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

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

and the energetic norm

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

If the embedding $X_A\subset X$ is compact, then the following Poincaré’s inequalities hold:

where

and the last $inf$ is reached, i.e., ${|\varphi |}_{X_A}^2=\lambda$ for some $\varphi \in X_A$ with ${|\varphi |}_X=1,$ and

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

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 $X_A^{\prime }$ and $X_A$, i.e.,

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

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

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

Note that for the embedding $H\subset X_A,$ the following inequality holds

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

Moreover using (2.2) we obtain

We also note that problem $(\text{<a href="\#S1.E1" title="In 1. Introduction ‣ Localization of solutions for semilinear problems with poly-Laplace type operators" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">1.1</span></a>})$ is equivalent to the fixed point equation

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

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 semi-norm on $Z$ for which there is a constant $\sigma >0$ such that $\Vert u\Vert \le \sigma {\left|u\right|}_Z$ 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 $\lambda \in {ℝ}_{+},$ 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

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 $K_{R_0{R}_1}.$

We also have a three solutions existence result.

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

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 $v-u\in K_0$ and let $\Vert u\Vert :Y\to {ℝ}_{+}$ be a semi-norm on $Y$ for which there exists $C>0$ such that for every $u\in Y$

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

$$u{\le }_{K_0}{\left|u\right|}_Y\psi ;$$

(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.,

$$0{\le }_{K_0}u{\le }_{K_0}v\text{implies}0{\le }_{K_0}F(u){\le }_{K_0}F(v).$$

Now we consider a cone in $Y,$

Clearly in view of (h2), $A^{-1}\left(K_0\right)\subset K.$

Denote $N:Y\to Y$ the operator

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

one has $N\left(Y\right)\subset H.$ Thus, any fixed point from $Y$ of $N$ belongs to $H.$

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

whence $\Vert \phi \Vert \le 1.$

Theorem 2.2 yields the following three solutions existence result.

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

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 $A=-\mathrm{\Delta }$, $X_A=H_0^1(\mathrm{\Omega })$, $X=L^2(\mathrm{\Omega })$, $H={(-\mathrm{\Delta })}^{-(p-1)}(H_0^1\left(\mathrm{\Omega }\right))$ and $D(A)=C^2(\overline{\mathrm{\Omega }})\cap C_0(\overline{\mathrm{\Omega }})$. Also

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

that is,

Let ${\mathrm{\Omega }}_0$ be any bounded subdomain of $\mathrm{\Omega }$ with ${\overline{\mathrm{\Omega }}}_0\subset \mathrm{\Omega }$ (i.e. ${\mathrm{\Omega }}_0⋐\mathrm{\Omega }$). As shall need the following theorem:

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

In the following, we fix any ${\mathrm{\Omega }}_0⋐\mathrm{\Omega }$ and we let $Y=L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right).$ Clearly, one has

Next we consider the seminorm on $Y$ given by

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

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

Note that the norm $|\cdot {|}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}$ and the semi-norm $\parallel \cdot \parallel$ 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

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 order-preserving. 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

From inequality (3.10) we have

Then clearly

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

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 $f:ℝ\to ℝ$ is continuous, nondecreasing and with $f({ℝ}_{+})\subset {ℝ}_{+}.$

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

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

and $N(K)\subset K$.

Now in order to apply Theorem 3.2 we take ${\varphi }_0\equiv 1$. Then

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.,

Then $\Vert \varphi \Vert ={\left|{\varphi }_1\right||}_{L^q({\mathrm{\Omega }}_0)}.$

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