Higher order evolution inequalities involving Leray–Hardy potential singular on the boundary

Abstract

We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions.

Keywords

Higher order evolution inequalities Leray–Hardy potential half ball nonexistence result

1. Introduction

For $N ⩾ 2$ , let $\begin{matrix} B^{N} = {x = (x_{1}, x_{2}, \dots, x_{N}) \in R^{N} : | x | ⩽ 1} . \end{matrix}$ We denote by $B_{+}^{N}$ the half ball defined by $\begin{matrix} B_{+}^{N} = {x = (x_{1}, x_{2}, \dots, x_{N}) \in B^{N} : x_{N} ⩾ 0} . \end{matrix}$ The boundary of $B_{+}^{N}$ is denoted by $\partial B_{+}^{N} = Γ_{0}^{N} \cup Γ_{1}^{N}$ , where $\begin{matrix} Γ_{0}^{N} = {x \in B_{+}^{N} : x_{N} = 0} \end{matrix}$ and $\begin{matrix} Γ_{1}^{N} = {x \in B_{+}^{N} : x_{N} > 0, | x | = 1} . \end{matrix}$ For $μ \in R$ , we consider elliptic operators of the form $\begin{matrix} L_{μ} = - Δ + \frac{μ}{| x |^{2}}, x \in B_{+}^{N} ∖ {0}, \end{matrix}$ defined by the sum of a Laplace differential operator and a singular Leray–Hardy potential term. The Leray–Hardy potential is recognized as a key tool to study borderline situations or critical behavior in different contexts, as well as the study of existence of solutions to nonlinear problems.

In this paper, we establish the existence and nonexistence of weak solutions to higher order evolution inequalities of the form $\begin{matrix} (1.1) & \{\begin{array}{ll} \frac{\partial^{k} u}{\partial t^{k}} + L_{μ} u ⩾ | x |^{- a} | u |^{p} & in (0, \infty) \times B_{+}^{N} ∖ {0}, \\ u (t, x) ⩾ 0 & on (0, \infty) \times Γ_{0}^{N} ∖ {0}, \\ u (t, x) ⩾ f (x) & on (0, \infty) \times Γ_{1}^{N} \end{array} \end{matrix}$ and $\begin{matrix} (1.2) & \{\begin{array}{ll} \frac{\partial^{k} u}{\partial t^{k}} + L_{μ} u ⩾ | x |^{- a} I_{0}^{α} | u |^{p} & in (0, \infty) \times B_{+}^{N} ∖ {0}, \\ u (t, x) ⩾ 0 & on (0, \infty) \times Γ_{0}^{N} ∖ {0}, \\ u (t, x) ⩾ f (x) & on (0, \infty) \times Γ_{1}^{N}, \\ \frac{\partial^{i} u}{\partial t^{i}} (0, x) = u_{i} (x), 0 ⩽ i ⩽ k - 1 & in B_{+}^{N} ∖ {0}, \end{array} \end{matrix}$ where $k ⩾ 1$ is an integer, $μ > \frac{- N^{2}}{4}$ , $a \in R$ , $p > 1$ , $f \in L^{1} (Γ_{1}^{N})$ is a nontrivial function, $u_{i} \in L_{loc}^{1} (B_{+}^{N} ∖ {0})$ , $α > 0$ (namely, the memory term) and $\begin{matrix} I_{0}^{α} | u |^{p} (t, x) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - s)}^{α - 1} {| u (s, x) |}^{p} d s . \end{matrix}$ Here, $Γ (\cdot)$ denotes the Gamma function. Notice that the value $\frac{- N^{2}}{4}$ appears in the following Leray–Hardy inequality (see [9]) $\begin{matrix} (1.3) & \int_{R_{+}^{N}} | \nabla ϕ |^{2} d x - \frac{N^{2}}{4} \int_{R_{+}^{N}} \frac{ϕ^{2}}{| x |^{2}} d x ⩾ 0, ϕ \in C_{0}^{\infty} (R_{+}^{N}), \end{matrix}$ where $R_{+}^{N} = {x = (x_{1}, x_{2}, \dots, x_{N}) \in R^{N} : x_{N} > 0}$ .

We recall that the Leray–Hardy potential plays a significant role in the establishment of a Fujita exponent for nonlinear evolution problems with zero (non-zero) boundary data (namely, the nonexistence of solutions and related blow-up phenomena in a finite time). In fact, when $k = 1$ , $μ = a = 0$ and $u ⩾ 0$ , problem (1.1) (with equality instead of inequality) posed in the whole space, reduces to the following equation $\begin{matrix} (1.4) & \frac{\partial u}{\partial t} - Δ u = u^{p} in (0, \infty) \times R^{N} . \end{matrix}$ In his famous paper [13], Fujita proved that (1.4) admits a critical behavior in the following sense:

If $1 < p < 1 + \frac{2}{N}$ and $u (0, \cdot) > 0$ , then (1.4) does not have any global positive solution;

If $p > 1 + \frac{2}{N}$ and $u_{0}$ is smaller than a small Gaussian, then (1.4) admits global positive solutions.

We say that

p_{F} = 1 + \frac{2}{N}

is critical in the sense of Fujita. Later, it was shown that

p_{F}

belongs to the case (a) (see [15] for

N = 1, 2

and [21] for any

N ⩾ 1

). It is interesting to mention that

p_{F}

is still critical for the following parabolic inequality

\begin{matrix} (1.5) & \frac{\partial u}{\partial t} - Δ u ⩾ | u |^{p} in (0, \infty) \times R^{N} . \end{matrix}

For more details about (1.5), see e.g. [23]. When

k = 2

and

μ = a = 0

, problem (1.1) posed in the whole space, reduces to

\begin{matrix} (1.6) & \frac{\partial^{2} u}{\partial t^{2}} - Δ u ⩾ | u |^{p} in (0, \infty) \times R^{N} . \end{matrix}

Problem (1.6) was firstly studied by Kato [19]. Namely, he found another critical exponent

p_{K} = \frac{N + 1}{N - 1}

. Pohozaev and Véron [25] generalized Kato’s result and pointed out the sharpness of

p_{K}

for problem (1.6). When

a = 0

, problem (1.1) posed in the whole space, reduces to

\begin{matrix} (1.7) & \frac{\partial^{k} u}{\partial t^{k}} + L_{μ} u ⩾ | u |^{p} in (0, \infty) \times R^{N} . \end{matrix}

Problem (1.7) was studied by Hamidi and Laptev [10], in the case where

N ⩾ 3

and

μ ⩾ - {(\frac{N - 2}{2})}^{2}

. Adopting the notation

\begin{matrix} s^{*} = \frac{N - 2}{2} + \sqrt{μ + {(\frac{N - 2}{2})}^{2}} \end{matrix}

and

\begin{matrix} s_{*} = - \frac{N - 2}{2} + \sqrt{μ + {(\frac{N - 2}{2})}^{2}}, \end{matrix}

it was shown in [10] that under suitable initial conditions, if

$μ ⩾ 0$ and $1 < p ⩽ 1 + \frac{2}{s^{*} + 2 / k}$ ; or

$- {(\frac{N - 2}{2})}^{2} ⩽ μ < 0$ and $1 < p ⩽ 1 + \frac{2}{- s_{*} + 2 / k}$ ,

then (1.7) has no nontrivial global solution.

In [17], we considered problem (1.1) with $k = 2$ and $a = 0$ , posed in the exterior domain $R^{N} ∖ B_{1}$ , under an inhomogeneous Robin-type boundary condition, where $B_{1}$ is the unit ball. Namely, we investigated the existence and nonexistence of weak solutions to $\begin{matrix} (1.8) & \{\begin{array}{ll} \frac{\partial^{2} u}{\partial t^{2}} + L_{μ} u ⩾ | u |^{p} & in (0, \infty) \times R^{N} ∖ B_{1}, \\ α \frac{\partial u}{\partial ν} (t, x) + β u (t, x) ⩾ f (x) & on (0, \infty) \times \partial B_{1}, \end{array} \end{matrix}$ where $N ⩾ 2$ , $μ ⩾ - {(\frac{N - 2}{2})}^{2}$ , $α, β ⩾ 0$ and $(α, β) \neq (0, 0)$ . In the case $μ = - {(\frac{N - 2}{2})}^{2}$ , we proved that

if $N = 2$ and $\int_{\partial B_{1}} f (x) d σ > 0$ , then for all $p > 1$ , (1.8) admits no weak solution;

if $N ⩾ 3$ and $\int_{\partial B_{1}} f (x) d σ > 0$ , then for all $\begin{matrix} 1 < p < 1 + \frac{4}{N - 2}, \end{matrix}$ (1.8) admits no weak solution;

if $N ⩾ 3$ and $\begin{matrix} p > 1 + \frac{4}{N - 2}, \end{matrix}$ then (1.8) admits solutions for some $f > 0$ .

In the case

μ > - {(\frac{N - 2}{2})}^{2}

, we shown that

if $\int_{\partial B_{1}} f (x) d σ > 0$ , then for all $\begin{matrix} 1 < p < 1 + \frac{4}{N - 2 + 2 \sqrt{{(\frac{N - 2}{2})}^{2} + μ}}, \end{matrix}$ (1.8) admits no weak solution;

if $\begin{matrix} p > 1 + \frac{4}{N - 2 + 2 \sqrt{{(\frac{N - 2}{2})}^{2} + μ}}, \end{matrix}$ then (1.8) admits solutions for some $f > 0$ .

For additional results related to evolution equations and inequalities in exterior domains of

R^{N}

, see e.g. [16,18,26,29]. We also mention the works [22,28]. In [22], the authors consider certain variable Hardy spaces, in respect of a Schrödinger type operator. The potential involved in such operator is nonnegative and belongs to suitable reverse Hölder class. Hence the regularity properties of these functional spaces and of the operator are studied. In [28], the authors study certain parametric Schrödinger–Kirchhoff problems in

R^{N}

. The main equation is driven by a fractional magnetic operator and presents the combined effects of a critical Sobolev–Hardy nonlinearity and a nontrivial perturbation term. The existence of solutions is obtained by following a variational approach, for suitable values of the parameter.

The study of parabolic equations with Leray–Hardy potential in a bounded domain of $R^{N}$ has been considered by some authors. For instance, Abdellaoui et al. [3] considered parabolic equations of the form $\begin{matrix} (1.9) & \{\begin{array}{ll} \frac{\partial u}{\partial t} + L_{μ} u = u^{p} + f & in (0, \infty) \times Ω, u ⩾ 0, \\ u (t, x) = 0 & on (0, \infty) \times \partial Ω, \\ u (0, x) = u_{0} (x) & in Ω, \end{array} \end{matrix}$ where $Ω \subset R^{N}$ , $N ⩾ 3$ , is a bounded regular domain containing the origin, $p > 1$ , $μ < 0$ , and $u_{0} ⩾ 0$ , $f ⩾ 0$ belong to a suitable class of functions. Namely, it was shown the existence of a critical exponent $p_{+} (μ)$ such that for $p ⩾ p_{+} (μ)$ , there is no distributional solution to (1.9), while for $p < p_{+} (μ)$ , and under some additional conditions on the data, (1.9) admits solutions. Notice that in [3], the positivity of u is essential in the proof of the obtained results. Moreover, in this reference, the authors used the comparison principle for the heat equation, which cannot be applied for our problems when $k ⩾ 2$ . For other contributions related to the study of parabolic equations and inequalities with Leray–Hardy potential in a bounded domain, see e.g. [4,5,7,14,27]. For the study of elliptic equations involving Leray–Hardy potential, see e.g. [1,2,6,8,9,11,12].

Notice that in the limit case $α \to 0^{+}$ , problem (1.2) reduces to problem (1.1). Our aim for considering problem (1.2) is to study the influence of the parameter α on the critical behavior of problem (1.1).

A feature of our results is that we do not impose any restrictions on the sign of solutions. To the best of our knowledge, the study of higher order (in time) evolution inequalities with Leray–Hardy potential in the half ball has not previously considered in the literature, even in the parabolic case with nonnegative solutions.

Before stating our main results, we need to introduce the notions of weak solutions to the considered problems (namely, problems (1.1) and (1.2)). We consider the following sets $\begin{matrix} Ω = (0, \infty) \times B_{+}^{N} ∖ {0}, Γ_{0} = (0, \infty) \times Γ_{0}^{N} ∖ {0}, Γ_{1} = (0, \infty) \times Γ_{1}^{N}, \end{matrix}$ and hence $Γ_{i} \subset Ω$ , $i = 0, 1$ . The appropriate setting to introduce the definition of weak solution to (1.1) requires the following functional space (namely, the test function space Φ).

Definition 1.1.
A function $φ = φ (t, x)$ belongs to Φ, if the following conditions are satisfied:
$φ \in C_{t, x}^{k, 2} (Ω)$ , $φ ⩾ 0$ ;

$supp (φ) \subset \subset Ω$ ;

$φ |_{Γ_{i}} = 0$ , $i = 0, 1$ ;

$\frac{\partial φ}{\partial ν_{i}} |_{Γ_{i}} ⩽ 0$ , $i = 0, 1$ , where $ν_{i}$ is the outward unit normal vector on $Γ_{i}^{N}$ .

Hence, using standard integration by parts, we define weak solutions to (1.1) as follows.
Definition 1.2.
We say that $u \in L_{loc}^{p} (Ω)$ is a weak solution to (1.1) if $\begin{matrix} (1.10) & \int_{Ω} | x |^{- a} | u |^{p} φ d x d t - \int_{Γ_{1}} f (x) \frac{\partial φ}{\partial ν_{1}} d σ d t ⩽ {(- 1)}^{k} \int_{Ω} u \frac{\partial^{k} φ}{\partial t^{k}} d x d t + \int_{Ω} u L_{μ} φ d x d t \end{matrix}$ for every $φ \in Φ$ .

In order to define weak solutions to (1.2), we need to recall some basic properties on fractional calculus. For more details, see e.g. [20].

Let $T > 0$ be fixed, then for $α > 0$ and $g \in L^{1} ([0, T])$ , we consider the integral operators $\begin{matrix} I_{0}^{α} g (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - s)}^{α - 1} g (s) d s \end{matrix}$ and $\begin{matrix} I_{T}^{α} g (t) = \frac{1}{Γ (α)} \int_{t}^{T} {(s - t)}^{α - 1} g (s) d s . \end{matrix}$ The operators $I_{0}^{α}$ and $I_{T}^{α}$ are called respectively the left-sided and right-sided Riemann–Liouville fractional integrals of order α.

If $g, h \in C ([0, T])$ , then we have the following equality (see [20, Lemma 2.7]) $\begin{matrix} (1.11) & \int_{0}^{T} I_{0}^{α} g (t) h (t) d t = \int_{0}^{T} g (t) I_{T}^{α} h (t) d t . \end{matrix}$

Now, we consider the following sets $\begin{matrix} Ω_{T} = [0, T] \times B_{+}^{N} ∖ {0}, Γ_{0, T} = [0, T] \times Γ_{0}^{N} ∖ {0}, Γ_{1, T} = [0, T] \times Γ_{1}^{N}, \end{matrix}$ and introduce a second functional space (namely, the test function space $Ψ_{T}$ ).
Definition 1.3.
A function $ψ = ψ (t, x)$ belongs to $Ψ_{T}$ , if the following conditions are satisfied:
$ψ \in C_{t, x}^{k, 2} (Ω_{T})$ , $ψ ⩾ 0$ ;

$supp (ψ) \subset \subset Ω_{T}$ ;

$ψ |_{Γ_{i, T}} = 0$ , $i = 0, 1$ ;

$\frac{\partial ψ}{\partial ν_{i}} |_{Γ_{i, T}} ⩽ 0$ , $i = 0, 1$ ;

$\frac{\partial^{i} ψ}{\partial t^{i}} (T, \cdot) = 0$ , $0 ⩽ i ⩽ k - 1$ .

Using the equality (1.11), we can define weak solution to (1.2) as follows.
Definition 1.4.
We say that $u \in L_{loc}^{p} ([0, \infty) \times B_{+}^{N} ∖ {0})$ is a weak solution to (1.2) if $\begin{matrix} (1.12) & \begin{aligned} \int_{Ω_{T}} | x |^{- a} | u |^{p} I_{T}^{α} ψ d x d t - \int_{Γ_{1, T}} f (x) \frac{\partial ψ}{\partial ν_{1}} d σ d t + \sum_{i = 0}^{k - 1} {(- 1)}^{i} \int_{B_{1}^{c}} u_{k - i - 1} (x) \frac{\partial^{i} ψ}{\partial t^{i}} (0, x) d x \\ ⩽ {(- 1)}^{k} \int_{Ω_{T}} u \frac{\partial^{k} ψ}{\partial t^{k}} d x d t + \int_{Ω_{T}} u L_{μ} ψ d x d t \end{aligned} \end{matrix}$ for all $T > 0$ and $ψ \in Ψ_{T}$ .

For $μ > \frac{- N^{2}}{4}$ , let us introduce the parameter $\begin{matrix} (1.13) & τ = - \frac{N}{2} + \sqrt{μ + {(\frac{N}{2})}^{2}}, \end{matrix}$ and, for all $f \in L^{1} (Γ_{1}^{N})$ , we set $\begin{matrix} (1.14) & I_{f} = \int_{Γ_{1}^{N}} f (x) x_{N} d σ . \end{matrix}$ We also denote by $L^{1, +} (Γ_{1}^{N})$ the functional space given by $\begin{matrix} L^{1, +} (Γ_{1}^{N}) = {f \in L^{1} (Γ_{1}^{N}) : I_{f} > 0} . \end{matrix}$

Our main results for problem (1.1) are stated in the following theorem.
Theorem 1.5.
Let $k ⩾ 1$ , $N ⩾ 2$ , $μ > \frac{- N^{2}}{4}$ , $a \in R$ and $p > 1$ . We distinguish the following cases:
Let $f \in L^{1, +} (Γ_{1}^{N})$ . If $\begin{matrix} (1.15) & (τ + 1) p < a + τ - 1, \end{matrix}$ then ( 1.1 ) admits no weak solution.

If $\begin{matrix} (1.16) & (τ + 1) p > a + τ - 1, \end{matrix}$ then ( 1.1 ) admits nonnegative (stationary) solutions for some $f \in L^{1, +} (Γ_{1}^{N})$ .

Remark 1.6.
Theorem 1.5 leaves open the issue of existence and nonexistence in the critical case: $\begin{matrix} (τ + 1) p = a + τ - 1 . \end{matrix}$
Remark 1.7.
We point out the following facts:
Let $μ = 1 - N$ . In this case, $τ = - 1$ and (1.15) reduces to $a > 2$ .

Let $μ > 1 - N$ . In this case, $τ + 1 > 0$ and (1.15) reduces to $\begin{matrix} a > 2, 1 < p < 1 + \frac{a - 2}{τ + 1} . \end{matrix}$

Let $\frac{- N^{2}}{4} < μ < 1 - N$ and $N ⩾ 3$ . In this case, $τ + 1 < 0$ and (1.15) reduces to $\begin{matrix} p > max {1, 1 + \frac{a - 2}{τ + 1}} = \{\begin{array}{ll} 1 & if a ⩾ 2, \\ 1 + \frac{a - 2}{τ + 1} & if a < 2 . \end{array} \end{matrix}$

Condition (1.15) is independent on k.

Clearly, Theorem 1.5 yields existence and nonexistence results for the corresponding elliptic problem $\begin{array}{r} (1.17) & \{\begin{array}{ll} L_{μ} u ⩾ | x |^{- a} | u |^{p} & in B_{+}^{N} ∖ {0}, \\ u (x) ⩾ 0 & on Γ_{0}^{N} ∖ {0}, \\ u (x) ⩾ f (x) & on Γ_{1}^{N} . \end{array} \end{array}$
Corollary 1.8.
Let $N ⩾ 2$ , $μ > \frac{- N^{2}}{4}$ , $a \in R$ and $p > 1$ . We distinguish the following cases:
Let $f \in L^{1, +} (Γ_{1}^{N})$ . If ( 1.15 ) holds, then ( 1.17 ) admits no weak solution.

If ( 1.16 ) holds, then ( 1.17 ) admits nonnegative solutions for some $f \in L^{1, +} (Γ_{1}^{N})$ .

Our main result for problem (1.2) is stated in the following theorem.
Theorem 1.9.
Let $k ⩾ 1$ , $N ⩾ 2$ , $μ > \frac{- N^{2}}{4}$ , $a \in R$ , $p > 1$ and $u_{i} \in L_{loc}^{1} (B_{+}^{N} ∖ {0})$ , $u_{i} ⩾ 0$ , for all $0 ⩽ i ⩽ k - 1$ . If $f \in L^{1, +} (Γ_{1}^{N})$ , then for all $α > 0$ , ( 1.2 ) admits no weak solution.
Remark 1.10.
We point out the following facts:
By Theorem 1.9, we deduce that for any $α > 0$ , the Fujita critical exponent of (1.2) is equal to ∞.

By Theorem 1.5 (see also Remark 1.7), we deduce that, if $\begin{matrix} μ = 1 - N, a > 2; or \frac{- N^{2}}{4} < μ < 1 - N, a ⩾ 2, \end{matrix}$ then the memory term (namely, the parameter $α > 0$ ) has no effect on the critical behavior of problem (1.1).

By Theorem 1.5 (see also Remark 1.7), we deduce that, if $\begin{matrix} μ > 1 - N, a > 2, \end{matrix}$ then the Fujita critical exponent of problem (1.2) jumps from $1 + \frac{a - 2}{τ + 1}$ (the Fujita critical exponent of problem (1.1)) to ∞.

The proofs of Theorem 1.5 (I) and Theorem 1.9 rely on nonlinear capacity estimates specifically adapted to the domain $B_{+}^{N}$ , the operator $L_{μ}$ and the considered boundary conditions. The existence result given by Theorem 1.5 (II) is established by the construction of explicit solutions.

The rest of the paper is organized as follows. In Section 2, we first construct two families of test functions belonging respectively to the test function spaces Φ and $Ψ_{T}$ . Next, we establish some useful integral estimates involving the constructed test functions. Section 3 is devoted to the proofs of Theorems 1.5 and 1.9. A short Section 4 concludes the paper.
2. Preliminaries

Throughout this paper, the letter C denotes always a generic positive constant whose value is unimportant and may vary at different occurrences.

Let $k ⩾ 1$ , $N ⩾ 2$ , $a \in R$ , $p > 1$ , $α > 0$ and $μ > \frac{- N^{2}}{4}$ . We introduce the nonnegative function ϑ defined by $\begin{matrix} (2.1) & ϑ (x) = x_{N} | x |^{τ} (| x |^{- N - 2 τ} - 1), x \in B_{+}^{N} ∖ {0}, \end{matrix}$ where τ is given by (1.13). It is not difficult to show that ϑ is a solution to $\begin{array}{r} (2.2) & \{\begin{array}{ll} L_{μ} ϑ = 0 & in B_{+}^{N} ∖ {0}, \\ ϑ = 0 & on Γ_{0}^{N} ∖ {0} \cup Γ_{1}^{N} . \end{array} \end{array}$ In the sequel, we need two cut-off functions $η, ξ \in C^{\infty} ([0, \infty))$ satisfying the following conditions $\begin{matrix} (2.3) & η ⩾ 0, supp (η) \subset \subset (0, 1) \end{matrix}$ and $\begin{matrix} (2.4) & 0 ⩽ ξ ⩽ 1, ξ (s) = 0 if 0 ⩽ s ⩽ \frac{1}{2}, ξ (s) = 1 if s ⩾ 1 . \end{matrix}$ Moreover, for $T > 0$ and sufficiently large ℓ, we introduce the functions $\begin{array}{c} (2.5) & η_{T} (t) = η {(\frac{t}{T})}^{ℓ}, t ⩾ 0, \\ (2.6) & γ_{T} (t) = T^{- ℓ} {(T - t)}^{ℓ}, 0 ⩽ t ⩽ T . \end{array}$ For $R > 1$ , we define the following function $\begin{matrix} (2.7) & ξ_{R} (x) = ϑ (x) ξ {(R | x |)}^{ℓ}, x \in B_{+}^{N} ∖ {0}, \end{matrix}$ that is, $\begin{matrix} (2.8) & ξ_{R} (x) = \{\begin{array}{ll} 0 & if x_{N} ⩾ 0, 0 < | x | ⩽ \frac{1}{2 R}, \\ ϑ (x) ξ {(R | x |)}^{ℓ} & if x_{N} ⩾ 0, \frac{1}{2 R} ⩽ | x | ⩽ \frac{1}{R}, \\ ϑ (x) & if x_{N} ⩾ 0, \frac{1}{R} ⩽ | x | ⩽ 1 . \end{array} \end{matrix}$ On this basis, we consider two test functions of the form $\begin{matrix} (2.9) & φ (t, x) = η_{T} (t) ξ_{R} (x), (t, x) \in Ω \end{matrix}$ and $\begin{matrix} (2.10) & ψ (t, x) = γ_{T} (t) ξ_{R} (x), (t, x) \in Ω_{T} . \end{matrix}$

In the following lemma, we show that $φ \in Φ$ .

Lemma 2.1.
The function φ defined by ( 2.9 ) belongs to the test function space Φ.
Proof.
Clearly, for sufficiently large ℓ, we have that $η_{T} \in C^{k} ((0, \infty))$ and $ξ_{R} \in C^{2} (B_{+}^{N} ∖ {0})$ , which imply that $φ \in C_{t, x}^{k, 2} (Ω)$ . Moreover, combining the information in formulas (2.3), (2.4), (2.5) and (2.8), we deduce that $\begin{matrix} supp (φ) = [c_{1} T, c_{2} T] \times {x \in R^{N} : x_{N} ⩾ 0, \frac{1}{2 R} ⩽ | x | ⩽ 1}, \end{matrix}$ for some $0 < c_{1} < c_{2} < 1$ . On the other hand, by (2.2) we note that $\begin{matrix} φ |_{Γ_{i}} = 0, i = 0, 1 . \end{matrix}$ So, we have just to show that $\begin{matrix} (2.11) & \frac{\partial φ}{\partial ν_{i}} (t, x) ⩽ 0, (t, x) \in Γ_{i}, i = 0, 1 . \end{matrix}$ We first prove that the inequality in (2.11) is true in the case where $i = 0$ . In fact, by (2.1) and (2.7), we get $\begin{aligned} \frac{\partial ξ_{R}}{\partial x_{N}} (x) = & \frac{\partial ϑ}{\partial x_{N}} (x) ξ {(R | x |)}^{ℓ} + ϑ (x) \frac{\partial}{\partial x_{N}} [ξ {(R | x |)}^{ℓ}] \\ = & | x |^{τ} (| x |^{- N - 2 τ} - 1) ξ {(R | x |)}^{ℓ} + x_{N} \frac{\partial}{\partial x_{N}} (| x |^{τ} (| x |^{- N - 2 τ} - 1)) ξ {(R | x |)}^{ℓ} \\ + x_{N} | x |^{τ} (| x |^{- N - 2 τ} - 1) \frac{\partial}{\partial x_{N}} [ξ {(R | x |)}^{ℓ}] . \end{aligned}$ Hence, for $x \in Γ_{0}^{N} ∖ {0}$ , we obtain that $\begin{matrix} \frac{\partial ξ_{R}}{\partial ν_{0}} (x) = - \frac{\partial ξ_{R}}{\partial x_{N}} (x) = - | x |^{τ} (| x |^{- N - 2 τ} - 1) ξ {(R | x |)}^{ℓ}, \end{matrix}$ which implies, by using (2.3) together with (2.5) and (2.9), that $\begin{aligned} \frac{\partial φ}{\partial ν_{0}} (t, x) & = η_{T} (t) \frac{\partial ξ_{R}}{\partial ν_{0}} (x) \\ = - | x |^{τ} (| x |^{- N - 2 τ} - 1) ξ {(R | x |)}^{ℓ} η_{T} (t) ⩽ 0 \end{aligned}$ for all $(t, x) \in Γ_{0}$ . This proves the inequality in (2.11) in the case where $i = 0$ . On the other hand, by (2.1), (2.4) and (2.8), for sufficiently small $ε > 0$ and $x \in B_{+}^{N} ∖ {0}$ with $1 - ε < | x | < 1$ , we have $\begin{aligned} \nabla ξ_{R} (x) & = \nabla ϑ (x) \\ = | x |^{τ} (| x |^{- N - 2 τ} - 1) \nabla x_{N} + x_{N} ((- N - τ) | x |^{- N - τ - 1} - τ | x |^{τ - 1}) \frac{x}{| x |}, \end{aligned}$ which implies that for all $x \in Γ_{1}^{N}$ , the following is the case $\begin{aligned} \frac{\partial ξ_{R}}{\partial ν_{1}} (x) & = \nabla ξ_{R} (x) \cdot \frac{x}{| x |} \\ = - (2 τ + N) x_{N}, \end{aligned}$ where “·” denotes as usual the inner product in $R^{N}$ . Hence, involving the test function φ defined in (2.9), we deduce that $\begin{matrix} (2.12) & \frac{\partial φ}{\partial ν_{1}} (t, x) = - (2 τ + N) x_{N} η_{T} (t) ⩽ 0 \end{matrix}$ for all $(t, x) \in Γ_{1}$ , which proves the inequality in (2.11) also in the case where $i = 1$ . □

Next lemma is devoted to characterize the test function ψ defined by (2.10) in the sense that we show that $ψ \in Ψ_{T}$ .
Lemma 2.2.
For all $T > 0$ , the function ψ defined by ( 2.10 ) belongs to the test function space $Ψ_{T}$ .
Proof.
For sufficiently large values of ℓ, we have that $γ_{T} \in C^{k} ([0, T])$ and $ξ_{R} \in C^{2} (B_{+}^{N} ∖ {0})$ , which imply that $ψ \in C_{t, x}^{k, 2} (Ω_{T})$ . Moreover, we have $\begin{matrix} supp (ψ) = [0, T] \times {x \in R^{N} : x_{N} ⩾ 0, \frac{1}{2 R} ⩽ | x | ⩽ 1} \end{matrix}$ and by (2.2) we conclude that $\begin{matrix} ψ |_{Γ_{i}} = 0, i = 0, 1 . \end{matrix}$ On the other hand, for all $0 ⩽ i ⩽ k - 1$ , we have $\frac{d^{i} γ_{T}}{d t^{i}} (T) = 0$ , which implies that $\frac{\partial^{i} ψ}{\partial t^{i}} (T, \cdot) = 0$ . Finally, proceeding as in the proof of Lemma 2.1, we obtain the conditions $\begin{matrix} \frac{\partial ψ}{\partial ν_{0}} (t, x) = - | x |^{τ} (| x |^{- N - 2 τ} - 1) ξ {(R | x |)}^{ℓ} γ_{T} (t) ⩽ 0, (t, x) \in Γ_{0} \end{matrix}$ and $\begin{matrix} (2.13) & \frac{\partial ψ}{\partial ν_{1}} (t, x) = - (2 τ + N) x_{N} γ_{T} (t) ⩽ 0, (t, x) \in Γ_{1} . \end{matrix}$ □

In the following, we shall give some integral estimates involving the functions φ and ψ defined by (2.9) and (2.10), respectively.
2.1. Estimates involving φ

The first estimate follows immediately from (2.3) and (2.5), and hence we do not give a proof of the lemma.

Lemma 2.3.
For all $T > 0$ and sufficiently large ℓ, there holds $\begin{matrix} \int_{supp (η_{T})} η_{T} {(t)}^{\frac{- 1}{p - 1}} {| \frac{d^{k} η_{T} (t)}{d t^{k}} |}^{\frac{p}{p - 1}} d t ⩽ C T^{1 - \frac{k p}{p - 1}} . \end{matrix}$

The second estimate mainly incorporates the effects of the truncation function $ξ_{R}$ in the integral term.
Lemma 2.4.
For sufficiently large ℓ and R, there holds $\begin{matrix} (2.14) & \int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} (x) d x ⩽ C (ln R + R^{τ - 1 - \frac{a}{p - 1}}) . \end{matrix}$
Proof.
Starting from the definition of the function ϑ in (2.1) and involving the condition (2.4) and the function $ξ_{R}$ defined by (2.7), we obtain the following chain of inequalities $\begin{aligned} \int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} (x) d x \\ = \int_{\frac{1}{2 R} < | x | < 1, x_{N} > 0} | x |^{\frac{a}{p - 1}} x_{N} | x |^{τ} (| x |^{- N - 2 τ} - 1) ξ {(R | x |)}^{ℓ} d x \\ ⩽ \int_{\frac{1}{2 R} < | x | < 1, x_{N} > 0} x_{N} | x |^{τ + \frac{a}{p - 1}} (| x |^{- N - 2 τ} - 1) d x \\ ⩽ \int_{\frac{1}{2 R} < | x | < 1} | x |^{- τ + 1 - N + \frac{a}{p - 1}} d x \\ = C \int_{\frac{1}{2 R}}^{1} r^{\frac{a}{p - 1} - τ} d r \\ = \{\begin{array}{ll} C & if (τ - 1) (p - 1) < a, \\ C ln R & if (τ - 1) (p - 1) = a, \\ C R^{τ - 1 - \frac{a}{p - 1}} & if (τ - 1) (p - 1) > a . \end{array} \end{aligned}$ It follows that the estimate (2.14) is established. □

Next estimate relies on the test function φ defined by (2.9) and its k-th derivative in time variable.
Lemma 2.5.
For all $T > 0$ and sufficiently large ℓ and R, there holds $\begin{matrix} (2.15) & \int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} {| \frac{\partial^{k} φ}{\partial t^{k}} |}^{\frac{p}{p - 1}} d x d t ⩽ C T^{1 - \frac{k p}{p - 1}} (ln R + R^{τ - 1 - \frac{a}{p - 1}}), \end{matrix}$ where φ is the function defined by ( 2.9 ).
Proof.
By the definition of φ in (2.9), we get that $\begin{aligned} \int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} {| \frac{\partial^{k} φ}{\partial t^{k}} |}^{\frac{p}{p - 1}} d x d t \\ = (\int_{supp (η_{T})} η_{T} {(t)}^{\frac{- 1}{p - 1}} {| \frac{d^{k} η_{T} (t)}{d t^{k}} |}^{\frac{p}{p - 1}} d t) (\int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} (x) d x) . \end{aligned}$ Hence, we can use Lemmas 2.3 and 2.4 to obtain the estimate (2.15). □

Now, we consider the elliptic operator $L_{μ}$ and construct the following estimate.
Lemma 2.6.
For sufficiently large ℓ and R, there holds $\begin{matrix} (2.16) & \int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} {(x)}^{\frac{- 1}{p - 1}} {| L_{μ} ξ_{R} (x) |}^{\frac{p}{p - 1}} d x ⩽ C R^{\frac{(τ + 1) p + 1 - a - τ}{p - 1}} . \end{matrix}$
Proof.
By the definition of $ξ_{R}$ (recall (2.7)), for $x \in supp (ξ_{R})$ we obtain the representation formula $\begin{aligned} - L_{μ} ξ_{R} (x) = & Δ (ϑ (x) ξ {(R | x |)}^{ℓ}) - \frac{μ}{| x |^{2}} ϑ (x) ξ {(R | x |)}^{ℓ} \\ = & ξ {(R | x |)}^{ℓ} Δ ϑ (x) + ϑ (x) Δ [ξ {(R | x |)}^{ℓ}] + 2 \nabla ϑ (x) \cdot \nabla [ξ {(R | x |)}^{ℓ}] \\ - \frac{μ}{| x |^{2}} ϑ (x) ξ {(R | x |)}^{ℓ} \\ = & - ξ {(R | x |)}^{ℓ} L_{μ} ϑ (x) + ϑ (x) Δ [ξ {(R | x |)}^{ℓ}] + 2 \nabla ϑ (x) \cdot \nabla [ξ {(R | x |)}^{ℓ}], \end{aligned}$ which implies by (2.2) that $\begin{matrix} (2.17) & - L_{μ} ξ_{R} (x) = ϑ (x) Δ [ξ {(R | x |)}^{ℓ}] + 2 \nabla ϑ (x) \cdot \nabla [ξ {(R | x |)}^{ℓ}] . \end{matrix}$ Hence, by condition (2.4), we deduce that $\begin{matrix} (2.18) & \begin{aligned} \int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} {(x)}^{\frac{- 1}{p - 1}} {| L_{μ} ξ_{R} (x) |}^{\frac{p}{p - 1}} d x \\ = \int_{\frac{1}{2 R} < | x | < \frac{1}{R}, x_{N} > 0} | x |^{\frac{a}{p - 1}} ξ_{R} {(x)}^{\frac{- 1}{p - 1}} {| L_{μ} ξ_{R} (x) |}^{\frac{p}{p - 1}} d x . \end{aligned} \end{matrix}$ On the other hand, the same condition (2.4), for $\frac{1}{2 R} < | x | < \frac{1}{R}$ , $x_{N} > 0$ , gives us $\begin{matrix} | Δ [ξ {(R | x |)}^{ℓ}] | ⩽ C R^{2} ξ {(R | x |)}^{ℓ - 2}, \end{matrix}$ which implies by (2.1) that $\begin{aligned} ϑ (x) | Δ [ξ {(R | x |)}^{ℓ}] | & ⩽ C R^{2} x_{N} | x |^{τ} (| x |^{- N - 2 τ} - 1) ξ {(R | x |)}^{ℓ - 2} \\ (2.19) & ⩽ C R^{N + τ + 2} x_{N} ξ {(R | x |)}^{ℓ - 2} . \end{aligned}$ Moreover, we have $\begin{aligned} \nabla ϑ (x) \cdot \nabla [ξ {(R | x |)}^{ℓ}] \\ = [| x |^{τ} (| x |^{- N - 2 τ} - 1) e_{N} + x_{N} ((- N - τ) | x |^{- N - τ - 1} - τ | x |^{τ - 1}) \frac{x}{| x |}] \\ \cdot 2 ℓ R ξ {(R | x |)}^{ℓ - 1} ξ^{'} (R | x |) \frac{x}{| x |} \\ = 2 ℓ R ξ {(R | x |)}^{ℓ - 1} ξ^{'} (R | x |) x_{N} | x |^{τ - 1} [(1 - N - τ) | x |^{- N - 2 τ} - 1 - τ], \end{aligned}$ where $e_{N} = (0, \dots, 0, 1) \in R^{N}$ , which implies (since $0 ⩽ ξ ⩽ 1$ ) that $\begin{matrix} (2.20) & | \nabla ϑ (x) \cdot \nabla [ξ {(R | x |)}^{ℓ}] | ⩽ C R^{N + τ + 2} x_{N} ξ {(R | x |)}^{ℓ - 2} . \end{matrix}$ Hence, in view of (2.17) and the obtained inequalities in (2.19) and (2.20), we deduce that $\begin{matrix} (2.21) & | L_{μ} ξ_{R} (x) | ⩽ C R^{N + τ + 2} x_{N} ξ {(R | x |)}^{ℓ - 2}, \frac{1}{2 R} < | x | < \frac{1}{R}, x_{N} > 0 . \end{matrix}$ Making use of (2.1), (2.4), (2.7), (2.18) and (2.21), we obtain the chain of inequalities $\begin{aligned} \int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} {(x)}^{\frac{- 1}{p - 1}} {| L_{μ} ξ_{R} (x) |}^{\frac{p}{p - 1}} d x \\ ⩽ C R^{\frac{(N + τ + 2) p}{p - 1}} \int_{\frac{1}{2 R} < | x | < \frac{1}{R}, x_{N} > 0} x_{N} | x |^{\frac{a - τ}{p - 1}} {(| x |^{- N - 2 τ} - 1)}^{\frac{- 1}{p - 1}} ξ {(R | x |)}^{ℓ - \frac{2 p}{p - 1}} d x \\ ⩽ C R^{\frac{(N + τ + 2) p}{p - 1}} \int_{\frac{1}{2 R} < | x | < \frac{1}{R}} | x |^{1 + \frac{a + τ + N}{p - 1}} d x \\ ⩽ C R^{\frac{(N + τ + 2) p}{p - 1}} R^{- 1 - \frac{a + τ + N}{p - 1} - N}, \end{aligned}$ which leads to the estimate (2.16). □

The following is the last estimate in this sub-section. Lemma 2.7.
For all $T > 0$ and sufficiently large R and ℓ, there holds $\begin{matrix} (2.22) & \int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} | L_{μ} φ |^{\frac{p}{p - 1}} d x d t ⩽ C T R^{\frac{(τ + 1) p + 1 - a - τ}{p - 1}}, \end{matrix}$ where φ is the function defined by ( 2.9 ).
Proof.
Starting from the definition of the function φ given by (2.9), we obtain that $\begin{matrix} (2.23) & \begin{aligned} \int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} | L_{μ} φ |^{\frac{p}{p - 1}} d x d t \\ = (\int_{supp (η_{T})} η_{T} (t) d t) (\int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} {(x)}^{\frac{- 1}{p - 1}} {| L_{μ} ξ_{R} (x) |}^{\frac{p}{p - 1}} d x) . \end{aligned} \end{matrix}$ On the other hand, using condition (2.3) together with the definition of the function $η_{T}$ given by (2.5), we deduce that $\begin{aligned} \int_{supp (η_{T})} η_{T} (t) d t & = \int_{0}^{T} η {(\frac{t}{T})}^{ℓ} d t \\ (2.24) & = T \int_{0}^{1} η {(s)}^{ℓ} d s . \end{aligned}$ Involving Lemma 2.6 and using formulas (2.23) and (2.24), we obtain the estimate (2.22). □

2.2. Estimates involving ψ

The following result follows from elementary calculations, and hence it is stated without a proof.

Lemma 2.8.
Let $T > 0$ . For sufficiently large ℓ, there holds $\begin{matrix} I_{T}^{α} γ_{T} (t) = \frac{Γ (ℓ + 1)}{Γ (α + ℓ + 1)} T^{- ℓ} {(T - t)}^{ℓ + α}, 0 ⩽ t ⩽ T, \end{matrix}$ where $γ_{T}$ is the function defined by ( 2.6 ).

Based on the right-sided Riemann–Liouville fractional integral of order α, namely $I_{T}^{α}$ , we establish the next estimate.
Lemma 2.9.
For all $T > 0$ and sufficiently large R and ℓ, there holds $\begin{matrix} (2.25) & \int_{supp (ψ)} | x |^{\frac{a}{p - 1}} {| I_{T}^{α} ψ |}^{\frac{- 1}{p - 1}} {| \frac{\partial^{k} ψ}{\partial t^{k}} |}^{\frac{p}{p - 1}} d x d t ⩽ C T^{1 - \frac{k p + α}{p - 1}} (ln R + R^{τ - 1 - \frac{a}{p - 1}}), \end{matrix}$ where ψ is the function defined by ( 2.10 ).
Proof.
For the function ψ given by (2.10), we obtain $\begin{matrix} (2.26) & \begin{aligned} \int_{supp (ψ)} | x |^{\frac{a}{p - 1}} {| I_{T}^{α} ψ |}^{\frac{- 1}{p - 1}} {| \frac{\partial^{k} ψ}{\partial t^{k}} |}^{\frac{p}{p - 1}} d x d t \\ = (\int_{0}^{T} {| I_{T}^{α} γ_{T} (t) |}^{\frac{- 1}{p - 1}} {| \frac{d^{k} γ_{T} (t)}{d t^{k}} |}^{\frac{p}{p - 1}} d t) (\int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} (x) d x) . \end{aligned} \end{matrix}$ On the other hand, we have $\begin{matrix} | \frac{d^{k} γ_{T} (t)}{d t^{k}} | = C T^{- ℓ} {(T - t)}^{ℓ - k}, 0 ⩽ t ⩽ T, \end{matrix}$ which implies by Lemma 2.8 that $\begin{matrix} {| I_{T}^{α} γ_{T} (t) |}^{\frac{- 1}{p - 1}} {| \frac{d^{k} γ_{T} (t)}{d t^{k}} |}^{\frac{p}{p - 1}} ⩽ C T^{- ℓ} {(T - t)}^{ℓ - \frac{k p + α}{p - 1}}, 0 ⩽ t ⩽ T . \end{matrix}$ Integrating over $(0, T)$ , we obtain $\begin{matrix} (2.27) & \int_{0}^{T} {| I_{T}^{α} γ_{T} (t) |}^{\frac{- 1}{p - 1}} {| \frac{d^{k} γ_{T} (t)}{d t^{k}} |}^{\frac{p}{p - 1}} d t ⩽ C T^{1 - \frac{k p + α}{p - 1}} . \end{matrix}$ Thus, making use of Lemma 2.4, (2.26) and (2.27), we conclude the estimate (2.25). □

The last lemma involves the elliptic operator $L_{μ}$ in the integrand. Lemma 2.10.
For all $T > 0$ and sufficiently large R and ℓ, there holds $\begin{matrix} (2.28) & \int_{supp (ψ)} | x |^{\frac{a}{p - 1}} {| I_{T}^{α} ψ |}^{\frac{- 1}{p - 1}} | L_{μ} ψ |^{\frac{p}{p - 1}} d x d t ⩽ C T^{1 - \frac{α}{p - 1}} R^{\frac{(τ + 1) p + 1 - a - τ}{p - 1}}, \end{matrix}$ where ψ is the function defined by ( 2.10 ).
Proof.
By (2.10), we get $\begin{matrix} (2.29) & \begin{aligned} \int_{supp (ψ)} | x |^{\frac{a}{p - 1}} {| I_{T}^{α} ψ |}^{\frac{- 1}{p - 1}} | L_{μ} ψ |^{\frac{p}{p - 1}} d x d t \\ = (\int_{0}^{T} {| I_{T}^{α} γ_{T} (t) |}^{\frac{- 1}{p - 1}} γ_{T} {(t)}^{\frac{p}{p - 1}} (t) d t) (\int_{supp (ξ_{R})} | x |^{\frac{a}{p - 1}} ξ_{R} {(x)}^{\frac{- 1}{p - 1}} {| L_{μ} ξ_{R} (x) |}^{\frac{p}{p - 1}} d x) . \end{aligned} \end{matrix}$ On the other hand, taking $k = 0$ in (2.27), we obtain $\begin{matrix} (2.30) & \int_{0}^{T} {| I_{T}^{α} γ_{T} (t) |}^{\frac{- 1}{p - 1}} γ_{T} {(t)}^{\frac{p}{p - 1}} (t) d t ⩽ C T^{1 - \frac{α}{p - 1}} . \end{matrix}$ Thus, making use of Lemma 2.6, (2.29) and (2.30), we establish (2.28). □

3. Proof of the main results

The main strategy of proof is based on the test functions method, which leads to self-contained and easy-to-follow proofs. We also develop the proofs by contradiction, using the estimates in the previous section.

Proof of Theorem 1.5.
(I) We argue by contradiction by supposing that $u \in L_{loc}^{p} (Ω)$ is a weak solution to (1.1). Thus, making use of the formula (1.10) and Lemma 2.1, for sufficiently large ℓ, T and R, we deduce the following inequality $\begin{matrix} (3.1) & \int_{Ω} | x |^{- a} | u |^{p} φ d x d t - \int_{Γ_{1}} f (x) \frac{\partial φ}{\partial ν_{1}} d σ d t ⩽ \int_{Ω} | u | | \frac{\partial^{k} φ}{\partial t^{k}} | d x d t + \int_{Ω} | u | | L_{μ} φ | d x d t, \end{matrix}$ where $φ \in Φ$ is the function defined by (2.9). On the other hand, using Young’s inequality, we get two more inequalities in the following form $\begin{matrix} (3.2) & \begin{aligned} \int_{Ω} | u | | \frac{\partial^{k} φ}{\partial t^{k}} | d x d t \\ ⩽ \frac{1}{2} \int_{Ω} | x |^{- a} | u |^{p} φ d x d t + C \int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} {| \frac{\partial^{k} φ}{\partial t^{k}} |}^{\frac{p}{p - 1}} d x d t \end{aligned} \end{matrix}$ and $\begin{matrix} (3.3) & \begin{aligned} \int_{Ω} | u | | L_{μ} φ | d x d t \\ ⩽ \frac{1}{2} \int_{Ω} | x |^{- a} | u |^{p} φ d x d t + C \int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} | L_{μ} φ |^{\frac{p}{p - 1}} d x d t . \end{aligned} \end{matrix}$ Then, combining the above inequalities (3.1), (3.2) and (3.3), we obtain the following inequality $\begin{matrix} (3.4) & \begin{aligned} - \int_{Γ_{1}} f (x) \frac{\partial φ}{\partial ν_{1}} d σ d t \\ ⩽ C (\int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} {| \frac{\partial^{k} φ}{\partial t^{k}} |}^{\frac{p}{p - 1}} d x d t + \int_{supp (φ)} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} | L_{μ} φ |^{\frac{p}{p - 1}} d x d t) . \end{aligned} \end{matrix}$ Moreover, by using (2.12), we deduce that $\begin{aligned} - \int_{Γ_{1}} f (x) \frac{\partial φ}{\partial ν_{1}} d σ d t & = (2 τ + N) \int_{0}^{\infty} \int_{Γ_{1}^{N}} f (x) x_{N} η_{T} (t) d σ d t \\ = (2 τ + N) (\int_{supp (η_{T})} η_{T} (t) d t) I_{f} (recall (1.14)) . \end{aligned}$ In view of (2.24), the following is the case (notice that $2 τ + N > 0$ ) $\begin{matrix} (3.5) & - \int_{Γ_{1}} f (x) \frac{\partial φ}{\partial ν_{1}} d σ d t = C T I_{f} . \end{matrix}$ Hence, making use of Lemma 2.5, Lemma 2.7, (3.4) and (3.5), we obtain the following inequality $\begin{matrix} T I_{f} ⩽ C (T^{1 - \frac{k p}{p - 1}} (ln R + R^{τ - 1 - \frac{a}{p - 1}}) + T R^{\frac{(τ + 1) p + 1 - a - τ}{p - 1}}), \end{matrix}$ that is, $\begin{matrix} (3.6) & I_{f} ⩽ C (T^{- \frac{k p}{p - 1}} (ln R + R^{τ - 1 - \frac{a}{p - 1}}) + R^{\frac{(τ + 1) p + 1 - a - τ}{p - 1}}) . \end{matrix}$ Taking $T = R^{ζ}$ , where $\begin{matrix} ζ > max {0, \frac{p - 1}{k p} (τ - 1 - \frac{a}{p - 1})}, \end{matrix}$ inequality (3.6) reduces to the following one $\begin{matrix} (3.7) & I_{f} ⩽ C (R^{- \frac{k p ζ}{p - 1}} ln R + R^{ρ_{1}} + R^{ρ_{2}}), \end{matrix}$ where $\begin{matrix} ρ_{1} = τ - 1 - \frac{a}{p - 1} - \frac{k p ζ}{p - 1} \end{matrix}$ and $\begin{matrix} ρ_{2} = \frac{(τ + 1) p + 1 - a - τ}{p - 1} . \end{matrix}$ Observe that due to (1.15) and the above choice of ζ, one has $\begin{matrix} ρ_{i} < 0, i = 1, 2 . \end{matrix}$ Hence, passing to the limit as $R \to \infty$ in (3.7), we obtain that $I_{f} ⩽ 0$ , which contradicts the positivity of $I_{f}$ . Consequently, (1.1) admits no weak solution. This proves part (I) of Theorem 1.5.

(II) For δ and ϵ satisfying, respectively, the restrictions $\begin{matrix} (3.8) & - τ < δ < min {N + τ, 1 + \frac{2 - a}{p - 1}} \end{matrix}$ and $\begin{matrix} (3.9) & 0 < ϵ < {(- δ^{2} + N δ + μ)}^{\frac{1}{p - 1}}, \end{matrix}$ consider the function $u_{δ, ϵ}$ defined by $\begin{matrix} (3.10) & u_{δ, ϵ} (x) = ϵ x_{N} | x |^{- δ}, x \in B_{+}^{N} ∖ {0} . \end{matrix}$ Observe that $N + 2 τ > 0$ . Moreover, by (1.16), one has $\begin{matrix} - τ < 1 + \frac{2 - a}{p - 1} . \end{matrix}$ This shows that the set of values δ satisfying (3.8) is nonempty. Observe also that $- τ$ and $N + τ$ are the roots of the polynomial function $\begin{matrix} P (δ) = - δ^{2} + N δ + μ . \end{matrix}$ Hence, for $- τ < δ < N + τ$ , one has the positivity condition $P (δ) > 0$ . This shows that $P {(δ)}^{\frac{1}{p - 1}}$ is well-defined and the set of values of ϵ satisfying (3.9) is nonempty. On the other hand, elementary calculations yield to the formula $\begin{matrix} L_{μ} u_{δ, ϵ} (x) = ϵ x_{N} P (δ) | x |^{- δ - 2} . \end{matrix}$ Then, using (3.8), (3.9) and (3.10), for all $x \in B_{+}^{N} ∖ {0}$ , we obtain that $\begin{aligned} L_{μ} u_{δ, ϵ} (x) & ⩾ ϵ^{p} x_{N} | x |^{- δ - 2} \\ = | x |^{- a} ϵ^{p} x_{N}^{p} | x |^{- δ p} (x_{N}^{1 - p} | x |^{- δ - 2 + δ p + a}) \\ ⩾ | x |^{- a} u_{δ, ϵ}^{p} (x) | x |^{δ (p - 1) + a - p - 1} \\ ⩾ | x |^{- a} u_{δ, ϵ}^{p} (x) . \end{aligned}$ Hence, for any δ and ϵ satisfying respectively (3.8) and (3.9), functions of the form (3.10) are nonnegative stationary solutions to (1.1) with $\begin{matrix} f (x) = ϵ x_{N}, x \in Γ_{1}^{N} . \end{matrix}$ This proves part (II) of Theorem 1.5. □
Proof of Theorem 1.9.
We also use the contradiction argument. Suppose that $u \in L_{loc}^{p} ([0, \infty) \times B_{+}^{N} ∖ {0})$ is a weak solution to (1.2). By using the formula (1.12) and Lemma 2.2, for sufficiently large ℓ, T and R, we deduce the following inequality $\begin{aligned} \int_{Ω_{T}} | x |^{- a} | u |^{p} I_{T}^{α} ψ d x d t - \int_{Γ_{1, T}} f (x) \frac{\partial ψ}{\partial ν_{1}} d σ d t + \sum_{i = 0}^{k - 1} {(- 1)}^{i} \int_{B_{1}^{c}} u_{k - i - 1} (x) \frac{\partial^{i} ψ}{\partial t^{i}} (0, x) d x \\ ⩽ \int_{Ω_{T}} | u | | \frac{\partial^{k} ψ}{\partial t^{k}} | d x d t + \int_{Ω_{T}} | u | | L_{μ} ψ | d x d t, \end{aligned}$ where $ψ \in Ψ_{T}$ is the test function defined by (2.10). Proceeding as in the proof of Theorem 1.5 (I), by means of Young’s inequality, we obtain that $\begin{matrix} (3.11) & \begin{aligned} - \int_{Γ_{1, T}} f (x) \frac{\partial ψ}{\partial ν_{1}} d σ d t + \sum_{i = 0}^{k - 1} {(- 1)}^{i} \int_{B_{1}^{c}} u_{k - i - 1} (x) \frac{\partial^{i} ψ}{\partial t^{i}} (0, x) d x \\ ⩽ C (\int_{supp (ψ)} | x |^{\frac{a}{p - 1}} {| I_{T}^{α} ψ |}^{\frac{- 1}{p - 1}} {| \frac{\partial^{k} ψ}{\partial t^{k}} |}^{\frac{p}{p - 1}} d x d t \\ + \int_{supp (ψ)} | x |^{\frac{a}{p - 1}} {| I_{T}^{α} ψ |}^{\frac{- 1}{p - 1}} | L_{μ} ψ |^{\frac{p}{p - 1}} d x d t) . \end{aligned} \end{matrix}$ On the other hand, using (2.13) we get that $\begin{aligned} - \int_{Γ_{1, T}} f (x) \frac{\partial ψ}{\partial ν_{1}} d σ d t & = (2 τ + N) (\int_{0}^{T} T^{- ℓ} {(T - t)}^{ℓ} d t) I_{f} \\ (3.12) & = C T I_{f} . \end{aligned}$ Moreover, since $u_{i} ⩾ 0$ for all $0 ⩽ i ⩽ k - 1$ , we deduce that $\begin{aligned} \sum_{i = 0}^{k - 1} {(- 1)}^{i} \int_{B_{1}^{c}} u_{k - i - 1} (x) \frac{\partial^{i} ψ}{\partial t^{i}} (0, x) d x & = \sum_{i = 0}^{k - 1} {(- 1)}^{i} \frac{d^{i} γ_{T} (0)}{d t^{i}} \int_{B_{1}^{c}} u_{k - i - 1} (x) ξ_{R} (x) d x \\ = C \sum_{i = 0}^{k - 1} T^{- i} \int_{B_{1}^{c}} u_{k - i - 1} (x) ξ_{R} (x) d x \\ (3.13) & ⩾ 0 . \end{aligned}$ Hence, using Lemma 2.9, Lemma 2.10, (3.11), (3.12) and (3.13), we obtain the following inequality $\begin{matrix} T I_{f} ⩽ C (T^{1 - \frac{k p + α}{p - 1}} (ln R + R^{τ - 1 - \frac{a}{p - 1}}) + T^{1 - \frac{α}{p - 1}} R^{\frac{(τ + 1) p + 1 - a - τ}{p - 1}}), \end{matrix}$ that is, $\begin{matrix} (3.14) & I_{f} ⩽ C (T^{- \frac{k p + α}{p - 1}} ln R + T^{- \frac{k p + α}{p - 1}} R^{τ - 1 - \frac{a}{p - 1}} + T^{- \frac{α}{p - 1}} R^{\frac{(τ + 1) p + 1 - a - τ}{p - 1}}) . \end{matrix}$ Finally, taking $T = R^{θ}$ , where $\begin{matrix} θ > max {0, \frac{p - 1}{k p} (τ - 1 - \frac{a}{p - 1}), \frac{(τ + 1) p + 1 - a - τ}{α}}, \end{matrix}$ and passing to the limit as $R \to \infty$ in (3.14), we arrive to contradiction with the positivity condition $I_{f} > 0$ . This completes the proof of Theorem 1.9. □

4. Conclusions

The knowledge of intrinsic properties to any physical system is constrained by uncertainty relations among the constituents of the same system. Referring to quantum mechanics, the uncertainty principle (namely, Heisenberg’s principle) bounds the accuracy with which the outcomes of a pair of certain measurements (for example, position and momentum of a particle) can be predicted. A variety of mathematical inequalities appeared in the literature to model this situation, showing the relevance to deal with singular potential terms and memory effects. Now, the Hardy inequality (refer to the type inequality (1.3)) can be considered as a first attempt to represent the above Heisenberg’s principle. We recall that the Hardy inequality was originally given in one dimension and Leray provided a analogous inequality in the case $R^{3}$ (refer again to (1.3)) and in the context of Navier–Stokes equations; see the recent book [24] for more information. So, (1.3) gives us a dimensional homogeneity (equivalence) condition between the inverse square potential and the gradient. This means that the Leray–Hardy potential possesses the same homogeneity of the Dirichlet integral for the Laplace differential operator (see again [24]). In such sense, this singular potential is significant enough to approach the process of finding solutions to various differential equations (also dealing with energy eigenfunctions and eigenvalues). As already mentioned in Section 1, a main feature of our results here is the fact that we study the existence of solutions without using any restrictions on the sign of solutions to problems (1.1) and (1.2). Moreover, since problem (1.2) reduces to problem (1.1) when $α \to 0^{+}$ (recall that $α > 0$ represents a memory term), then solving (1.2) means to observe the influence of this “memory” on the critical behavior of problem (1.1).

Footnotes

Acknowledgements

The authors thank the referee and editor for useful remarks. The first author is supported by Researchers Supporting Project number (RSP2023R57), King Saud University, Riyadh, Saudi Arabia.

Statements & declarations

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. The authors have no relevant financial or non-financial interests to disclose.

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