Semilinear hyperbolic inequalities with Hardy potential in a bounded domain

Abstract

We consider hyperbolic inequalities with Hardy potential $\begin{matrix} \{\begin{matrix} u_{t t} - Δ u + \frac{λ}{| x |^{2}} u ⩾ | x |^{- a} | u |^{p} in (0, \infty) \times B_{1} ∖ {0}, \\ u (t, x) ⩾ f (x) on (0, \infty) \times \partial B_{1}, \end{matrix} \end{matrix}$ where $B_{1}$ is the unit ball in $R^{N}$ , $N ⩾ 3$ , $λ > - {(\frac{N - 2}{2})}^{2}$ , $a ⩾ 0$ , $p > 1$ and f is a nontrivial $L^{1}$ -function. We study separately the cases: $λ = 0$ , $- {(\frac{N - 2}{2})}^{2} < λ < 0$ and $λ > 0$ . For each case, we obtain an optimal criterium for the nonexistence of weak solutions. Our study yields naturally optimal nonexistence results for the corresponding stationary problem. The novelty of this work lies in two facts: (i) To the best of our knowledge, in all previous works dealing with nonexistence results for evolution equations with Hardy potential in a bounded domain, only the parabolic case has been investigated, making use of some comparison principles. (ii) To the best of our knowledge, in all previous works, the issue of nonexistence has been studied only in the case of positive solutions. In this paper, there is no restriction on the sign of solutions.

Keywords

Hyperbolic inequalities Hardy potential bounded domain weak solution nonexistence

1. Introduction

This work is concerned with the study of existence and nonexistence of weak solutions to hyperbolic inequalities with Hardy potential $\begin{matrix} (1) & \{\begin{matrix} u_{t t} - Δ u + \frac{λ}{| x |^{2}} u ⩾ | x |^{- a} | u |^{p} in (0, \infty) \times B_{1} ∖ {0}, \\ u (t, x) ⩾ f (x) on (0, \infty) \times \partial B_{1}, \end{matrix} \end{matrix}$ where $B_{1} = {x \in R^{N} : | x | < 1}$ , $N ⩾ 3$ , $λ > - {(\frac{N - 2}{2})}^{2}$ , $a ⩾ 0$ , $p > 1$ and $f \in L^{1} (\partial B_{1})$ , $f ≢ 0$ .

Problems of type (1) (with $a = 0$ ) posed in $(0, \infty) \times R^{N}$ have been considered in [8]. Namely, the authors studied the nonexistence of weak solutions to higher-order semilinear evolution inequalities of the form $\begin{matrix} (2) & \frac{\partial^{k} u}{\partial t^{k}} - Δ u + \frac{λ}{| x |^{2}} u ⩾ | u |^{p} in (0, \infty) \times R^{N}, \end{matrix}$ where $N ⩾ 3$ , $λ ⩾ - {(\frac{N - 2}{2})}^{2}$ , $k ⩾ 1$ (a natural number) and $p > 1$ . In particular, in the case $k = 2$ and $u_{t} (0, \cdot) ⩾ 0$ , it was shown that,

if $λ ⩾ 0$ and $1 < p ⩽ 1 + \frac{2}{1 - μ_{1}}$ , then (2) admits no weak solution;

if $- {(\frac{N - 2}{2})}^{2} ⩽ λ < 0$ and $1 < p ⩽ 1 + \frac{2}{1 - μ_{2}}$ , then (2) admits no weak solution,

where

μ_{1}

and

μ_{2}

are the roots of the polynomial function

\begin{matrix} P (s) = s^{2} + (N - 2) s - λ = 0 \end{matrix}

with

μ_{1} ⩽ μ_{2}

. The proof of the above results is based on the test function method introduced by Mitidieri and Pohozaev (see e.g. [17]). Notice that in the case

λ = 0

, we have

μ_{1} = 2 - N

, and then condition (i) reduces to

\begin{matrix} 1 < p ⩽ p_{K} (N), \end{matrix}

where

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

is the Kato exponent obtained by Kato [15] for the hyperbolic inequality

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

Recently, in [13], the authors considered problem (1) (with $a = 0$ ) posed in $(0, \infty) \times R^{N} ∖ B_{1}$ under an inhomogeneous Robin-type boundary condition. Namely, they investigated the existence and nonexistence of weak solutions to $\begin{matrix} (3) & \{\begin{matrix} u_{t t} - Δ u + \frac{λ}{| x |^{2}} u ⩾ | u |^{p} in (0, \infty) \times R^{N} ∖ B_{1}, \\ α \frac{\partial u}{\partial ν} (t, x) + β u (t, x) ⩾ f (x) on (0, \infty) \times \partial B_{1}, \end{matrix} \end{matrix}$ where $N ⩾ 2$ , $λ ⩾ - {(\frac{N - 2}{2})}^{2}$ , $α, β ⩾ 0$ and $(α, β) \neq (0, 0)$ . In the case $λ = - {(\frac{N - 2}{2})}^{2}$ , it was shown that,

if $N = 2$ and $\int_{\partial B_{1}} f (x) d σ > 0$ , then for all $p > 1$ , (3) admits no weak solution;

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

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

In the case

λ > - {(\frac{N - 2}{2})}^{2}

, it was shown that,

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

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

The study of parabolic equations with hardy potential in a bounded domain was considered by some authors. For instance, in [3], the authors considered parabolic equations of the form $\begin{matrix} (4) & \{\begin{matrix} u_{t} - Δ u = \frac{λ}{| x |^{2}} u + u^{p} + f in (0, \infty) \times Ω, u ⩾ 0, \\ u (t, x) = 0 on (0, \infty) \times \partial Ω, \\ u (0, x) = u_{0} (x), in Ω, \end{matrix} \end{matrix}$ where $Ω \subset R^{N}$ , $N ⩾ 3$ , is a bounded regular domain containing the origin, $p > 1$ , $λ > 0$ , and $u_{0} ⩾ 0$ , $f ⩾ 0$ belong to a suitable class of functions. Namely, it was shown the existence of a critical exponent $p_{+} (λ)$ such that for $p ⩾ p_{+} (λ)$ , there is no distributional solution to (4), while for $p < p_{+} (λ)$ , and under some additional conditions on the data, (4) 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 problem (1). For other contributions related to the study of parabolic equations with Hardy potential in a bounded domain, see e.g. [4,5,7,11,20] and the references therein. For the study of existence and nonexistence of solutions to elliptic equations involving Hardy potential, see e.g. [1,2,6,9,10] and the references therein.

From the above mentioned works, it is a natural question to ask about the critical behavior of hyperbolic inequalities involving hardy potential in a bounded domain. This motivates our study in the present paper.

Before stating our main results, we need to define weak solutions to (1). For $δ > 0$ , we denote by $\overline{B_{δ}}$ the closed ball centered at the origin with radius δ, i.e., $\begin{matrix} \overline{B_{δ}} = {x \in R^{N} : | x | ⩽ δ} . \end{matrix}$ Let $\begin{matrix} Q = (0, \infty) \times \overline{B_{1}} and Γ = (0, \infty) \times \partial B_{1} . \end{matrix}$

Definition 1.1 (Admissible test functions).

A function $φ = φ (t, x)$ is said to be admissible, if it satisfies the following conditions: (A ₁ )

$φ \in C^{2} (Q)$ , $φ ⩾ 0$ .

(A ₂ )

$supp (φ) \subset \subset (0, \infty) \times \overline{B_{1}} ∖ {0}$ .

(A ₃ )

$φ |_{Γ} = 0$ , $\frac{\partial φ}{\partial ν} |_{Γ} ⩽ 0$ , where ν denotes the outward unit normal vector on $\partial B_{1}$ .

We denote by

A

the space of admissible functions.

Definition 1.2 (Weak solutions).

We say that $u \in L_{loc}^{p} (Q)$ is a weak solution to (1), if for all $φ \in A$ , we have $\begin{matrix} (5) & \int_{Q} | x |^{- a} | u |^{p} φ d x d t - \int_{Γ} f (x) \frac{\partial φ}{\partial ν} d σ d t ⩽ \int_{Q} u (φ_{t t} - Δ φ + \frac{λ}{| x |^{2}} φ) d x d t . \end{matrix}$

Multiplying (1) by φ, where $φ \in A$ , an integrating by parts, it can be easily seen that any smooth solution to (1) is a weak solution in the sense of Definition 1.2.

For $f \in L^{1} (\partial B_{1})$ , let $\begin{matrix} I_{f} = \int_{\partial B_{1}} f (x) d σ . \end{matrix}$ For $λ > - {(\frac{N - 2}{2})}^{2}$ , we introduce the parameter β given by $\begin{matrix} (6) & β = - \frac{N - 2}{2} + \sqrt{{(\frac{N - 2}{2})}^{2} + λ} . \end{matrix}$

We first consider the case $λ = 0$ .

Theorem 1.3.
Let $N ⩾ 3$ and $λ = 0$ .
Let $f \in L^{1} (\partial B_{1})$ be such that $I_{f} > 0$ . If $a > 2$ , then for all $p > 1$ , ( 1 ) admits no weak solution.

If $a < 2$ , then for all $p > 1$ , ( 1 ) admits positive solutions for some $f \in L^{1} (\partial B_{1})$ .

We next consider the case $- {(\frac{N - 2}{2})}^{2} < λ < 0$ .
Theorem 1.4.
Let $N ⩾ 3$ and $- {(\frac{N - 2}{2})}^{2} < λ < 0$ .
Let $f \in L^{1} (\partial B_{1})$ be such that $I_{f} > 0$ . Then, for all $\begin{matrix} p > max {1, 1 + \frac{a - 2}{β}} = \{\begin{matrix} 1 & if a ⩾ 2, \\ 1 + \frac{a - 2}{β} & if 0 ⩽ a < 2, \end{matrix} \end{matrix}$ ( 1 ) admits no weak solution.

If $0 ⩽ a < 2$ , then for all $\begin{matrix} 1 0$ .
Theorem 1.5.
Let $N ⩾ 3$ and $λ > 0$ .
Let $f \in L^{1} (\partial B_{1})$ be such that $I_{f} > 0$ . If $a > 2$ , then for all $\begin{matrix} 1 1$ , ( 1 ) admits positive solutions for some $f \in L^{1} (\partial B_{1})$ .

If $a > 2$ , then for all $\begin{matrix} p > 1 + \frac{a - 2}{β}, \end{matrix}$ ( 1 ) admits positive solutions for some $f \in L^{1} (\partial B_{1})$ .

Remark 1.6.
Let $N ⩾ 3$ , $λ > - {(\frac{N - 2}{2})}^{2}$ , $a ⩾ 0$ , $p > 1$ and $\begin{matrix} δ > max {β, \frac{a - 2}{p - 1}} . \end{matrix}$ For sufficiently small $ϵ > 0$ , one can check that $\begin{matrix} u (x) = - ϵ | x |^{δ}, x \in B_{1} ∖ {0} \end{matrix}$ is a stationary solution to (1) with $f \equiv - ε < 0$ . This shows the necessity of the assumption $I_{f} > 0$ in the nonexistence results given by Theorems 1.3, 1.4 and 1.5, when $I_{f} \neq 0$ .
Remark 1.7.
Theorems 1.3, 1.4 and 1.5 leave open the issue of existence and nonexistence in the following two cases:
$λ = 0$ and $a = 2$ .

$λ > - {(\frac{N - 2}{2})}^{2}$ , $λ \neq 0$ and $p = 1 + \frac{a - 2}{β}$ , where $\begin{matrix} 0 ⩽ a < 2 if λ < 0; a > 2 if λ > 0 . \end{matrix}$

Clearly, Theorems 1.3, 1.4 and 1.5 yield nonexistence results for the corresponding stationary problem $\begin{matrix} (7) & \{\begin{matrix} - Δ u + \frac{λ}{| x |^{2}} u ⩾ | x |^{- a} | u |^{p} in B_{1} ∖ {0}, \\ u (x) ⩾ f (x) on \partial B_{1} . \end{matrix} \end{matrix}$
Corollary 1.8.
Let $N ⩾ 3$ , $λ > - {(\frac{N - 2}{2})}^{2}$ , $a ⩾ 0$ and $f \in L^{1} (\partial B_{1})$ with $I_{f} > 0$ .
If $λ = 0$ and $a > 2$ , then for all $p > 1$ , ( 7 ) admits no weak solution.

If $- {(\frac{N - 2}{2})}^{2} < λ < 0$ , then for all $\begin{matrix} p > max {1, 1 + \frac{a - 2}{β}}, \end{matrix}$ ( 7 ) admits no weak solution.

Let $λ > 0$ and $a > 2$ . Then, for all $\begin{matrix} 1 < p < 1 + \frac{a - 2}{β}, \end{matrix}$ ( 7 ) admits no weak solution.

In the proof of the existence results given by Theorems 1.3, 1.4 and 1.5 (see Section 3.2), we establish that (1) admits both stationary and time-dependent positive solutions for some $f \in L^{1} (\partial B_{1})$ . Hence, we also deduce the following existence results for (7).
Corollary 1.9.
Let $N ⩾ 3$ and $λ > - {(\frac{N - 2}{2})}^{2}$ .
If $λ = 0$ and $0 ⩽ a < 2$ , then for all $p > 1$ , ( 7 ) admits positive solutions for some $f \in L^{1} (\partial B_{1})$ .

If $- {(\frac{N - 2}{2})}^{2} < λ < 0$ and $0 ⩽ a < 2$ , then for all $\begin{matrix} 1 0$ .

If $0 ⩽ a ⩽ 2$ , then for all $p > 1$ , ( 7 ) admits positive solutions for some $f \in L^{1} (\partial B_{1})$ .

If $a > 2$ , then for all $\begin{matrix} p > 1 + \frac{a - 2}{β}, \end{matrix}$ ( 7 ) admits positive solutions for some $f \in L^{1} (\partial B_{1})$ .

Let us say some words about our approach:

Our nonexistence results are based on suitable test functions and integral estimates. This technique was introduced by Mitidieri and Pohozaev (see e.g. [17,18]). Next, it was used by many authors for studying the nonexistence of solutions for various nonlinear problems posed in infinite domains (see e.g. [8,12–14,16,19]). For instance, in the case of the whole space, we often use a family of cut-off functions ${ξ_{R}}_{R}$ satisfying $\begin{matrix} supp (ξ_{R}) \subset {x \in R^{N} : | x | < R} . \end{matrix}$ Next, by a contradiction argument, using an a priori estimate and taking the limit as $R \to \infty$ , we deduce the nonexistence of solutions. For problem (1), due to the boundedness of the domain, the above choice cannot be considered. To overcome this difficulty, we make use of test functions vanishing only in a neighborhood of the origin. Moreover, the considered test functions involve a function H satisfying $\begin{matrix} - Δ H + \frac{λ}{| x |^{2}} H = 0 in B_{1} ∖ {0}, H = 0 on \partial B_{1} . \end{matrix}$

In the most previous works dealing with nonexistence results for nonlinear problems involving Hardy potential in a bounded domain, only the case of positive solutions has been treated. In this paper, there is no restriction on the sign of solutions.

Our existence results are established by the construction of explicit positive solutions to (1) for some $f \in L^{1} (\partial B_{1})$ .

The rest of the paper is organized as follows. In Section 2, we establish some preliminary results that will be useful in the proof of the above theorems. Namely, we first establish an a priori estimate for problem (1). Next, we introduce a certain class of admissible test functions specifically adapted to our problem and the considered bounded domain, and prove some useful estimates involving such functions. Finally, Section 3 is devoted to the proof of Theorems 1.3, 1.4 and 1.5. Namely, we first establish the nonexistence results (Section 3.1). Next, we prove the existence results (Section 3.2).

Throughout this paper, the symbol C denotes always a generic positive constant, which is independent of the scaling parameters T, R and the solution u. Its value could be changed from one line to another.
2. Preliminaries

Let $N ⩾ 3$ , $λ > - {(\frac{N - 2}{2})}^{2}$ , $a ⩾ 0$ , $p > 1$ and $f \in L^{1} (\partial B_{1})$ .

2.1. A priori estimate

For any admissible function φ, let $\begin{matrix} (8) & K_{1} (φ) = \int_{Q} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} | φ_{t t} |^{\frac{p}{p - 1}} d x d t \end{matrix}$ and $\begin{matrix} (9) & K_{2} (φ) = \int_{Q} | x |^{\frac{a}{p - 1}} φ^{\frac{- 1}{p - 1}} {| - Δ φ + \frac{λ}{| x |^{2}} φ |}^{\frac{p}{p - 1}} d x d t . \end{matrix}$ We have the following a priori estimate.

Lemma 2.1.
Let $u \in L_{loc}^{p} (Q)$ be a weak solution to ( 1 ). Then, for all $φ \in A$ , there holds $\begin{matrix} (10) & - \int_{Γ} f (x) \frac{\partial φ}{\partial ν} d σ d t ⩽ C \sum_{i = 1}^{2} K_{i} (φ), \end{matrix}$ provided that $K_{i} (φ) < \infty$ , $i = 1, 2$ .
Proof.
Let $u \in L_{loc}^{p} (Q)$ be a weak solution to (1). In view of (5), we obtain $\begin{matrix} (11) & \int_{Q} | x |^{- a} | u |^{p} φ d x d t - \int_{Γ} f (x) \frac{\partial φ}{\partial ν} d σ d t ⩽ \int_{Q} | u | | φ_{t t} | d x d t + \int_{Q} | u | | - Δ φ + \frac{λ}{| x |^{2}} φ | d x d t, \end{matrix}$ for all $φ \in A$ . Making use of Young’s inequality, we obtain $\begin{array}{rcl} \int_{Q} | u | | φ_{t t} | d x d t & = & \int_{Q} (| x |^{\frac{- a}{p}} | u | φ^{\frac{1}{p}}) (| x |^{\frac{a}{p}} φ^{\frac{- 1}{p}} | φ_{t t} |) d x d t \\ (12) & ⩽ & \frac{1}{2} \int_{Q} | x |^{- a} | u |^{p} φ d x d t + C K_{1} (φ) \end{array}$ and $\begin{matrix} (13) & \int_{Q} | u | | - Δ φ + \frac{λ}{| x |^{2}} φ | d x d t ⩽ \frac{1}{2} \int_{Q} | x |^{- a} | u |^{p} φ d x d t + C K_{2} (φ) . \end{matrix}$ Thus, (10) follows from (11), (12) and (13). □

2.2. Admissible test functions

Let us introduce the radial function H defined in $\overline{B_{1}} ∖ {0}$ by $\begin{matrix} (14) & H (x) = | x |^{2 - N - β} (1 - | x |^{2 β + N - 2}), \end{matrix}$ where the parameter β is given by (6). An elementary calculation shows that H is a nonnegative solution to $\begin{matrix} \{\begin{matrix} - Δ H (x) + \frac{λ}{| x |^{2}} H (x) = 0, x \in B_{1} ∖ {0}, \\ H (x) = 0, x \in \partial B_{1} . \end{matrix} \end{matrix}$ Let η and ξ be two cut-off functions satisfying: $\begin{matrix} (15) & η \in C^{2} ([0, \infty)), η ⩾ 0, supp (η) \subset \subset (0, 1) \end{matrix}$ and $\begin{matrix} (16) & ξ \in C^{2} ([0, \infty)), 0 ⩽ ξ ⩽ 1, ξ (s) = 0 if 0 ⩽ s ⩽ \frac{1}{2}, ξ (s) = 1 if s ⩾ 1 . \end{matrix}$ For sufficiently large T, R and ℓ, let $\begin{matrix} (17) & ι (t) = η {(\frac{t}{T})}^{ℓ}, t ⩾ 0 \end{matrix}$ and $\begin{matrix} (18) & ψ_{R} (x) = H (x) ξ^{ℓ} (R | x |), x \in \overline{B_{1}}, \end{matrix}$ that is, $\begin{matrix} ψ_{R} (x) = \{\begin{matrix} 0 & if 0 ⩽ | x | ⩽ {(2 R)}^{- 1}, \\ H (x) ξ^{ℓ} (R | x |) & if {(2 R)}^{- 1} ⩽ | x | ⩽ R^{- 1}, \\ H (x) & if R^{- 1} ⩽ | x | ⩽ 1 . \end{matrix} \end{matrix}$ We introduce test functions of the form $\begin{matrix} (19) & φ (t, x) = ι (t) ψ_{R} (x), t ⩾ 0, x \in \overline{B_{1}} . \end{matrix}$

Lemma 2.2.
The function φ defined by ( 19 ) is admissible.
Proof.
Properties (A₁) and (A₂) follow immediately from (15), (16), (17), (18) and (19). On the other hand, using that $H |_{\partial B_{1}} = 0$ , we obtain $\begin{matrix} (20) & φ |_{Γ} = 0 . \end{matrix}$ Next, for $\frac{1}{R} ⩽ | x | ⩽ 1$ , it follows from (14) and (18) that $\begin{matrix} \nabla ψ_{R} (x) = \nabla H (x) = ((2 - N - β) | x |^{1 - N - β} - β | x |^{β - 1}) \frac{x}{| x |}, \end{matrix}$ which yields $\begin{matrix} \frac{\partial ψ_{R}}{\partial ν} |_{\partial B_{1}} (x) = 2 - N - 2 β < 0 \end{matrix}$ and $\begin{matrix} (21) & \frac{\partial φ}{\partial ν} |_{Γ} (t, x) = (2 - N - 2 β) ι (t) ⩽ 0 . \end{matrix}$ Hence, (20) and (21) show that condition (A₃) is satisfied. Consequently, the function φ belongs to $A$ . □

2.3. Further estimates

Lemma 2.3.
We have $\begin{array}{l} (22) & \int_{0}^{\infty} ι (t) d t = C T, \\ (23) & \int_{0}^{\infty} ι {(t)}^{\frac{- 1}{p - 1}} {| ι^{″} (t) |}^{\frac{p}{p - 1}} d t ⩽ C T^{- \frac{p + 1}{p - 1}} . \end{array}$
Proof.
By (15) and (17), we obtain $\begin{array}{rcl} \int_{0}^{\infty} ι (t) d t & = & \int_{0}^{\infty} η {(\frac{t}{T})}^{ℓ} d t \\ = & \int_{0}^{T} η {(\frac{t}{T})}^{ℓ} d t \\ = & T \int_{0}^{1} η {(s)}^{ℓ} d s \\ = & C T, \end{array}$ which proves (22). On the other hand, observing that $\begin{matrix} | ι^{″} (t) | ⩽ C T^{- 2} η {(\frac{t}{T})}^{ℓ - 2}, t > 0, \end{matrix}$ we obtain $\begin{array}{rcl} \int_{0}^{\infty} ι {(t)}^{\frac{- 1}{p - 1}} {| ι^{″} (t) |}^{\frac{p}{p - 1}} d t & ⩽ & C T^{\frac{- 2 p}{p - 1}} \int_{0}^{T} η {(\frac{t}{T})}^{ℓ - \frac{2 p}{p - 1}} d t \\ = & C T^{\frac{- 2 p}{p - 1}} T, \end{array}$ which proves (23). □
Lemma 2.4.
We have, as $R \to \infty$ , $\begin{matrix} (24) & \int_{B_{1}} | x |^{\frac{a}{p - 1}} ψ_{R} (x) d x = \{\begin{matrix} O (1) & if β < 2, \\ O (ln R) & if β = 2, \\ O (R^{β - 2}) & if β > 2 . \end{matrix} \end{matrix}$
Proof.
By (16) and (18), we obtain $\begin{array}{rcl} \int_{B_{1}} | x |^{\frac{a}{p - 1}} ψ_{R} (x) d x & = & \int_{\frac{1}{2 R} < | x | < 1} | x |^{\frac{a}{p - 1}} H (x) ξ {(R | x |)}^{ℓ} d x \\ (25) & ⩽ & \int_{\frac{1}{2 R} < | x | < 1} H (x) d x . \end{array}$ On the other hand, by (14), we get $\begin{array}{rcl} (26) & \begin{matrix} \int_{\frac{1}{2 R} < | x | < 1} H (x) d x & = \int_{\frac{1}{2 R} < | x | < 1} | x |^{2 - N - β} (1 - | x |^{2 β + N - 2}) d x \\ ⩽ C \int_{\frac{1}{2 R} < | x | < 1} | x |^{2 - N - β} d x \\ = C \int_{r = \frac{1}{2 R}}^{1} r^{1 - β} d r . \end{matrix} \end{array}$ Thus, (24) follows from (25) and (26). □
Lemma 2.5.
We have, as $R \to \infty$ , For sufficiently large ℓ, there holds, as $R \to \infty$ , $\begin{matrix} (27) & \int_{B_{1}} | x |^{\frac{a}{p - 1}} ψ_{R}^{\frac{- 1}{p - 1}} {| - Δ ψ_{R} + \frac{λ}{| x |^{2}} ψ_{R} |}^{\frac{p}{p - 1}} d x = O (R^{\frac{β p + 2 - β - a}{p - 1}}) . \end{matrix}$
Proof.
Using (18) and taking in consideration that $\begin{matrix} - Δ H (x) + \frac{λ}{| x |^{2}} H (x) = 0, x \in B_{1} ∖ {0}, \end{matrix}$ we obtain $\begin{matrix} (28) & - Δ ψ_{R} (x) + \frac{λ}{| x |^{2}} ψ_{R} (x) = - H (x) Δ (ξ^{ℓ} (R | x |)) - 2 \nabla H (x) \cdot \nabla (ξ^{ℓ} (R | x |)), \end{matrix}$ where “·” is the inner product in $R^{N}$ . Hence, by (16), it holds that $\begin{matrix} supp (- Δ ψ_{R} + \frac{λ}{| x |^{2}} ψ_{R}) \subset {x \in R^{N} : \frac{1}{2 R} < | x | < \frac{1}{R}} \end{matrix}$ and $\begin{matrix} (29) & \begin{matrix} \int_{B_{1}} | x |^{\frac{a}{p - 1}} ψ_{R}^{\frac{- 1}{p - 1}} {| - Δ ψ_{R} + \frac{λ}{| x |^{2}} ψ_{R} |}^{\frac{p}{p - 1}} d x \\ = \int_{\frac{1}{2 R} < | x | < \frac{1}{R}} | x |^{\frac{a}{p - 1}} ψ_{R}^{\frac{- 1}{p - 1}} {| - Δ ψ_{R} + \frac{λ}{| x |^{2}} ψ_{R} |}^{\frac{p}{p - 1}} d x . \end{matrix} \end{matrix}$ On the other hand, by (16), for $\frac{1}{2 R} < | x | < \frac{1}{R}$ , we obtain $\begin{matrix} (30) & | Δ (ξ^{ℓ} (R | x |)) | ⩽ C R^{2} ξ^{ℓ - 2} (R | x |), | \nabla (ξ^{ℓ} (R | x |)) | ⩽ C R ξ^{ℓ - 1} (R | x |) . \end{matrix}$ Moreover, by (14), we have $\begin{matrix} (31) & H (x) ⩽ C | x |^{2 - N - β}, | \nabla H (x) | ⩽ C | x |^{1 - N - β}, \frac{1}{2 R} < | x | < \frac{1}{R} . \end{matrix}$ Hence, using (28), (30), (31) and Cauchy–Schwartz inequality, we get $\begin{matrix} (32) & | - Δ ψ_{R} (x) + \frac{λ}{| x |^{2}} ψ_{R} (x) | ⩽ C R^{β + N} ξ^{ℓ - 2} (R | x |), \frac{1}{2 R} < | x | < \frac{1}{R} . \end{matrix}$ Next, using (14), (16), (29) and (32), we obtain $\begin{matrix} \int_{B_{1}} | x |^{\frac{a}{p - 1}} ψ_{R}^{\frac{- 1}{p - 1}} {| - Δ ψ_{R} + \frac{λ}{| x |^{2}} ψ_{R} |}^{\frac{p}{p - 1}} d x \\ ⩽ C R^{\frac{(β + N) p}{p - 1}} \int_{\frac{1}{2 R} < | x | < \frac{1}{R}} | x |^{\frac{a}{p - 1}} H^{\frac{- 1}{p - 1}} (x) ξ^{ℓ - \frac{2 p}{p - 1}} (R | x |) d x \\ ⩽ C R^{\frac{(β + N) p}{p - 1}} \int_{\frac{1}{2 R} < | x | < \frac{1}{R}} | x |^{\frac{a + β + N - 2}{p - 1}} d x \\ ⩽ C R^{\frac{(β + N) p}{p - 1}} R^{\frac{- a - β - N + 2}{p - 1}} R^{- N} \\ = C R^{\frac{β p + 2 - β - a}{p - 1}}, \end{matrix}$ which proves (27). □

The following estimate follows immediately form (8), (19), (23) and Lemma 2.4.
Lemma 2.6.
We have $\begin{matrix} K_{1} (φ) ⩽ C \{\begin{matrix} T^{- \frac{p + 1}{p - 1}} & if β < 2, \\ T^{- \frac{p + 1}{p - 1}} ln R & if β = 2, \\ T^{- \frac{p + 1}{p - 1}} R^{β - 2} & if β > 2, \end{matrix} \end{matrix}$ where φ is the test function given by ( 19 ).

The following estimate follows immediately form (9), (19), (22) and Lemma 2.5. Lemma 2.7.
We have $\begin{matrix} K_{2} (φ) ⩽ C T R^{\frac{β p + 2 - β - a}{p - 1}}, \end{matrix}$ where φ is the test function given by ( 19 ).

3. Proofs of the main results

In this section, we prove Theorems 1.3, 1.4 and 1.5.

3.1. Proofs of the nonexistence results

Proof of part (I) of Theorem 1.3.
We argue by contradiction by supposing that $u \in L_{loc}^{p} (Q)$ is a weak solution to (1). Then, by Lemma 2.1, (10) holds for any admissible function φ. For sufficiently large T, R and ℓ, let φ be the test function given by (19). Since $φ \in A$ by Lemma 2.2, there holds $\begin{matrix} (33) & - \int_{Γ} f (x) \frac{\partial φ}{\partial ν} d σ d t ⩽ C \sum_{i = 1}^{2} K_{i} (φ) . \end{matrix}$ On the other hand, in view of (15), (17) and (21), we obtain $\begin{array}{rcl} - \int_{Γ} f (x) \frac{\partial φ}{\partial ν} d σ d t & = & C \int_{0}^{\infty} \int_{B_{1}} ι (t) f (x) d σ d t \\ (34) & = & C (\int_{0}^{T} η {(\frac{t}{T})}^{ℓ} d t) I_{f} . \end{array}$ Using the change of variable $t = T s$ , we get $\begin{array}{rcl} \int_{0}^{T} η {(\frac{t}{T})}^{ℓ} d t & = & T \int_{0}^{1} η {(s)}^{ℓ} d s \\ (35) & = & C T . \end{array}$ Thus, combining (34) with (35), we obtain $\begin{matrix} (36) & - \int_{Γ} f (x) \frac{\partial φ}{\partial ν} d σ d t = C T I_{f} . \end{matrix}$ Moreover, by Lemmas 2.6 and 2.7, we have $\begin{matrix} (37) & \sum_{i = 1}^{2} K_{i} (φ) ⩽ C (T^{- \frac{p + 1}{p - 1}} F (R) + T R^{β + \frac{2 - a}{p - 1}}), \end{matrix}$ where $\begin{matrix} (38) & F (R) = \{\begin{matrix} 1 & if β < 2, \\ ln R & if β = 2, \\ R^{β - 2} & if β > 2 . \end{matrix} \end{matrix}$ Hence, it follows from (33), (36) and (37) that $\begin{matrix} T I_{f} ⩽ C (T^{- \frac{p + 1}{p - 1}} F (R) + T R^{β + \frac{2 - a}{p - 1}}), \end{matrix}$ that is, $\begin{matrix} (39) & I_{f} ⩽ C (T^{- \frac{2 p}{p - 1}} F (R) + R^{β + \frac{2 - a}{p - 1}}) . \end{matrix}$ Since $λ = 0$ , by (6), we have $β = 0$ . Then, by (38), $F (R) = 1$ and (39) reduces to $\begin{matrix} I_{f} ⩽ C (T^{- \frac{2 p}{p - 1}} + R^{\frac{2 - a}{p - 1}}) . \end{matrix}$ Taking $T = R$ in the above estimate, we obtain $\begin{matrix} (40) & I_{f} ⩽ C (R^{- \frac{2 p}{p - 1}} + R^{\frac{2 - a}{p - 1}}) . \end{matrix}$ Since $a > 2$ , passing to the limit as $R \to \infty$ in (40), we get $I_{f} ⩽ 0$ , which contradicts the condition $I_{f} > 0$ . Consequently, for all $p > 1$ , no weak solution exists. This proves part (I) of Theorem 1.3. □
Proof of part (I) of Theorem 1.4.
As previously, let us suppose that $u \in L_{loc}^{p} (Q)$ is a weak solution to (1). Following exactly the same argument used in the proof of part (I) of Theorem 1.3, we obtain (39). Since $- {(\frac{N - 2}{2})}^{2} < λ < 0$ , then by (6), we have $β < 0$ . Then, by (38), $F (R) = 1$ and (39) reduces to $\begin{matrix} (41) & I_{f} ⩽ C (T^{- \frac{2 p}{p - 1}} + R^{β + \frac{2 - a}{p - 1}}) . \end{matrix}$ Taking $T = R$ in the above inequality, we obtain $\begin{matrix} (42) & I_{f} ⩽ C (R^{- \frac{2 p}{p - 1}} + R^{β + \frac{2 - a}{p - 1}}) . \end{matrix}$ Case 1: $a ⩾ 2$ .

In this case, taking in consideration that $β < 0$ , we deduce that $\begin{matrix} (43) & β + \frac{2 - a}{p - 1} < 0 . \end{matrix}$ Hence, passing to the limit as $R \to \infty$ in (42), we obtain a contradiction with $I_{f} > 0$ .

Case 2: $0 ⩽ a < 2$ and $p > 1 + \frac{a - 2}{β}$ .

As in Case 1, (43) holds. Thus, proceeding as above, we reach a contradiction with $I_{f} > 0$ .

Consequently, in both cases, no weak solution exists, which proves part (I) of Theorem 1.4. □
Proof of part (I) of Theorem 1.5.
Again, we suppose that $u \in L_{loc}^{p} (Q)$ is a weak solution to (1). Then, the estimate (39) holds. Since $λ > 0$ , then by (6), we have $β > 0$ . Thus, we have three possible cases.

Case 1: $0 < β < 2$ .

In this case, by (38), $F (R) = 1$ and (39) reduces to (41). Taking $T = R$ in (41), we obtain (42). Hence, if $a > 2$ and $1 0$ .

Case 2: $β = 2$ .

In this case, by (38), $F (R) = ln R$ and (39) reduces to $\begin{matrix} I_{f} ⩽ C (T^{- \frac{2 p}{p - 1}} ln R + R^{\frac{2 p - a}{p - 1}}) . \end{matrix}$ Proceeding as above, taking $T = R$ in the above estimate and passing to the limit as $R \to \infty$ , under the condition $a > 2$ and $1 0$ follows.

Case 3: $β > 2$ .

In this case, by (38), $F (R) = R^{β - 2}$ and (39) reduces to $\begin{matrix} I_{f} ⩽ C (T^{- \frac{2 p}{p - 1}} R^{β - 2} + R^{β + \frac{2 - a}{p - 1}}) . \end{matrix}$ Taking $T = R^{θ}$ in the above estimate, where $\begin{matrix} θ = \frac{a - 2 p}{2 p}, \end{matrix}$ we get $\begin{matrix} (44) & I_{f} ⩽ C R^{β + \frac{2 - a}{p - 1}} . \end{matrix}$ Notice that, since $a > 2$ , $1 2$ , then $θ > 0$ . Passing to the limit as $R \to \infty$ in (44), we reach a contradiction with $I_{f} > 0$ .

Thus, in all the above cases, no weak solution exists. This completes the proof of part (I) of Theorem 1.5. □
3.2. Proof of the existence results

Let us introduce the functional space $\begin{matrix} Λ = {ρ \in C^{2} ([0, \infty)) : inf_{t ⩾ 0} ρ (t) > 0, ρ ⩽ 1, ρ^{″} ⩾ 0} . \end{matrix}$ It is not difficult to show that $Λ \neq \emptyset$ . For instance, the function $ρ : [0, \infty) \to R$ defined by $\begin{matrix} ρ (t) = \frac{1}{2} (1 + \frac{1}{1 + t}), t ⩾ 0, \end{matrix}$ belongs to Λ.

The following lemma will be used later in the proof of our existence results.

Lemma 3.1.

Assume that $ϑ = ϑ (x) \in C^{2} (B_{1} ∖ {0})$ is a stationary positive solution to ( 1 ) for some $f = F > 0$ . Let $ρ = ρ (t) \in Λ$ . Then $u = u (t, x) = ρ (t) ϑ (x)$ is a positive solution to ( 1 ) with $f = ρ_{0} F$ , where $ρ_{0} = {inf}_{t ⩾ 0} ρ (t)$ .

Proof.

Let $ϑ = ϑ (x) \in C^{2} (B_{1} ∖ {0})$ be a positive solution to $\begin{matrix} (45) & - Δ ϑ (x) + \frac{λ}{| x |^{2}} ϑ (x) ⩾ | x |^{- a} {| ϑ (x) |}^{p}, x \in B_{1} ∖ {0} \end{matrix}$ and $\begin{matrix} (46) & ϑ (x) ⩾ F (x), x \in \partial B_{1}, \end{matrix}$ where $F > 0$ . Let $ρ = ρ (t) \in Λ$ and $\begin{matrix} u (t, x) = ρ (t) ϑ (x), t ⩾ 0, x \in B_{1} ∖ {0} . \end{matrix}$ Then, using (45), (46) and the properties of the function ρ, we obtain $\begin{matrix} (47) & \begin{matrix} u_{t t} (t, x) - Δ u (t, x) + \frac{λ}{| x |^{2}} u (t, x) \\ = ρ^{″} (t) ϑ (x) + ρ (t) (- Δ ϑ (t, x)) + \frac{λ}{| x |^{2}} ρ (t) ϑ (x) \\ ⩾ ρ (t) (- Δ ϑ (t, x) + \frac{λ}{| x |^{2}} ϑ (x)) \\ ⩾ ρ (t) | x |^{- a} {| ϑ (x) |}^{p} \\ ⩾ | x |^{- a} {| ρ (t) ϑ (x) |}^{p} \\ = | x |^{- a} {| u (t, x) |}^{p}, t > 0, x \in B_{1} ∖ {0} \end{matrix} \end{matrix}$ and $\begin{matrix} (48) & u (t, x) = ρ (t) ϑ (x) ⩾ ρ (t) F (x) ⩾ ρ_{0} F (x), x \in \partial B_{1} . \end{matrix}$ Hence, by (47) and (48), we deduce that u is a positive solution to (1) with $f = ρ_{0} F$ . □

Proof of part (II) of Theorem 1.3.

Let $λ = 0$ and $a < 2$ . Let $\begin{matrix} (49) & ϑ (x) = ϵ | x |^{- δ}, x \in B_{1} ∖ {0}, \end{matrix}$ where $\begin{matrix} (50) & 0 < δ < min {N - 2, \frac{2 - a}{p - 1}} \end{matrix}$ and $\begin{matrix} (51) & 0 < ϵ ⩽ {[δ (N - δ - 2)]}^{\frac{1}{p - 1}} . \end{matrix}$ Notice that, since $a < 2$ , $N ⩾ 3$ and $p > 1$ , the set of δ satisfying (50) is nonempty. Moreover, by (50), we have $δ (N - δ - 2) > 0$ , which shows that the set of ϵ satisfying (51) is nonempty. An elementary calculation shows that $\begin{matrix} - Δ ϑ (x) = ϵ δ (N - δ - 2) | x |^{- δ - 2}, x \in B_{1} ∖ {0} . \end{matrix}$ Hence, using (50) and (51), we obtain $\begin{array}{rcl} - Δ ϑ (x) & ⩾ & ϵ δ (N - δ - 2) | x |^{- a - δ p} \\ ⩾ & | x |^{- a} ϵ ϵ^{p - 1} | x |^{- δ p} \\ = & | x |^{- a} ϑ^{p} (x) . \end{array}$ This shows that ϑ is a stationary positive solution to (1) with $f \equiv ϵ$ . Then, by Lemma 3.1, we deduce that for any function $ρ \in Λ$ , functions of the form $\begin{matrix} (52) & u (t, x) = ρ (t) ϑ (x), t > 0, x \in B_{1} ∖ {0} \end{matrix}$ are positive solutions to (1) with $f \equiv ρ_{0} ϵ$ . This proves part (II) of Theorem 1.3. □

Proof of part (II) of Theorem 1.4.

Let $- {(\frac{N - 2}{2})}^{2} < λ < 0$ , $0 ⩽ a < 2$ and $\begin{matrix} (53) & 1 < p < 1 + \frac{a - 2}{β} . \end{matrix}$ Consider functions ϑ of the form (49), where $\begin{matrix} (54) & - β < δ < min {N - 2 + β, \frac{2 - a}{p - 1}} \end{matrix}$ and $\begin{matrix} (55) & 0 < ϵ ⩽ {[δ (N - δ - 2) + λ]}^{\frac{1}{p - 1}} . \end{matrix}$ Notice that, since $λ < 0$ (so $β < 0$ ) and p satisfies (53), the set of δ satisfying (54) is nonempty. Moreover, by (54), we have $δ (N - δ - 2) + λ > 0$ , which shows that the set of ϵ satisfying (55) is nonempty. An elementary calculation shows that $\begin{matrix} (56) & - Δ ϑ (x) + \frac{λ}{| x |^{2}} ϑ (x) = ϵ [δ (N - δ - 2) + λ] | x |^{- δ - 2}, x \in B_{1} ∖ {0} . \end{matrix}$ Hence, in view of (54) and (55), we obtain $\begin{array}{rcl} - Δ ϑ (x) + \frac{λ}{| x |^{2}} ϑ (x) & ⩾ & ϵ [δ (N - δ - 2) + λ] | x |^{- a - δ p} \\ ⩾ & | x |^{- a} ϵ^{p} | x |^{- δ p} \\ = & | x |^{- a} ϑ^{p} (x) . \end{array}$ Then, ϑ is a stationary positive solution to (1) with $f \equiv ϵ$ . Thus, by Lemma 3.1, we deduce that for any function $ρ \in Λ$ , functions of the form (52) are positive solutions to (1) with $f \equiv ρ_{0} ϵ$ . This proves part (II) of Theorem 1.4. □

Proof of part (II) of Theorem 1.5.

(i) Let $λ > 0$ and $0 ⩽ a ⩽ 2$ . Consider functions ϑ of the form (49), where $\begin{matrix} (57) & - β < δ < 0 \end{matrix}$ and the parameter ϵ satisfies (55). Observe that, since $λ > 0$ (so $β > 0$ ), the set of δ satisfying (57) is nonempty. Moreover, by (57), we have $δ (N - δ - 2) + λ > 0$ , which shows that the set of ϵ satisfying (55) is nonempty. Hence, using (55), (56), (57), and taking in consideration that $0 ⩽ a ⩽ 2$ , we obtain $\begin{matrix} - Δ ϑ (x) + \frac{λ}{| x |^{2}} ϑ (x) ⩾ | x |^{- a} ϑ^{p} (x), \end{matrix}$ which proves that ϑ is a stationary positive solution to (1) with $f \equiv ϵ$ . Hence, for any function $ρ \in Λ$ , functions of the form (52) are positive solutions to (1) with $f \equiv ρ_{0} ϵ$ .

(ii) Let $λ > 0$ , $a > 2$ and $\begin{matrix} (58) & p > 1 + \frac{a - 2}{β} . \end{matrix}$ In this case, we consider functions ϑ of the form (49), where $\begin{matrix} (59) & - β < δ < \frac{2 - a}{p - 1} \end{matrix}$ and the parameter ϵ satisfies (55). Notice that, due to (58) and the fact that $λ > 0$ , the set of δ satisfying (59) is nonempty. Moreover, due to (59), the set of ϵ satisfying (55) is nonempty. Proceeding as in the case (i), we obtain that for any function $ρ \in Λ$ , functions of the form (52) are positive solutions to (1) with $f \equiv ρ_{0} ϵ$ . Then, part (II) of Theorem 1.5 is proved. □

Footnotes

Acknowledgement

The first author is supported by Researchers Supporting Project number (RSP2023R57), King Saud University, Riyadh, Saudi Arabia.

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