, and let ${\mathrm{\Omega}}_{0}\u22d0\mathrm{\Omega}$. Then there exists a constant $M=M(n,q,\mathrm{\Omega},{\mathrm{\Omega}}_{0})>0$ such that for every nonnegative superharmonic function $u$ in $\mathrm{\Omega}$, the following inequality holds:
(3.10)  $$u(x)\ge M{u}_{{L}^{q}({\mathrm{\Omega}}_{0})}\phantom{\rule{1.5em}{0ex}}\text{for}\phantom{\rule{1.5em}{0ex}}x\in {\mathrm{\Omega}}_{0}.$$ 
This result was obtained in [13] as a consequence of the MoserHarnack inequality (see [15, p. 305]).
In the following, we fix any ${\mathrm{\Omega}}_{0}\u22d0\mathrm{\Omega}$ and we let $Y={L}^{\mathrm{\infty}}\left(\mathrm{\Omega}\right).$ Clearly, one has
(3.11)  $$Y={L}^{\mathrm{\infty}}(\mathrm{\Omega})\subset {L}^{q}(\mathrm{\Omega})\subset {L}^{q}({\mathrm{\Omega}}_{0}).$$ 
Next we consider the seminorm on $Y$ given by
$$\Vert u\Vert ={u}_{{L}^{q}({\mathrm{\Omega}}_{0})}={\left(\underset{{\mathrm{\Omega}}_{0}}{\int}{u}^{q}\mathit{d}x\right)}^{\frac{1}{q}}\phantom{\rule{1em}{0ex}}(u\in {L}^{\mathrm{\infty}}(\mathrm{\Omega})).$$ 
From (3.11) it follows that there exists a constant $C>0$ such that for every $u\in {L}^{\mathrm{\infty}}(\mathrm{\Omega}),$
$${u}_{{L}^{q}({\mathrm{\Omega}}_{0})}\le C{u}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})},$$ 
which is the analogue of the formula (3.1). It easy to see that $C={\left(\text{mes}\left({\mathrm{\Omega}}_{0}\right)\right)}^{1/q}.$
As cone ${K}_{0}$ we have
$${K}_{0}=\{u\in {L}^{\mathrm{\infty}}(\mathrm{\Omega}):u\ge 0\text{on}\mathrm{\Omega}\}.$$ 
Note that the norm $\cdot {}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}$ and the seminorm $\parallel \cdot \parallel $ are monotone, i.e., if $0\le u\le v$ $\left(u,v\in {L}^{\mathrm{\infty}}(\mathrm{\Omega})\right)$, then ${u}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}\le {v}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}$ and $\Vert u\Vert \le \Vert v\Vert $.
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 $\mathrm{\Omega}$ is a bounded regular domain of class ${C}^{1,\beta}$ for some $\beta \in (0,1)$ and $g\in {L}^{\mathrm{\infty}}(\mathrm{\Omega})$, then the weak solution in ${H}_{0}^{1}(\mathrm{\Omega})$ of the problem
(3.12)  $$\{\begin{array}{cc}\mathrm{\Delta}u=g\hfill & \text{in}\mathrm{\Omega},\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega}\hfill \end{array}$$ 
belongs to ${C}^{1}(\overline{\mathrm{\Omega}})$. Also the linear solution operator ${(\mathrm{\Delta})}^{1}:{L}^{\mathrm{\infty}}(\mathrm{\Omega})\to {C}^{1}(\overline{\mathrm{\Omega}})$ assigning to each $g\in {L}^{\mathrm{\infty}}(\mathrm{\Omega})$, the corresponding solution of (3.12), is continuous, compact and orderpreserving. Therefore, all the more it is compact as an operator from ${L}^{\mathrm{\infty}}(\mathrm{\Omega})$ to ${L}^{\mathrm{\infty}}(\mathrm{\Omega}).$ Thus (h1) is fulfilled.
Assumption (h2) (abstract Harnack inequality): We need to show that there exists $\phi \in {K}_{0}\setminus \{0\}$ such that for every $f\in {K}_{0}$ one has
$${(\mathrm{\Delta})}^{1}f\ge {{(\mathrm{\Delta})}^{1}f}_{{L}^{q}({\mathrm{\Omega}}_{0})}\phi \phantom{\rule{1.5em}{0ex}}\text{on}\mathrm{\Omega}.$$ 
From inequality (3.10) we have
$${(\mathrm{\Delta})}^{1}f\ge M{{(\mathrm{\Delta})}^{1}f}_{{L}^{q}({\mathrm{\Omega}}_{0})}\phantom{\rule{1.5em}{0ex}}\text{on}{\mathrm{\Omega}}_{0}.$$ 
Then clearly
$${(\mathrm{\Delta})}^{1}f\ge M{{(\mathrm{\Delta})}^{1}f}_{{L}^{q}({\mathrm{\Omega}}_{0})}{\chi}_{{\mathrm{\Omega}}_{0}}\phantom{\rule{1.5em}{0ex}}\text{on}\mathrm{\Omega},$$ 
where ${\chi}_{{\mathrm{\Omega}}_{0}}$ is the characteristic function of ${\mathrm{\Omega}}_{0}$, i.e. ${\chi}_{{\mathrm{\Omega}}_{0}}(x)=1$ if $x\in {\mathrm{\Omega}}_{0}$ and ${\chi}_{{\mathrm{\Omega}}_{0}}(x)=0$ otherwise. So, (h2) is fulfilled with $\phi =M{\chi}_{{\mathrm{\Omega}}_{0}}.$
Assumption (h3) (norm estimate): There exists $\psi \in {K}_{0}\setminus \{0\}$ such that for every $u\in {K}_{0}$ $(u\in {L}^{\mathrm{\infty}}(\mathrm{\Omega}),u\ge 0)$ one has
$$u\le {\leftu\right}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}\psi .$$ 
Obviously, this condition is fulfilled with $\psi \equiv 1$ (constant function $1$).
Assumption (h4) (positivity and monotonicity): This hypothesis is clearly fulfilled if we assume that $f:\mathbb{R}\to \mathbb{R}$ is continuous, nondecreasing and with $f({\mathbb{R}}_{+})\subset {\mathbb{R}}_{+}.$
Notice that in virtue of (h2), in this case, the cone $K$ is given by
$$K=\{u\in {L}^{\mathrm{\infty}}(\mathrm{\Omega}):\phantom{\rule{1.278em}{0ex}}u\ge u{}_{{L}^{q}({\mathrm{\Omega}}_{0})}M{\chi}_{{\mathrm{\Omega}}_{0}}\phantom{\rule{1em}{0ex}}\text{on}\phantom{\rule{1.5em}{0ex}}\mathrm{\Omega}\},$$ 
where constant $M>0$ comes from inequality (3.10). Also, the operator $N$ is welldefined from ${L}^{\mathrm{\infty}}(\mathrm{\Omega})$ to ${L}^{\mathrm{\infty}}(\mathrm{\Omega}),$ by
$$N(u)={(\mathrm{\Delta})}^{p}f(u(\cdot )),$$ 
and $N(K)\subset K$.
Now in order to apply Theorem 3.2 we take ${\varphi}_{0}\equiv 1$. Then
$$\Vert {\varphi}_{0}\Vert ={{\varphi}_{0}}_{{L}^{q}({\mathrm{\Omega}}_{0})}={\left(\underset{{\mathrm{\Omega}}_{0}}{\int}{1}^{q}\mathit{d}x\right)}^{\frac{1}{q}}={\left(\text{mes}({\mathrm{\Omega}}_{0})\right)}^{\frac{1}{q}}.$$ 
Note in addition that for this case, $\lambda ={\lambda}_{1}$ and $\varphi ={\varphi}_{1},$ where ${\lambda}_{1}$ is the first eigenvalue of the Dirichlet problem for $\mathrm{\Delta}$ and ${\varphi}_{1}$ is the corresponding positive eigenfunction with ${\varphi }_{{L}^{2}(\mathrm{\Omega})}=1,$ i.e.,
$$\{\begin{array}{cc}\mathrm{\Delta}\varphi ={\lambda}_{1}\varphi \hfill & \text{in}\mathrm{\Omega},\hfill \\ \varphi =0\hfill & \text{on}\partial \mathrm{\Omega}.\hfill \end{array}$$ 
Then $\Vert \varphi \Vert ={\left{\varphi}_{1}\right}_{{L}^{q}({\mathrm{\Omega}}_{0})}.$
With the above specifications, Theorem 3.2 gives the following result.
Theorem 3.6.
Let $\mathrm{\Omega}$ is a bounded regular domain of class ${C}^{1,\beta}$ for some $\beta \in (0,1)$ and $f:\mathbb{R}\to \mathbb{R}$ continuous, nondecreasing and with $f({\mathbb{R}}_{+})\subset {\mathbb{R}}_{+}.$ Assume that there exist numbers ${R}_{0}$, ${R}_{1}$ with
$$ 
such that
$$\frac{f({R}_{1})}{{R}_{1}}\le \frac{1}{{{(\mathrm{\Delta})}^{p}1}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}},$$ 
(3.13)  $${\left(\mathrm{\Delta}\right)}^{1}f({R}_{0}M{\chi}_{{\mathrm{\Omega}}_{0}})\ge \frac{{R}_{0}{\lambda}_{1}^{p1}}{M{\left(\mathrm{mes}({\mathrm{\Omega}}_{0})\right)}^{\frac{1}{q}}\Vert {\varphi}_{1}\Vert}{\varphi}_{1}.$$ 
Then (1.2) has at least one solution $u\in K\cap H$ with
$$ 
Remark 2.
A sufficient condition not involving $A,$ for (3.13) to hold is
$$f({R}_{0}M{\chi}_{{\mathrm{\Omega}}_{0}})\ge \frac{{R}_{0}{\lambda}_{1}^{p}}{M{\left(\mathrm{mes}({\mathrm{\Omega}}_{0})\right)}^{\frac{1}{q}}\Vert {\varphi}_{1}\Vert}{\varphi}_{1}.$$ 
Since ${\varphi}_{1}\left(x\right)>0$ for all $x\in \mathrm{\Omega}$ and ${\chi}_{{\mathrm{\Omega}}_{0}}\left(x\right)=0$ on $\mathrm{\Omega}\setminus {\mathrm{\Omega}}_{0},$ this inequality excludes the posibility to have $f\left(0\right)=0.$ However, this is possible under condition (3.13).
4. Conclusions
Acknowledgments
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