On a class of infinite semipositone problems for ( p,q ) Laplace operator

Abstract

We analyze a non-linear elliptic boundary value problem that involves $(p, q)$ Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in $(0, \infty)$ which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving $(p, q)$ Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.

Keywords

infinite semipositone problem maximal solution asymptotic estimate singular problem

1. Introduction

We consider the following boundary value problem $\begin{aligned} - Δ_{p} u - Δ_{q} u = λ \frac{f (u)}{u^{β}} in Ω \\ (P_{λ}) & u > 0 in Ω \\ u = 0 on \partial Ω \end{aligned}$ where Ω is a bounded domain in $R^{N}$ with a smooth boundary, $0 < β < 1$ , $1 < q < p < \infty$ and $λ > 0$ is a parameter. The term $Δ_{s} u : = div (| \nabla u |^{s - 2} \nabla u)$ . We make the following assumptions on f:

$f : [0, \infty) \to R$ is a continuous monotonically increasing function with $f (0) < 0$ and $\frac{f (t)}{t^{β}}$ is monotonically increasing.

$\frac{f (t)}{t^{β}} > 0$ for some $t > 0$ .

Note that here

{lim}_{t \to 0 +} \frac{f (t)}{t^{β}} = - \infty

. The type of nonlinear functions analyzed in this article leads to infinite semipositone problems. Establishing the existence of a positive solution to such problems is challenging due to the presence of the singular term which is negative. Moreover, due to the non-homogeneous nature of the associated operator, many conventional methods are not directly applicable.

The motivation to study the non-homogeneous operator $(p, q)$ Laplacian emanates from pure mathematical interest and also due to its applications in several scientific fields. The operator along with the associated energy functional was introduced first by Zhikov [39] in the field of elasticity theory. It can also be viewed as a generalization of the p-Laplacian and as an approximation of the mean curvature operator in Minkowski space which arises in the study of Born–Infeld equation, see [7] and [35]. The structure of the energy functional associated with $(p, q)$ Laplacian drives its application into diverse fields, for instance in the study of the geometry of anisotropic materials, modeling of elementary particles, chemical reaction design [10], biophysics [19], plasma physics etc. Though the non-homogeneity of the operator makes the mathematical treatments strenuous, it has been extensively studied in the literature, see [10,18,20,31] and [21] among others. Also active research on the existence and multiplicity of solutions for $(p, q)$ Laplace problems continues to unfold, even incorporating weight factors within the operator, see [32] and [34].

In general, the problem $\begin{aligned} - Δ_{p} u - Δ_{q} u & = h (u) in Ω \end{aligned}$ with $u > 0$ in Ω and $u = 0$ on $\partial Ω$ is called semipositone problem if h is a monotonically increasing continuous function such that $h (0) \in (- \infty, 0)$ and h is eventually positive, i.e. $h (t) > 0$ for large t. If $h (0) = - \infty$ , we call this an infinite semipositone problem. Semipositone problems exhibit certain unique characteristics which are not expected for positone problems (i.e. when $h (0) > 0$ ). A notable phenomenon is the occurrence of interior zeros in non-negative solutions of semipositone problems, making the proof of the existence of positive solutions more challenging (see [8]). However topological tools such as degree theory, fixed point methods, bifurcation theory etc. are found reliable in obtaining positive solutions to semipositone problems (see [11,24,28] and [14]). But all these methods depend strongly on the qualitative properties of the solutions, such as regularity, asymptotic behavior, a priori estimates, stability etc. Due to the non-homogeneity of the operator, such results were not available for the $(p, q)$ Laplace operator until recently. In [22], Giacomoni et.al. proved $C^{1, α}$ boundary regularity for the solutions of singular elliptic problems involving $(p, q)$ Laplacian and in [15] authors proved an asymptotic behavior of solutions for such problems. In light of these latest developments, we address the question of the existence of a positive solution to an infinite semipositone problem using a sub-super solution approach.

In recent years, semipositone problems for non-homogeneous operators are gaining considerable attention. Das et al. showed in [13] that a positive solution for such a problem exists in any bounded domain, as long as the reaction term is non-singular. In [37], authors study an infinite semipositone problem for $(p, q)$ Laplacian in an interval using a fixed point theorem. Hai and co-authors in [27] have recently examined a similar equation in a ball by assuming the radial nature of the solution and transforming it to an ODE. By using a representation formula, they derived necessary estimates and proved the existence of a positive solution. This approach, however, is limited to symmetrical domains and cannot be extended to arbitrary bounded open sets. Nevertheless, Hai et al., in a series of papers [1,24,25] and [26], explored similar problems in arbitrary bounded open sets in $R^{N}$ , but for the p-Laplace operator. Interested readers may also refer to [2,4,12] and [23] for semilinear or quasilinear elliptic problems with indefinite sign non-linearity in bounded domains. The significance of our paper is that it presents the first results of their kind for a $(p, q)$ Laplace operator in a general domain with an infinite semipositone non-linearity.

In Section 2, we focus on a general theory regarding the existence of positive minimal and maximal solutions and in Sections 3 and 4, we discuss specific examples. Our main result is stated below.

Theorem 1.1.
Suppose f satisfies (H1) and (H2). Also assume that there exists an ordered pair $(ψ, ϕ)$ of sub-supersolutions for ( P λ ) such that $ψ, ϕ \in C_{0}^{1} (\overline{Ω})$ and there exists a positive constant $c_{0}$ satisfying $c_{0} d (x) ⩽ ψ (x) ⩽ ϕ (x)$ and $‖ ψ ‖_{\infty} > β_{0}$ . Here $d (x)$ denotes the distance of x to $\partial Ω$ and $β_{0} = inf {t > 0 : \frac{f (t)}{t^{β}} > 0}$ . Then the problem ( P λ ) admits a maximal and a minimal solution within the ordered interval $[ψ, ϕ]$ .

The above theorem is rather a straightforward extension of results in [16] to the $(p, q)$ Laplace operator, thanks to the global regularity result proved in [22]. Rather the nontrivial step is to prove the existence of sub-supersolutions that satisfy the assumptions mentioned in the above theorem. Singular semipositone non-linearity combined with the non-homogeneity of the operator makes this task harder. In this context, we wish to emphasize that the results we prove here are not yet known even for the p-Laplace operator. In Section 3, under two more hypotheses on the non-linear function f, we prove the existence of an ordered pair of sub-supersolutions bounded below by $c_{0} d (x)$ . We assume that
there exist constants $σ \in (0, β + 1)$ and $A > 0$ such that $f (t) ⩾ A t^{σ}$ for $t ≫ 1$ .

there exist $γ > 0$ and $B > 0$ such that $β ⩽ γ < β + 1$ and $f (t) ⩽ B t^{γ}$ for $t ⩾ 0$ .
The condition (H4) implies ${lim}_{t \to \infty} \frac{f (t)}{t^{β + 1}} = 0$ . Now as an application of Theorem 1.1 we prove the next result.
Theorem 1.2.
Suppose that f satisfies hypotheses (H1)–(H4) and $2 < q < p < \infty$ . Then for $λ ≫ 1$ , there exists a positive solution for the problem ( P λ ).

Motivated by the constructions given in [16] and [13], Theorem 1.2 is proved in Section 3 by the sub-supersolution approach. The additional hypothesis $q > 2$ helps us to circumvent the non-homogeneity of the operator in a clever manner. If $p = q$ , this restriction of $q > 2$ is not necessary as the operator is no more inhomogeneous.

In Section 4, we prove that ( P λ ) admits a maximal solution for a specific non-linear function f. The existence of a global minimal or maximal solution is the foremost step in the bifurcation analysis of a semilinear elliptic problem. In [14], the author has shown the existence of a continuum of a branch of positive solutions emanating from infinity and a multiplicity result for an infinite semipositone problem for Laplacian under certain hypotheses. This phenomenon of multiple positive solutions for non-linearities with an indefinite sign is difficult to obtain and is understood completely only in the case of Laplacian in dimension one [17] and for the strictly semipositone problems in higher dimensions [9]. We believe that Theorem 1.3, combined with the understanding of the weighted eigenvalue problem for p-Laplacian, can provide insight into the multiplicity results for the related quasilinear semipositone problem.

Our second main result in this paper is proved in Theorem 1.3, Section 4. Before stating the theorem we shall make a note of the function space where we seek a positive solution and its properties as given in [3]. Let $P = {u \in C_{0}^{1} (\overline{Ω}) : u ⩾ 0 in Ω}$ denote the positive cone in $C_{0}^{1} (\overline{Ω})$ . Then the interior of $P$ denoted by $P^{\circ}$ consists of set of all ${u \in C_{0}^{1} (\overline{Ω}) : \frac{\partial u}{\partial ν} < 0}$ where ν denotes the outward normal vector on $\partial Ω$ . We shall prove that ( P λ ) admits a maximal positive solution in $P^{\circ}$ .
Theorem 1.3.
Let $β \in (0, 1), 2 < q < p < \infty$ , $σ \in (0, q - 1)$ and $f (s) = (s^{σ + β} - 1)$ . Then the problem ( P λ ) admits maximal solution $u_{λ} \in P^{\circ}$ for large λ. Also, ( P λ ) does not admit any positive solution when λ is small.

Proof of the above theorem essentially depends on the construction of a proper supersolution $Φ_{λ}$ . In Section 5 of the paper we define $Φ_{λ}$ to be the unique solution of $\begin{aligned} - Δ_{p} Φ_{λ} - Δ_{q} Φ_{λ} = λ Φ_{λ}^{σ} in Ω \\ (Q_{λ}) & Φ_{λ} > 0 in Ω \\ Φ_{λ} = 0 on \partial Ω \end{aligned}$ where $σ \in (0, q - 1)$ . The existence and uniqueness of the positive solution $Φ_{λ}$ can be established using standard methods. It is also evident through a comparison principle that $Φ_{λ_{1}} (x) ⩽ Φ_{λ_{2}} (x)$ if $λ_{1} ⩽ λ_{2}$ . Though we expect, $‖ Φ_{λ} ‖_{\infty} \to \infty$ as $λ \to \infty$ , proving this result is challenging due to the non-homogeneous nature of the $(p, q)$ Laplace operator.

Furthermore, the precise behavior of the solution for large λ remains unknown. In a recent work [15], inspired by discussions in [31], an explicit behavior of solutions for a singular elliptic problem is obtained using suitable scaling arguments. However, applying the same approach to establish the asymptotic behavior of solutions for ( Q λ ) is not feasible. This is due to the possibility that the solution of the limiting problem corresponding to ( Q λ ) might be the zero function, making it impossible to obtain a positive lower bound.

Consequently, we employ an alternative approach, similar to the technique used in the proof of Lemma 3.1. This approach yields a parameter estimate for ( Q λ ), allowing us to accurately characterize the behavior of $Φ_{λ}$ for large λ. The significance of Theorem 1.4 is noteworthy, holding implications beyond the current investigation. By applying relevant comparison principles, this result can aid in establishing the existence or non-existence of solutions for non-linear elliptic problems involving the $(p, q)$ Laplacian. A similar estimate derived in [15] for a double phase elliptic problem with a singular non-linearity is applied to demonstrate a uniqueness result. Theorem 1.4.
Let $Φ_{λ}$ be the solution of the boundary value problem given in ( Q λ ) and $2 < q < p < \infty$ . Then there exist positive constants $c_{1}, c_{2}, λ_{0}$ such that for all $λ ⩾ λ_{0}$ and for all $x \in Ω$ , $\begin{matrix} c_{1} λ^{\frac{1}{p - 1 - σ}} d (x) ⩽ Φ_{λ} (x) ⩽ c_{2} λ^{\frac{1}{p - 1 - σ}} d (x) . \end{matrix}$

2. Preliminaries

Definition 2.1.
We define $w \in W_{0}^{1, p} (Ω), w > 0$ to be a weak solution of ( P λ ) if $\begin{aligned} \int_{Ω} | \nabla w |^{p - 2} \nabla w \cdot \nabla ϕ d x + \int_{Ω} | \nabla w |^{q - 2} \nabla w \cdot \nabla ϕ d x \\ = λ \int_{Ω} \frac{f (w)}{w^{β}} ϕ d x, \forall ϕ \in C_{c}^{\infty} (Ω) \end{aligned}$

We say $\overline{w} \in W^{1, p} (Ω), \overline{w} ⩾ 0$ on $\partial Ω$ is a supersolution of ( P λ ) if the following inequality holds for all $ϕ \in C_{c}^{\infty} (Ω)$ and $ϕ ⩾ 0$ , $\begin{matrix} \int_{Ω} | \nabla \overline{w} |^{p - 2} \nabla \overline{w} \cdot \nabla ϕ d x + \int_{Ω} | \nabla \overline{w} |^{q - 2} \nabla \overline{w} \cdot \nabla ϕ d x ⩾ λ \int_{Ω} \frac{f (\overline{w})}{{\overline{w}}^{β}} ϕ d x . \end{matrix}$ and a similar definition holds for subsolution.

Notations: By $d (x)$ we denote the distance of $\partial Ω$ from the point x, i.e, $d (x) = inf {‖ x - y ‖ : y \in \partial Ω}$ . We write $Ω_{ϵ}$ for the set ${x \in Ω : d (x) < ϵ}$ .
Lemma 2.2.
Given $h \in C (Ω)$ such that $| h (x) | ⩽ \frac{C}{d {(x)}^{β}}$ for some $β \in (0, 1)$ there exists a unique $w \in W_{0}^{1, p} (Ω)$ solving $\begin{matrix} (2.1) & \begin{aligned} - Δ_{p} w - Δ_{q} w = h in Ω \\ w = 0 on \partial Ω \end{aligned} \end{matrix}$ in the weak sense.
Proof.
We will prove the existence of a solution to (2.1) by the direct minimization technique. The energy functional $E : W_{0}^{1, p} (Ω) \to R$ corresponding to (2.1) is weakly lower semi-continuous on the space $W_{0}^{1, p} (Ω)$ . Using Hölder’s and Hardy’s inequality we can show that E is coercive. Thus E possesses a global minimizer $w \in W_{0}^{1, p} (Ω)$ and by weak comparison principle this minimizer w is the unique weak solution of the required problem. □
Lemma 2.3.
Let h and w be as in Lemma 2.2 . Then there exist constants $α \in (0, 1)$ and $M > 0$ , depending only on $C, β$ and Ω such that $w \in C^{1, α} (\overline{Ω})$ and $‖ w ‖_{C^{1, α} (\overline{Ω})} < M$ .
Proof.
Let u be the solution of $\begin{matrix} (2.2) & \begin{aligned} - Δ u = h in Ω \\ u = 0 on \partial Ω \end{aligned} \end{matrix}$ and by lemma 3.1 from [25], there exists $α^{'} \in (0, 1)$ such that $u \in C^{1, α^{'}} (\overline{Ω})$ .

Also, let $\tilde{w}$ be the solution of $\begin{aligned} - Δ_{p} \tilde{w} - Δ_{q} \tilde{w} = | h | in Ω \\ \tilde{w} = 0 on \partial Ω \end{aligned}$ Then, by weak comparison principle and proposition 2.7 from [22], there exists a constant $C_{1}$ such that $w ⩽ \tilde{w} ⩽ C_{1} d (x)$ in Ω. Again $\begin{matrix} - Δ_{p} (- w) - Δ_{q} (- w) = - h ⩽ | h | = - Δ_{p} \tilde{w} - Δ_{q} \tilde{w} in Ω \end{matrix}$ and $- w = \tilde{w} = 0 on \partial Ω$ . By a similar reasoning $- w ⩽ \tilde{w} ⩽ C_{1} d (x)$ in Ω and therefore, we have, $| w | ⩽ C_{1} d (x) in Ω$ . Then from (2.1) and (2.2), we have, $\begin{matrix} (2.3) & div (| \nabla w |^{p - 2} \nabla w + | \nabla w |^{q - 2} \nabla w - \nabla u) = 0 in Ω \end{matrix}$ and $w = 0 on \partial Ω$ . Now we conclude the proof by applying Lieberman’s boundary regularity result given in Theorem 1.7 of [29]. □

In this section, from now on, we assume that ( P λ ) admits a positive subsolution $ψ \in C_{0}^{1} (\overline{Ω})$ . Also we assume that $ψ (x) ⩾ c_{0} d (x)$ for some positive constant $c_{0}$ and $‖ ψ ‖_{\infty} > β_{0}$ where $β_{0}$ is given in Theorem 1.1. We define the set $\begin{array}{r} A = {u \in C_{0}^{1} (\overline{Ω}) : u (x) ⩾ ψ (x)} \end{array}$ Fix a $λ > 0$ and let $u \in A$ and $g (u) : = λ \frac{f (u)}{u^{β}}$ . Then $| g (u) | ⩽ \frac{c_{1}}{d {(x)}^{β}}$ for some $c_{1} > 0$ . Let w be the unique weak solution of $- Δ_{p} w - Δ_{q} w = g (u)$ in Ω, $w = 0$ on $\partial Ω$ which is now guaranteed to exist by Lemma 2.2. Since g is monotonically increasing and $u ⩾ ψ$ , we have $\begin{array}{r} - Δ_{p} w - Δ_{q} w = g (u) ⩾ g (ψ) ⩾ - Δ_{p} ψ - Δ_{q} ψ in Ω and w = ψ = 0 on \partial Ω \end{array}$ Now by weak comparison principle, we have $w ⩾ ψ$ . Also, by Lemma 2.3, $w \in C^{1, α} (\overline{Ω})$ , for some $α \in (0, 1)$ . Hence, we define the map $T_{g}$ as follows. Definition 2.4.
We define the map $T_{g} : A \to A$ as $T_{g} (u) = w$ iff w is the weak solution of $\begin{matrix} (2.4) & \begin{aligned} - Δ_{p} w - Δ_{q} w = g (u) in Ω \\ w = 0 on \partial Ω \end{aligned} \end{matrix}$ Clearly, $T_{g}$ is a well-defined map from $A$ to itself. Next, we show that the map $T_{g} : A \to C_{0}^{1} (\overline{Ω})$ is completely continuous.
Lemma 2.5.
$T_{g} : A \to C_{0}^{1} (\overline{Ω})$ is completely continuous.
Proof.
Fix $u \in A$ , then $u (x) ⩾ c_{0} d (x)$ and $‖ u ‖_{\infty} > β_{0}$ . Let $h \in C_{0}^{1} (\overline{Ω})$ such that $‖ h ‖_{C_{0}^{1} (\bar{Ω})} < δ$ . Choose δ small enough so that $u (x) + h (x) ⩾ \frac{c_{0}}{2} d (x)$ and $‖ u + h ‖_{\infty} > β_{0}$ . Let $w_{h}$ be the unique solution of $\begin{matrix} (2.5) & \begin{aligned} - Δ_{p} w_{h} - Δ_{q} w_{h} = g (u + h) in Ω \\ w_{h} = 0 on \partial Ω \end{aligned} \end{matrix}$ and w be the unique solution of $\begin{matrix} (2.6) & \begin{aligned} - Δ_{p} w - Δ_{q} w = g (u) in Ω \\ w = 0 on \partial Ω \end{aligned} \end{matrix}$ Note that $g (u + h) \to g (u)$ pointwise as $‖ h ‖_{C_{0}^{1} (\bar{Ω})} \to 0$ . As $| g (u + h) | ⩽ C_{1} d {(x)}^{- β}$ , by Lemma 2.3 there exists an $α \in (0, 1)$ such that $‖ w_{h} ‖_{C^{1, α} (\bar{Ω})} < M$ . Now by Ascoli Arzela theorem, upto a subsequence, $w_{h_{k}} \to \tilde{w}$ in $C^{1, α - ϵ} (\overline{Ω})$ . Also for all $ϕ \in W_{0}^{1, p} (Ω)$ , $\begin{matrix} (2.7) & \int_{Ω}^{} | \nabla w_{h_{k}} |^{p - 2} \nabla w_{h_{k}} \cdot \nabla ϕ d x + \int_{Ω}^{} | \nabla w_{h_{k}} |^{q - 2} \nabla w_{h_{k}} \cdot \nabla ϕ d x = \int_{Ω}^{} g (u + h_{k}) ϕ d x \end{matrix}$ Passing through the limit in the equation (2.7), we get $\begin{matrix} (2.8) & \int_{Ω}^{} | \nabla \tilde{w} |^{p - 2} \nabla \tilde{w} \cdot \nabla ϕ d x + \int_{Ω}^{} | \nabla \tilde{w} |^{q - 2} \nabla \tilde{w} \cdot \nabla ϕ d x = \int_{Ω}^{} g (u) ϕ d x \end{matrix}$ Thus $\tilde{w}$ is a weak solution of (2.6) and by uniqueness $w = \tilde{w}$ . In a standard way, we can show that every subsequence of the original sequence ${w_{h}}$ converges to w and hence $T_{g} : A \to C_{0}^{1, α - ϵ} (\overline{Ω})$ is continuous. In other words, $T_{g} (u + h) \to T_{g} (u)$ in $C^{1, α - ϵ} (\overline{Ω})$ as $‖ h ‖_{C_{0}^{1} (\bar{Ω})} \to 0$ . Using the fact that $C_{0}^{1, α - ϵ} (\bar{Ω}) \subset \subset C_{0}^{1} (\bar{Ω})$ , we conclude that $T_{g} : A \to C_{0}^{1} (\bar{Ω})$ is completely continuous. □
Proof of Theorem 1.1.
Since g is monotonically increasing, we know that $T_{g}$ maps X into itself where $X = {u \in C_{0}^{1} (\overline{Ω}) : ψ ⩽ u ⩽ ϕ}$ . Also, the map $T_{g} : X \to X$ is completely continuous. Now our result follows due to the celebrated Theorem 6.1 of Amann [3]. □

3. Sub-super solution construction

In this section, we consider $2 < q < p < \infty$ and focus on constructing an ordered pair of sub and supersolutions for the problem ( P λ ) bounded below by a constant multiple of the distance function. Here we assume f satisfies all the hypotheses $(H 1)$ – $(H 4)$ mentioned in the introduction.

Lemma 3.1.
Assume f satisfies (H1)–(H3). Let $ϕ_{1}$ be the first eigen function of $- Δ$ with zero Dirichlet boundary condition. Define $\begin{matrix} ψ : = λ^{r} (ϕ_{1} + ϕ_{1}^{\frac{2}{1 + β}}) \end{matrix}$ where $\frac{1}{p - 1 + β} < r < \frac{1}{p - 1 + β - σ}$ . Then, for large λ, $ψ \in C_{0}^{1} (\overline{Ω})$ is a subsolution for the problem ( P λ ) satisfying $ψ (x) ⩾ c_{λ} d (x)$ .
Proof.
From the definition, since $β < 1$ it is clear that $ψ \in C_{0}^{1} (\overline{Ω}) \cap C^{2} (Ω)$ and $ψ (x) ⩾ c_{λ} d (x)$ . We only need to show that ψ is a subsolution for ( P λ ) when λ is large. We have, for $s = p, q$ , $\begin{aligned} Δ_{s} (ϕ_{1} + ϕ_{1}^{\frac{2}{1 + β}}) = & div [{(1 + \frac{2}{1 + β} ϕ_{1}^{\frac{2}{1 + β} - 1})}^{s - 1} | \nabla ϕ_{1} |^{s - 2} \nabla ϕ_{1}] \\ = & {(1 + \frac{2}{1 + β} ϕ_{1}^{\frac{2}{1 + β} - 1})}^{s - 1} div (| \nabla ϕ_{1} |^{s - 2} \nabla ϕ_{1}) \\ + \frac{2}{1 + β} (\frac{2}{1 + β} - 1) (s - 1) | \nabla ϕ_{1} |^{s} {(1 + \frac{2}{1 + β} ϕ_{1}^{\frac{2}{1 + β} - 1})}^{s - 2} ϕ_{1}^{\frac{2}{1 + β} - 2} \\ = & {(1 + \frac{2}{1 + β} ϕ_{1}^{\frac{2}{1 + β} - 1})}^{s - 1} [Δ_{s} ϕ_{1}] \\ + \frac{2}{1 + β} (\frac{2}{1 + β} - 1) (s - 1) | \nabla ϕ_{1} |^{s} {(1 + \frac{2}{1 + β} ϕ_{1}^{\frac{2}{1 + β} - 1})}^{s - 2} ϕ_{1}^{\frac{2}{1 + β} - 2} \end{aligned}$ Thus we write, $\begin{matrix} (3.1) & - Δ_{s} ψ = λ^{r (s - 1)} ϕ_{1}^{\frac{- 2 β}{1 + β}} L_{s} (ϕ_{1}) \end{matrix}$ where $\begin{array}{rcl} L_{s} (ϕ_{1}) & = & {(1 + \frac{2}{1 + β} ϕ_{1}^{\frac{1 - β}{1 + β}})}^{s - 1} ϕ_{1}^{\frac{2 β}{1 + β}} [- Δ_{s} ϕ_{1}] \\ - \frac{2}{1 + β} (\frac{1 - β}{1 + β}) (s - 1) | \nabla ϕ_{1} |^{s} {(1 + \frac{2}{1 + β} ϕ_{1}^{\frac{1 - β}{1 + β}})}^{s - 2} \end{array}$ Now as $s > 2$ , we have $Δ_{s} ϕ_{1} \in L^{\infty} (Ω)$ and by Vasquez [38], $| \nabla ϕ_{1} | > 0$ on $\partial Ω$ . Thus there exist $δ > 0$ and $m > 0$ such that for $x \in Ω_{δ}$ , $\begin{matrix} (3.2) & L_{p} (ϕ_{1}) < - m and L_{q} (ϕ_{1}) < 0 \end{matrix}$ Using (3.1) and (3.2) we have $\begin{matrix} (3.3) & - Δ_{p} ψ - Δ_{q} ψ ⩽ - \frac{m λ^{r (p - 1)}}{ϕ_{1}^{\frac{2 β}{1 + β}}} ⩽ - \frac{m λ^{r (p - 1 + β)}}{ψ^{β}} for all x \in Ω_{δ} . \end{matrix}$ The last inequality is obtained directly from the definition of ψ. Now that $f (0)$ is negative and f is monotone increasing, so for $r > \frac{1}{p - 1 + β}$ and $λ ≫ 1$ , we have, in $Ω_{δ}$ , $\begin{matrix} - \frac{m λ^{r (p - 1 + β)}}{ψ^{β}} ⩽ \frac{λ f (0)}{ψ^{β}} ⩽ \frac{λ f (ψ)}{ψ^{β}} \end{matrix}$ Hence, $\begin{matrix} (3.4) & - Δ_{p} ψ - Δ_{q} ψ ⩽ \frac{λ f (ψ)}{ψ^{β}} in Ω_{δ} \end{matrix}$ In $Ω ∖ Ω_{δ}$ , there exists a positive constant μ such that $ϕ_{1} > μ$ and thus $ψ > λ^{r} μ$ . Also using the monotonicity of f and the assumption (H3) on f, we have for $λ ≫ 1$ , $\begin{matrix} f (ψ) > f (λ^{r} μ) ⩾ A λ^{r σ} μ^{σ} \end{matrix}$ Further, without loss of generality, we assume $‖ ϕ_{1} ‖_{\infty} = 1$ . Therefore, in $Ω ∖ Ω_{δ}$ , we have, for $s = p, q$ , $\begin{matrix} (3.5) & - Δ_{s} ψ ⩽ \frac{λ^{r (s - 1)} c_{s}}{\overline{μ}} ⩽ \frac{λ [A λ^{r σ} μ^{σ}]}{2 ψ^{β}} ⩽ \frac{λ f (ψ)}{2 ψ^{β}} \end{matrix}$ whenever $r < \frac{1}{s - 1 + β - σ}$ and $c_{s} = {(1 + \frac{2}{1 + β})}^{s - 1} ‖ Δ_{s} ϕ_{1} ‖_{\infty}$ and $\overline{μ} = μ^{\frac{2 β}{1 + β}}$ . Since $q < p$ , by taking $r < \frac{1}{p - 1 + β - σ}$ we have, $\begin{matrix} (3.6) & - Δ_{p} ψ - Δ_{q} ψ ⩽ \frac{λ f (ψ)}{ψ^{β}}, in Ω ∖ Ω_{δ} \end{matrix}$ Now from (3.4) and (3.6) we have the required result. □
Lemma 3.2.
Assume f satisfies (H1), (H2) and (H4). Then the problem ( P λ ) admits a super-solution for all $λ > 0$ .
Proof.
Let $R > 0$ be such that $\overline{Ω} \subset B_{R} (0)$ where $B_{R} (0)$ is the open ball of radius R around origin and e be the unique solution of $\begin{matrix} (3.7) & \begin{aligned} - Δ_{p} e = 1 in B_{R} (0) \\ e = 0 on \partial B_{R} (0) \end{aligned} \end{matrix}$ We know that $e (x)$ must be a radial function and $e (x) = R^{\frac{p}{p - 1}} η (| x |)$ where $\begin{matrix} (3.8) & η (r) = \frac{1 - {(\frac{r}{R})}^{p^{'}}}{p^{'}} . \end{matrix}$ $0 ⩽ r ⩽ R$ and $p^{'}$ is the p-conjugate. Since $γ ⩾ β$ , we can choose a constant $m (λ) ≫ 1$ such that $\begin{matrix} m {(λ)}^{p - 1 + β - γ} ⩾ λ B e^{γ - β} \end{matrix}$ We claim that the function $ϕ : = m (λ) e$ is a supersolution for ( P λ ).

Again using (H4), $\begin{matrix} (3.9) & - Δ_{p} ϕ = m {(λ)}^{p - 1} ⩾ \frac{B λ {(m (λ) e)}^{γ}}{{(m (λ) e)}^{β}} ⩾ \frac{λ f (ϕ)}{ϕ^{β}}, in Ω \end{matrix}$ Also, $\begin{matrix} (3.10) & - Δ_{q} η = - {({| η^{'} (r) |}^{q - 2} η^{'} (r))}^{'} = {(\frac{r^{(p^{'} - 1) (q - 1)}}{R^{p^{'} (q - 1)}})}^{'} ⩾ 0 in B_{R} (0), \end{matrix}$ and $- Δ_{q} ϕ = - k Δ_{q} η$ in Ω for some $k > 0$ . Thus, from (3.9) and (3.10) we have $\begin{matrix} - Δ_{p} ϕ - Δ_{q} ϕ ⩾ - Δ_{p} ϕ ⩾ \frac{λ f (ϕ)}{ϕ^{β}} in Ω and ϕ ⩾ 0 on \partial Ω . \end{matrix}$ Hence, ϕ is a super-solution for ( P λ ). □
Proof of Theorem 1.2.
We note that under the hypotheses $(H 1)$ – $(H 4)$ using Lemma 3.1 and Lemma 3.2, the problem ( P λ ) admits a pair of sub-supersolutions for large λ. Also $ϕ > 0$ in $\overline{Ω}$ and hence $m (λ)$ can be chosen large enough so that $ϕ ⩾ ψ$ in $\overline{Ω}$ . We can now adapt the idea of the proof of Theorem 1.1 to show the existence of a minimal and a maximal solution within the ordered interval $[ψ, ϕ]$ . □

4. Existence of maximal solution

In this section, we are interested in the following boundary value problem: $\begin{aligned} - Δ_{p} u - Δ_{q} u = λ (u^{σ} - \frac{1}{u^{β}}) in Ω \\ ({\tilde{P}}_{λ}) & u > 0 in Ω \\ u = 0 on \partial Ω \end{aligned}$ where $0 < β < 1, 0 < σ < q - 1$ and $2 < q < p < \infty$ . It is evident that $f (s) = s^{σ + β} - 1$ satisfies the hypotheses $(H 1) - (H 2)$ . We define the solution set $S$ to be $\begin{matrix} S = {u \in C_{0}^{1} (\overline{Ω}) : u (x) ⩾ k d (x) for some k > 0} . \end{matrix}$ Our aim in this section is to prove the existence of a positive maximal (global maximal) solution for the Dirichlet problem ( P ˜ λ ) belonging to the set $S$ . We can verify that $Ψ_{λ} = λ^{r} (ϕ_{1} + ϕ_{1}^{\frac{2}{1 + β}})$ defined in Lemma 3.1 is a positive subsolution of ( P ˜ λ ). If there exists a supersolution $Φ_{λ}$ such that $Ψ_{λ} ⩽ Φ_{λ}$ , then Theorem 1.2 can be applied and it guarantees a maximal solution in the ordered interval $[Ψ_{λ}, Φ_{λ}]$ . Our aim in this section is to redefine our supersolution appropriately so that the resulting solution obtained by Theorem 1.2 is in fact a maximal solution (global maximal solution). We make the following definition.

Definition 4.1.
We say that $Φ_{λ}$ is a global supersolution to ( P ˜ λ ) if
$Φ_{λ}$ is a supersolution of ( P ˜ λ ) and

if u is any solution of ( P ˜ λ ), then $u ⩽ Φ_{λ}$ .

We now prove that the solution of the following BVP $\begin{aligned} - Δ_{p} z - Δ_{q} z = λ z^{σ} in Ω \\ (4.1) & z > 0 in Ω \\ z = 0 on \partial Ω \end{aligned}$ is a global supersolution to ( P ˜ λ ). The existence and uniqueness of the positive solution of (4.1) can be proved via a global minimization technique and Lemma A.1. Let us denote $Φ_{λ}$ to be the unique solution of (4.1) and since $- Δ_{p} Φ_{λ} - Δ_{q} Φ_{λ} ⩾ λ (Φ_{λ}^{σ} - \frac{1}{Φ_{λ}^{β}})$ , $Φ_{λ}$ serves as a supersolution for ( P ˜ λ ). Using the comparison principle proved in Lemma A.1, we can show that if u is any solution of ( P ˜ λ ), then $u ⩽ Φ_{λ}$ . Thus, $Φ_{λ}$ is a global supersolution of ( P ˜ λ ). Proof of Theorem 1.3.
We first show that for large λ, $Ψ_{λ} ⩽ Φ_{λ}$ in Ω. Since $Ψ_{λ}$ is a positive subsolution of ( P ˜ λ ), we have, $\begin{matrix} (4.2) & - Δ_{p} Ψ_{λ} - Δ_{q} Ψ_{λ} ⩽ λ (Ψ_{λ}^{σ} - \frac{1}{Ψ_{λ}^{β}}) ⩽ λ Ψ_{λ}^{σ} in Ω \end{matrix}$ and $Ψ_{λ} = 0$ on $\partial Ω$ , i.e, $Ψ_{λ}$ is a positive subsolution of (4.1) also. Consequently, by Lemma A.1, we have $Ψ_{λ} ⩽ Φ_{λ}$ in Ω for $λ ≫ 1$ .

Now Theorem 1.1 ensures the existence of a positive maximal solution for the problem ( P ˜ λ ) within the ordered interval $[Ψ_{λ}, Φ_{λ}]$ . Let us call this maximal solution $u_{λ}$ . Again if u is any other positive solution of ( P ˜ λ ), then Lemma A.1 gives $u ⩽ Φ_{λ}$ in $\overline{Ω}$ . By the construction of $u_{λ}$ , clearly $u ⩽ u_{λ}$ i.e, $u_{λ}$ is the global maximal solution of ( P ˜ λ ).

Next suppose λ is small enough and there exists a positive solution $u_{λ}$ for ( P ˜ λ ). Since $λ \mapsto Φ_{λ}$ is monotone increasing, so there exists some $M > 0$ such that in Ω, $\begin{matrix} - Δ_{p} u_{λ} - Δ_{q} u_{λ} = λ (u_{λ}^{σ} - \frac{1}{u_{λ}^{β}}) ⩽ λ u_{λ}^{σ} ⩽ λ Φ_{λ}^{σ} ⩽ λ Φ_{1}^{σ} ⩽ λ M \end{matrix}$ Let $v_{λ}$ be the solution of $- Δ_{p} v_{λ} - Δ_{q} v_{λ} = λ M in Ω$ and $v_{λ} = 0 on \partial Ω$ . Then by weak comparison principle, $u_{λ} ⩽ v_{λ}$ in Ω. Now by Proposition 10 of [33], as $λ \to 0 +$ , $v_{λ} \to 0$ in $C_{0}^{1} (\overline{Ω})$ . Thus by comparison principle any such solution $u_{λ}$ must be negative which is a contradiction and proves Theorem 1.3. □

5. Asymptotic estimates

In this section we analyze the behavior of the function $Φ_{λ}$ in detail when $λ \to \infty$ . To get a deeper understanding of the solution of equation (4.1), we define ${\tilde{Φ}}_{λ} : = λ^{\frac{- 1}{p - 1 - σ}} Φ_{λ}$ so that $\begin{matrix} (5.1) & \begin{aligned} - Δ_{p} {\tilde{Φ}}_{λ} - γ Δ_{q} {\tilde{Φ}}_{λ} = {\tilde{Φ}}_{λ}^{σ} in Ω \\ {\tilde{Φ}}_{λ} = 0 on \partial Ω \end{aligned} \end{matrix}$ where $γ = λ^{\frac{q - p}{p - 1 - σ}}$ which tends to 0 as $λ \to \infty$ . Note that there exists a constant $M_{0} > 0$ , independent of λ, such that $‖ {\tilde{Φ}}_{λ} ‖_{L^{p} (Ω)} ⩽ M_{0}$ . The uniform $L^{\infty}$ upper bound we obtain here is true for any $1 < q < p < \infty$ .

Lemma 5.1.

There exists a positive constant M, independent of λ, such that $‖ {\tilde{Φ}}_{λ} ‖_{\infty} ⩽ M$ .

Proof.

We adapt the idea of De Giorgi–Stampacchia iteration method to obtain the uniform $L^{\infty}$ estimate. First of all we define $v^{(λ)} : = ϵ_{0} {\tilde{Φ}}_{λ}$ where $ϵ_{0} = \frac{δ^{\frac{1}{p}}}{M_{0}} ⩽ 1$ , so that $‖ v^{(λ)} ‖_{L^{p} (Ω)} ⩽ δ^{\frac{1}{p}}$ where δ is to be chosen later. Hereafter, for convenience, we simply write v for $v^{(λ)}$ . Then v satisfies: $\begin{aligned} (5.2) & - \frac{1}{ϵ_{0}^{p - 1 - σ}} Δ_{p} v - \frac{γ}{ϵ_{0}^{q - 1 - σ}} Δ_{q} v & = v^{σ} in Ω \end{aligned}$ and $v = 0$ on $\partial Ω$ . For $k \in N$ , we define $C_{k} : = 1 - 2^{- k}$ , $v_{k} : = v - C_{k}$ , $w_{k} : = v_{k}^{+}$ and $U_{k} : = ‖ w_{k} ‖_{L^{p} (Ω)}^{p}$ . Then, $0 ⩽ w_{k} ⩽ | v | + 1$ in Ω and $w_{k} = 0$ on $\partial Ω$ . Clearly, $w_{k} \in W_{0}^{1, p} (Ω)$ and $w_{k + 1} ⩽ w_{k}, \forall k \in N$ . Also we note that ${lim}_{k \to \infty} w_{k} = {(v - 1)}^{+}$ and so by Lebesgue Dominated Convergence Theorem, ${lim}_{k \to \infty} U_{k} = \int_{Ω} {[{(v - 1)}^{+}]}^{p} d x$ .

Now define $A_{k} : = \frac{C_{k + 1}}{C_{k + 1} - C_{k}} = 2^{k + 1} - 1, k \in N$ . Then $v < A_{k} w_{k}$ on the set ${w_{k + 1} > 0}$ . Taking $w_{k + 1}$ as test function in the weak formulation of (5.2), we get, $\begin{matrix} \frac{1}{ϵ_{0}^{p - 1 - σ}} \int_{Ω} | \nabla w_{k + 1} |^{p} d x + \frac{γ}{ϵ_{0}^{q - 1 - σ}} \int_{Ω} | \nabla w_{k + 1} |^{q} d x = \int_{Ω} v^{σ} w_{k + 1} d x \end{matrix}$ Since $γ > 0$ we have, $\begin{aligned} ‖ \nabla w_{k + 1} ‖_{L^{p} (Ω)}^{p} & ⩽ ϵ_{0}^{p - 1 - σ} \int_{Ω} v^{σ} w_{k + 1} d x ⩽ \int_{{w_{k + 1} > 0}} v^{σ} w_{k + 1} d x \\ ⩽ \int_{{w_{k + 1} > 0}} (1 + v^{p - 1}) w_{k + 1} d x ⩽ \int_{{w_{k + 1} > 0}} (1 + A_{k}^{p - 1} w_{k}^{p - 1}) w_{k} d x \\ (5.3) & ⩽ {| {w_{k + 1} > 0} |}^{1 - \frac{1}{p}} U_{k}^{\frac{1}{p}} + 2^{(k + 1) (p - 1)} U_{k} \end{aligned}$ Now, as in (3.64) of Proposition 3.4 from [5], we get, $\begin{matrix} (5.4) & U_{k} ⩾ 2^{- (k + 1) p} | {w_{k + 1} > 0} | \end{matrix}$ Using the above estimate in (5.3) we get, $\begin{matrix} (5.5) & ‖ \nabla w_{k + 1} ‖_{L^{p} (Ω)}^{p} ⩽ c \cdot 2^{(k + 1) (p - 1)} U_{k} . \end{matrix}$ By Hölder’s inequality with the exponents $\frac{N}{N - p}$ and $\frac{N}{p}$ we have, $\begin{matrix} U_{k + 1} ⩽ ‖ w_{k + 1} ‖_{L^{\frac{N p}{N - p}} (Ω)}^{p} {| {w_{k + 1} > 0} |}^{\frac{p}{N}} ⩽ c_{0} {(2^{\frac{p^{2}}{N} + p})}^{k} U_{k}^{1 + \frac{p}{N}} \end{matrix}$ To derive the last inequality in the previous line we have used (5.4) and (5.5). Therefore for some $C > 1$ and $α = \frac{p}{N}$ , $\begin{matrix} (5.6) & U_{k + 1} ⩽ C^{k} U_{k}^{1 + α} \end{matrix}$ Now we shall define $δ : = C^{\frac{- 1}{α^{2}}}$ . Then following the induction argument as in [5], Proposition 3.4 we have ${lim}_{k \to 0} U_{k} = 0$ . We also know that ${lim}_{k \to \infty} U_{k} = \int_{Ω} {[{(v - 1)}^{+}]}^{p} d x$ , thus ${(v - 1)}^{+} = 0$ a.e. in Ω or $v ⩽ 1$ a.e. in Ω. Hence ${\tilde{Φ}}_{λ} ⩽ \frac{1}{ϵ_{0}}$ a.e. in Ω and since ${\tilde{Φ}}_{λ} > 0$ we have the required $L^{\infty}$ estimate. □

Proof of Theorem 1.4.

From Lemma 5.1, there exists a positive constant $M_{1}$ such that ${\tilde{Φ}}_{λ}^{σ} ⩽ M_{1}$ a.e. in Ω. Let w be the weak solution of $\begin{aligned} - Δ_{p} w - γ Δ_{q} w = M_{1} in Ω \\ w = 0 on \partial Ω \end{aligned}$ Therefore, by weak comparison principle, ${\tilde{Φ}}_{λ} (x) ⩽ w (x)$ , $\forall x \in Ω$ . Now following the ideas of the proof in Proposition 2.1 of [15], we get that for some positive constant $c_{2}$ , independent of γ, and for all $x \in Ω$ , ${\tilde{Φ}}_{λ} (x) ⩽ w (x) ⩽ c_{2} d (x)$ . i.e $\begin{aligned} (5.7) & Φ_{λ} (x) & ⩽ c_{2} λ^{\frac{1}{p - 1 - σ}} d (x) \end{aligned}$ Now we let $2 < q < p < \infty$ . In order to obtain a lower bound we resonate some of the ideas used in Lemma 3.1. Define $\begin{matrix} (5.8) & ξ_{λ} : = ϵ λ^{\frac{1}{p - 1 - σ}} (ϕ_{1} + ϕ_{1}^{α}) = ϵ λ^{r_{0}} (ϕ_{1} + ϕ_{1}^{α}) \end{matrix}$ where $1 < α < 2$ , $r_{0} = \frac{1}{p - 1 - σ}$ and $ϕ_{1}$ is the first eigen function of $- Δ$ such that $‖ ϕ_{1} ‖_{\infty} = 1$ and $ϵ > 0$ is to be chosen later.

Claim: $ξ_{λ}$ is a subsolution of (4.1) for some admissible range of λ.

Proceeding same as Lemma 3.1 we get, for $s = p, q$ , $\begin{matrix} (5.9) & - Δ_{s} ξ_{λ} = ϵ^{s - 1} λ^{r_{0} (s - 1)} ϕ_{1}^{α - 2} L_{s} (ϕ_{1}) \end{matrix}$ where $L_{s} (ϕ_{1}) = {(1 + α ϕ_{1}^{α - 1})}^{s - 1} ϕ_{1}^{2 - α} [- Δ_{s} ϕ_{1}] - α (α - 1) (s - 1) | \nabla ϕ_{1} |^{s} {(1 + α ϕ_{1}^{α - 1})}^{s - 2}$ and since $s > 2$ , there exist $δ > 0$ and $m > 0$ such that $L_{p} (ϕ_{1}) < - m$ and $L_{q} (ϕ_{1}) < 0$ in $Ω_{δ}$ . Therefore, $ξ_{λ}$ being non-negative by definition, we have, $\begin{matrix} (5.10) & - Δ_{p} ξ_{λ} - Δ_{q} ξ_{λ} ⩽ - \frac{ϵ^{s - 1} λ^{r_{0} (s - 1)} m}{ϕ_{1}^{2 - α}} ⩽ λ ξ_{λ}^{σ} in Ω_{δ} \end{matrix}$ On the other hand, in $Ω ∖ Ω_{δ}$ , there exists $μ > 0$ such that $ϕ_{1} > μ$ . Then since $Δ_{s} ϕ_{1} \in L^{\infty} (Ω)$ as $2 < q < p$ , from (5.9) we get, $\begin{matrix} (5.11) & - Δ_{s} ξ_{λ} ⩽ \frac{ϵ^{s - 1} λ^{r_{0} (s - 1)} c_{s}}{\overline{μ}} in Ω ∖ Ω_{δ} \end{matrix}$ for some positive constants $c_{s}$ (which depends on s only) and $\overline{μ} = μ^{2 - α}$ . Since in $Ω ∖ Ω_{δ}$ , $ϕ_{1} > μ$ , so $ξ_{λ} ⩾ ϵ λ^{r_{0}} μ$ and consequently $ϵ^{σ} λ^{r_{0} σ} μ^{σ} ⩽ ξ_{λ}^{σ}$ . First we choose $ϵ \in (0, 1)$ such that $\frac{2 c_{s} ϵ^{s - 1 - σ}}{\overline{μ} μ^{σ}} ⩽ 1$ for $s = p, q$ . Also note that $1 + r_{0} (σ - s + 1) ⩾ 0$ for $s = p, q$ . Therefore from (5.11) we have for $λ ⩾ 1$ , $\begin{aligned} - Δ_{s} ξ_{λ} ⩽ \frac{ϵ^{s - 1} λ^{r_{0} (s - 1)} c_{s}}{\overline{μ}} ⩽ \frac{λ}{2} (ϵ^{σ} λ^{r_{0} σ} μ^{σ}) ⩽ \frac{λ ξ_{λ}^{σ}}{2} \\ (5.12) & \Rightarrow - Δ_{p} ξ_{λ} - Δ_{q} ξ_{λ} ⩽ λ ξ_{λ}^{σ} in Ω ∖ Ω_{δ} \end{aligned}$ Thus (5.10) and (5.12) together establish that for $λ ⩾ 1$ , $ξ_{λ}$ is a positive subsolution of (4.1). We know that $Φ_{λ}$ is defined to be the positive solution of the same equation (4.1). By the comparison Lemma A.1, we have for $λ ⩾ 1$ , $ξ_{λ} ⩽ Φ_{λ}$ or $\begin{matrix} Φ_{λ} ⩾ ϵ λ^{r_{0}} (ϕ_{1} + ϕ_{1}^{α}) ⩾ ϵ λ^{\frac{1}{p - 1 - σ}} ϕ_{1} \end{matrix}$ Now since $ϕ_{1}$ is the first eigen function of $- Δ$ , so taking $λ_{0} = 1$ , there exists a positive constant $c_{1}$ such that for all $λ ⩾ λ_{0}$ , $\begin{matrix} (5.13) & c_{1} λ^{\frac{1}{p - 1 - σ}} d (x) ⩽ Φ_{λ} (x) for all x \in Ω . \end{matrix}$ Thus combining (5.7) and (5.13) we get the desired estimate. □

Footnotes

Appendix

The following result provides an extension of a comparison lemma from [36] to $(p, q)$ Laplacian.

Acknowledgement

R. Dhanya was supported by SERB MATRICS grant MTR/2022/000780 when this project was being carried out. Sarbani Pramanik was supported by Prime Minister’s Research Fellowship.

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