Citation: Murali Ramdoss, Ponmana Selvan-Arumugam, Choonkil Park. Ulam stability of linear differential equations using Fourier transform[J]. AIMS Mathematics, 2020, 5(2): 766-780. doi: 10.3934/math.2020052
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This article studies the questions of existence and nonexistence of weak solutions to the system of polyharmonic wave inequalities
{utt+(−Δ)mu≥|x|a|v|p,(t,x)∈(0,∞)×RN∖¯B1,vtt+(−Δ)mv≥|x|b|u|q,(t,x)∈(0,∞)×RN∖¯B1. | (1.1) |
Here, (u,v)=(u(t,x),v(t,x)), N≥2, B1 is the open unit ball of RN, m≥1 is an integer, a,b≥−2m, (a,b)≠(−2m,−2m), and p,q>1. We will investigate (1.1) under the Navier-type boundary conditions
{(−Δ)iu≥fi(x),i=0,⋯,m−1,(t,x)∈(0,∞)×∂B1,(−Δ)iv≥gi(x),i=0,⋯,m−1,(t,x)∈(0,∞)×∂B1, | (1.2) |
where fi,gi∈L1(∂B1) and (−Δ)0 is the identity operator. Notice that no restriction on the signs of fi or gi is imposed.
The study of semilinear wave inequalities in RN was firstly considered by Kato [1] and Pohozaev & Véron [2]. It was shown that the problem
utt−Δu≥|u|p,(t,x)∈(0,∞)×RN | (1.3) |
possesses a critical exponent pK=N+1N−1 in the following sense:
(ⅰ) If N≥2 and 1<p≤pK, then (1.3) possesses no global weak solution, provided
∫RNut(0,x)dx>0. | (1.4) |
(ⅱ) If p>pK, there are global positive solutions satisfying (1.4).
Caristi [3] studied the higher-order evolution polyharmonic inequality
∂ju∂tj−|x|αΔmu≥|u|p,(t,x)∈(0,∞)×RN, | (1.5) |
where α≤2m. Caristi discussed separately the cases α=2m and α<2m. For instance, when j=2 and α=0, it was shown that, if N≥m+1 and 1<p≤N+mN−m, then (1.5) possesses no global weak solution, provided (1.4) holds. Other existence and nonexistence results for evolution inequalities involving the polyharmonic operator in the whole space can be found in [4,5,6].
The study of the blow-up for semilinear wave equations in exterior domains was firstly considered by Zhang [7]. Namely, among many other problems, Zhang investigated the equation
utt−Δu=|x|a|u|p,(t,x)∈(0,∞)×RN∖D, | (1.6) |
where N≥3, a>−2, and D is a smooth bounded subset of RN. It was shown that (1.6) under the Neumann boundary condition
∂u∂ν=f(x)≥0,(t,x)∈(0,∞)×∂D, |
admits a critical exponent N+aN−2 in the following sense:
(ⅰ) If 1<p<N+aN−2, then (1.6) admits no global solution, provided f≢0.
(ⅱ) If p>N+aN−2, then (1.6) admits global solutions for some f>0.
In [8,9], it was shown that the critical value p=N+aN−2 belongs to case (ⅰ). Furthermore, the same result holds true, if (1.6) is considered under the Dirichlet boundary condition
u=f(x)≥0,(t,x)∈(0,∞)×∂D, |
where D=¯B1.
In [10], the authors considered the system of wave inequalities (1.1) in the case m=1. The system was studied under different types of inhomogeneous boundary conditions. In particular, under the boundary conditions (1.2) with m=1 (Dirichlet-type boundary conditions), the authors obtained the following result: Assume that a,b≥−2, (a,b)≠(−2,−2), If0:=∫∂B1f0dSx≥0, Ig0:=∫∂B1g0dSx≥0, (If0,Ig0)≠(0,0), and p,q>1. If N=2; or N≥3 and
N<max{sign(If0)2p(q+1)+pb+apq−1,sign(Ig0)2q(p+1)+qa+bpq−1}, |
then (1.1)-(1.2) (with m=1) admits no weak solution. Moreover, the authors pointed out the sharpness of the above condition.
In the case m=2, the system (1.1) was recently studied in [11] under different types of boundary conditions. In particular, under the boundary conditions (1.2) with f0≡0 and g0≡0, i.e.,
{u≥0,−Δu≥f1(x),(t,x)∈(0,∞)×∂B1,v≥0,−Δv≥g1(x),(t,x)∈(0,∞)×∂B1. | (1.7) |
Namely, the following result was obtained: Let N≥2, a,b≥−4, (a,b)≠(−4,−4), ∫∂B1f1dSx>0, ∫∂B1g1dSx>0, and p,q>1. If N∈{2,3,4}; or
N≥5,N<max{4p(q+1)+pb+apq−1,4q(p+1)+qa+bpq−1}, |
then (1.1) (with m=2) under the boundary conditions (1.7) admits no weak solution. Moreover, it was shown that the above condition is sharp.
Further results related to the existence and nonexistence of solutions for evolution problems in exterior domains can be found in [12,13,14,15,16,17].
The present work aims to extend the obtained results in [10,11] from m∈{1,2} to an arbitrary m≥1. Before presenting our main results, we need to define weak solutions to the considered problem.
Let
Q=(0,∞)×RN∖B1,ΣQ=(0,∞)×∂B1. |
Notice that ΣQ⊂Q.
Definition 1.1. We say that φ is an admissible test function, if
(i) φ∈C2,2mt,x(Q);
(ii) supp(φ)⊂⊂Q (φ is compactly supported in Q);
(iii) φ≥0;
(iv) For all j=0,1,⋯,m−1,
Δjφ|ΣQ=0,(−1)j∂(Δjφ)∂ν|ΣQ≤0, |
where ν denotes the outward unit normal vector on ∂B1, relative to RN∖B1.
The set of all admissible test functions is denoted by Φ.
Definition 1.2. We say that the pair (u,v) is a weak solution to (1.1)-(1.2), if
(u,v)∈Lqloc(Q)×Lploc(Q),∫Q|x|a|v|pφdxdt−m−1∑i=0∫ΣQfi(x)∂((−Δ)m−1−iφ)∂νdσdt≤∫Qu(−Δ)mφdxdt+∫Quφttdxdt | (1.8) |
and
∫Q|x|b|u|qφdxdt−m−1∑i=0∫ΣQgi∂((−Δ)m−1−iφ)∂νdσdt≤∫Qv(−Δ)mφdxdt+∫Qvφttdxdt | (1.9) |
for every φ∈Φ.
Notice that, if (u,v) is a regular solution to (1.1)-(1.2), then (u,v) is a weak solution in the sense of Definition 1.2.
For every function f∈L1(∂B1), we set
If=∫∂B1f(x)dσ. |
Our first main result is stated in the following theorem.
Theorem 1.1. Let p,q>1, N≥2, and a,b≥−2m with (a,b)≠(−2m,−2m). Let fi,gi∈L1(∂B1) for every i=0,⋯,m−1. Assume that Ifm−1,Igm−1≥0 and (Ifm−1,Igm−1)≠(0,0). If N≤2m; or N≥2m+1 and
N<max{sign(Ifm−1)×2mp(q+1)+pb+apq−1,sign(Igm−1)×2mq(p+1)+qa+bpq−1}, | (1.10) |
then (1.1)-(1.2) possesses no weak solution.
Remark 1.1. Notice that (1.10) is equivalent to
N−2m<α,Ifm−1>0; orN−2m<β,Igm−1>0, | (1.11) |
where
α=a+2m+p(b+2m)pq−1 | (1.12) |
and
β=b+2m+q(a+2m)pq−1. | (1.13) |
On the other hand, due to the condition a,b≥−2m and (a,b)≠(−2m,−2m), we have α,β>0, which shows that, if N≤2m, then (1.10) is always satisfied.
The proof of Theorem 1.1 is based on the construction of a suitable admissible test function and integral estimates. The construction of the admissible test function is specifically adapted to the polyharmonic operator (−Δ)m, the geometry of the domain, and the Navier-type boundary conditions (1.2).
Remark 1.2. By Theorem 1.1, we recover the nonexistence result obtained in [10] in the case m=1. We also recover the nonexistence result obtained in [11] in the case m=2.
Next, we are concerned with the existence of solutions to (1.1)-(1.2). Our second main result shows the sharpness of condition (1.10).
Theorem 1.2. Let p,q>1 and a,b≥−2m with (a,b)≠(−2m,−2m). If
N−2m>max{α,β}, | (1.14) |
where α and β are given by (1.12) and (1.13), then (1.1)-(1.2) admits stationary solutions for some fi,gi∈L1(∂B1) (i=0,⋯,m−1) with Ifm−1,Igm−1>0.
Theorem 1.2 will be proved by the construction of explicit stationary solutions to (1.1)-(1.2).
Remark 1.3. At this moment, we don't know whether there is existence or nonexistence in the critical case N≥2m+1,
N=max{sign(Ifm−1)×2mp(q+1)+pb+apq−1,sign(Igm−1)×2mq(p+1)+qa+bpq−1}. |
This question is left open.
From Theorem 1.1, we deduce the following nonexistence result for the corresponding stationary polyharmonic system
{(−Δ)mu≥|x|a|v|p,x∈RN∖¯B1,(−Δ)mv≥|x|b|u|q,x∈RN∖¯B1, | (1.15) |
under the Navier-type boundary conditions
{(−Δ)iu≥fi(x),i=0,⋯,m−1,x∈∂B1,(−Δ)iv≥gi(x),i=0,⋯,m−1,x∈∂B1. | (1.16) |
Corollary 1.1. Let p,q>1, N≥2, and a,b≥−2m with (a,b)≠(−2m,−2m). Let fi,gi∈L1(∂B1) for every i=0,⋯,m−1. Assume that Ifm−1,Igm−1≥0 and (Ifm−1,Igm−1)≠(0,0). If N≤2m; or N≥2m+1 and (1.10) holds, then (1.15)-(1.16) possesses no weak solution.
The rest of this manuscript is organized as follows: Section 2 is devoted to some auxiliary results. Namely, we first construct an admissible test function in the sense of Definition 1.1. Next, we establish some useful integral estimates involving the constructed test function. The proofs of Theorems 1.1 and 1.2 are provided in Section 3.
Throughout this paper, the letter C denotes a positive constant that is independent of the scaling parameters T, τ, and the solution (u,v). The value of C is not necessarily the same from one line to another.
In this section, we establish some auxiliary results that will be used later in the proof of our main result.
Let us introduce the radial function H defined in RN∖B1 by
H(x)={ln|x|ifN=2,1−|x|2−NifN≥3. | (2.1) |
We collect below some useful properties of the function H.
Lemma 2.1. The function H satisfies the following properties:
(i) H≥0;
(ii) H∈C2m(RN∖B1);
(iii) H|∂B1=0;
(iv) ΔH=0 in RN∖B1;
(v) For all j≥1,
ΔjH|∂B1=∂(ΔjH)∂ν|∂B1=0; |
(vi) ∂H∂ν|∂B1=−C.
Proof. (ⅰ)–(ⅴ) follow immediately from (2.1). On the other hand, we have
∂H∂ν|∂B1={−1ifN=2,−(N−2)ifN≥3, |
which proves (ⅵ).
We next consider a cut-off function ξ∈C∞(R) satisfying the following properties:
0≤ξ≤1,ξ(s)=1 if |s|≤1,ξ(s)=0 if |s|≥2. | (2.2) |
For all τ≫1, let
ξτ(x)=ξ(|x|τ),x∈RN∖B1, |
that is (from (2.2)),
ξτ(x)={1if1≤|x|≤τ,ξ(|x|τ)ifτ≤|x|≤2τ,0if|x|≥2τ. | (2.3) |
For k≫1, we introduce the function
ζτ(x)=H(x)ξkτ(x),x∈RN∖B1. | (2.4) |
We now introduce a second cut-off function G∈C∞(R) satisfying the following properties:
G≥0,supp(G)⊂⊂(0,1). | (2.5) |
For T>0 and k≫1, let
GT(t)=Gk(tT),t≥0. | (2.6) |
Let φ be the function defined by
φ(t,x)=GT(t)ζτ(x),(t,x)∈Q. | (2.7) |
By Lemma 2.1, (2.3)–(2.7), we obtain the following result.
Lemma 2.2. The function φ belongs to Φ.
For all λ>1, μ≥−2m, and φ∈Φ, we consider the integral terms
J(λ,μ,φ)=∫Q|x|−μλ−1φ−1λ−1|(−Δ)mφ|λλ−1dxdt | (2.8) |
and
K(λ,μ,φ)=∫Q|x|−μλ−1φ−1λ−1|φtt|λλ−1dxdt. | (2.9) |
Lemma 2.3. Let φ be the admissible test function defined by (2.7). Assume that
(i) J(p,a,φ),J(q,b,φ),K(p,a,φ),K(q,b,φ)<∞;
(ii) Ifm−1,Igm−1≥0.
If (u,v) is a weak solution to (1.1)-(1.2), then
Ifm−1≤CT−1([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p)ppq−1([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q)pqpq−1 | (2.10) |
and
Igm−1≤CT−1([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q)qpq−1([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p)pqpq−1. | (2.11) |
Proof. Let (u,v) be a weak solution to (1.1)-(1.2) and φ be the admissible test function defined by (2.7). By (1.8), we have
∫Q|x|a|v|pφdxdt−m−1∑i=0∫ΣQfi(x)∂((−Δ)m−1−iφ)∂νdσdt≤∫Qu(−Δ)mφdxdt+∫Quφttdxdt. |
On the other hand, by Lemma 2.1: (ⅴ), (ⅵ), (2.5)–(2.7), we have
m−1∑i=0∫ΣQfi(x)∂((−Δ)m−1−iφ)∂νdσdt=∫ΣQfm−1(x)∂φ∂νdσdt=−C∫ΣQfm−1(x)GT(t)dσdt=−C(∫∞0GT(t)dt)∫∂B1fm−1(x)dσ=−C(∫∞0Gk(tT)dt)Ifm−1=−CT(∫10Gk(s)ds)Ifm−1=−CTIfm−1. |
Consequently, we obtain
∫Q|x|a|v|pφdxdt+CTIfm−1≤∫Qu(−Δ)mφdxdt+∫Quφttdxdt. | (2.12) |
Similarly, by (1.9), we obtain
∫Q|x|b|u|qφdxdt+CTIgm−1≤∫Qv(−Δ)mφdxdt+∫Qvφttdxdt. | (2.13) |
Furthermore, by Hölder's inequality, we have
∫Qu(−Δ)mφdxdt≤∫Q|u||(−Δ)mφ|dxdt=∫Q(|x|bq|u|φ1q)(|x|−bq|(−Δ)mφ|φ−1q)dxdt≤(∫Q|x|b|u|qφdxdt)1q(∫Q|x|−bq−1|(−Δ)mφ|qq−1φ−1q−1dxdt)q−1q, |
that is,
∫Qu(−Δ)mφdxdt≤(∫Q|x|b|u|qφdxdt)1q[J(q,b,φ)]q−1q. | (2.14) |
Similarly, we obtain
∫Quφttdxdt≤(∫Q|x|b|u|qφdxdt)1q[K(q,b,φ)]q−1q. | (2.15) |
Thus, it follows from (2.12), (2.14), and (2.15) that
∫Q|x|a|v|pφdxdt+CTIfm−1≤(∫Q|x|b|u|qφdxdt)1q([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q). | (2.16) |
Using (2.13) and proceeding as above, we obtain
∫Q|x|b|u|qφdxdt+CTIgm−1≤(∫Q|x|a|v|pφdxdt)1p([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p). | (2.17) |
Using (2.16)-(2.17), and taking into consideration that Igm−1≥0, we obtain
∫Q|x|a|v|pφdxdt+CTIfm−1≤(∫Q|x|a|v|pφdxdt)1pq([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p)1q([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q). |
Then, by Young's inequality, it holds that
∫Q|x|a|v|pφdxdt+CTIfm−1≤1pq∫Q|x|a|v|pφdxdt+pq−1pq([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p)pqq(pq−1)([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q)pqpq−1. |
Consequently, we have
(1−1pq)∫Q|x|a|v|pφdxdt+CTIfm−1≤pq−1pq([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p)ppq−1([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q)pqpq−1, |
which yields (2.10). Similarly, using (2.16)-(2.17), and taking into consideration that Ifm−1≥0, we obtain
∫Q|x|b|u|qφdxdt+CTIgm−1≤(∫Q|x|b|u|qφdxdt)1pq([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q)1p([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p), |
which implies by Young's inequality that
∫Q|x|b|u|qφdxdt+CTIgm−1≤1pq∫Q|x|b|u|qφdxdt+pq−1pq([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q)pqp(pq−1)([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p)pqpq−1. |
Thus, it holds that
(1−1pq)∫Q|x|b|u|qφdxdt+CTIgm−1≤pq−1pq([J(q,b,φ)]q−1q+[K(q,b,φ)]q−1q)qpq−1([J(p,a,φ)]p−1p+[K(p,a,φ)]p−1p)pqpq−1, |
which yields (2.11).
The aim of this subsection is to estimate the integral terms J(λ,μ,φ) and K(λ,μ,φ), where λ>1, μ≥−2m, and φ is the admissible test function defined by (2.7) with τ,k≫1.
The following result follows immediately from (2.5) and (2.6).
Lemma 2.4. We have
∫∞0GT(t)dt=CT. |
Lemma 2.5. We have
∫∞0G−1λ−1T|d2GTdt2|λλ−1dt≤CT1−2λλ−1. | (2.18) |
Proof. By (2.5) and (2.6), we have
\begin{equation} \int_{0}^\infty G_T^{\frac{-1}{\lambda-1}}\left|\frac{d^2G_T}{dt^2}\right|^{\frac{\lambda}{\lambda-1}}\, dt = \int_{0}^T G_T^{\frac{-1}{\lambda-1}}\left|\frac{d^2G_T}{dt^2}\right|^{\frac{\lambda}{\lambda-1}} \, dt \end{equation} | (2.19) |
and
\frac{d^2G_T}{dt^2}(t) = kT^{-2}G^{k-2}\left(\frac{t}{T}\right)\left((k-1)G'^2\left(\frac{t}{T}\right)+G\left(\frac{t}{T}\right)G''\left(\frac{t}{T}\right)\right) |
for all t\in (0, T) . The above inequality yields
\left|\frac{d^2G_T}{dt^2}(t)\right|\leq CT^{-2}G^{k-2}\left(\frac{t}{T}\right), \quad t\in (0, T), |
which implies that
G_T^{\frac{-1}{\lambda-1}}\left|\frac{d^2G_T}{dt^2}\right|^{\frac{\lambda}{\lambda-1}} \leq C T^{\frac{-2\lambda}{\lambda-1}} G^{k-\frac{2\lambda}{\lambda-1}}\left(\frac{t}{T}\right), \quad t\in (0, T). |
Then, by (2.19), it holds that
\begin{aligned} \int_{0}^\infty G_T^{\frac{-1}{\lambda-1}}\left|\frac{d^2G_T}{dt^2}\right|^{\frac{\lambda}{\lambda-1}}\, dt& \leq CT^{\frac{-2\lambda}{\lambda-1}}\int_0^T G^{k-\frac{2\lambda}{\lambda-1}}\left(\frac{t}{T}\right)\, dt\\ & = C T^{1-\frac{2\lambda}{\lambda-1}}\int_0^1 G^{k-\frac{2\lambda}{\lambda-1}}(s)\, ds\\ & = C T^{1-\frac{2\lambda}{\lambda-1}}, \end{aligned} |
which proves (2.18).
To estimate J(\lambda, \mu, \varphi) and K(\lambda, \mu, \varphi) , we consider separately the cases N\geq 3 and N = 2 .
Lemma 2.6. We have
\begin{equation} \int_{\mathbb{R}^N\backslash B_1} |x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\, dx\leq C\tau^{N-\frac{\mu+2m\lambda}{\lambda-1}}. \end{equation} | (2.20) |
Proof. Since H and \xi_\tau are radial functions (see (2.1) and (2.3)), to simplify writing, we set
H(x) = H(r), \, \, \xi_\tau(x) = \xi_\tau(r), |
where r = |x| . By (2.4) and making use of Lemma 2.1 (ⅳ), one can show that for all x\in \mathbb{R}^N\backslash B_1 , we have
\begin{aligned} \Delta^m \zeta_\tau(x) & = \Delta^m \left(H(x) \xi_\tau^k(x) \right)\\ & = \sum\limits_{i = 0}^{2m-1} \frac{d^i H}{dr^{i}}(r)\sum\limits_{j = 1}^{2m-i}C_{i, j}\frac{d^{j} \xi_\tau^k}{dr^{j}}(r)r^{i+j-2m}, \end{aligned} |
where C_{i, j} are some constants, which implies by (2.3) that
\begin{equation} {\rm{supp}}\left(\Delta^m\zeta_\tau\right)\subset \left\{x\in \mathbb{R}^N: \tau\leq |x|\leq 2\tau\right\} \end{equation} | (2.21) |
and
\begin{equation} |\Delta ^m\zeta_\tau(x)|\leq C \sum\limits_{i = 0}^{2m-1} \left|\frac{d^i H}{dr^{i}}(r)\right|\sum\limits_{j = 1}^{2m-i} \left|\frac{d^{j} \xi_\tau^k}{dr^{j}}(r)\right|r^{i+j-2m}, \quad x\in {\rm{supp}}\left(\Delta^m\zeta_\tau\right). \end{equation} | (2.22) |
On the other hand, for all x\in {\rm{supp}}\left(\Delta^m\zeta_\tau\right) , we have by (2.1) and (2.3) that
\begin{equation} \left|\frac{d^i H}{dr^{i}}(r)\right| = \left\{\begin{array}{llll} H(r) &\text{if}& i = 0, \\[10pt] C r^{2-N-i} &\text{if}& i = 1, \cdots, 2m-1 \end{array} \right. \end{equation} | (2.23) |
and (we recall that 0\leq \xi_\tau\leq 1 )
\begin{equation} \begin{aligned} \left|\frac{d^{j} \xi_\tau^k}{dr^{j}}(r)\right| &\leq C \tau^{-j} \xi_\tau^{k-j}(r)\\ &\leq C \tau^{-j} \xi_\tau^{k-2m}(r), \, \, j = 1, \cdots, 2m-i. \end{aligned} \end{equation} | (2.24) |
Then, in view of (2.1), (2.21)–(2.24), we have
\begin{aligned} |\Delta^m\zeta_\tau(x)|&\leq C \xi_\tau^{k-2m}(r)\left(H(r) \sum\limits_{j = 1}^{2m}\tau^{-j} r^{j-2m}+ r^{2-N}\sum\limits_{i = 1}^{2m-1} \sum\limits_{j = 1}^{2m-i}\tau^{-j} r^{j-2m}\right)\\ &\leq C \xi_\tau^{k-2m}(r)\left(\tau^{-2m}+\tau^{2-N-2m}\right)\\ &\leq C \tau^{-2m}\xi_\tau^{k-2m}(x) \end{aligned} |
for all x\in {\rm{supp}}\left(\Delta^m\zeta_\tau\right) . Taking into consideration that H\geq C for all x\in {\rm{supp}}\left(\Delta^m\zeta_\tau\right) , the above estimate yields
\begin{equation} |x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\leq C \tau^{\frac{-2m\lambda-\mu}{\lambda-1}}\xi_\tau^{k-\frac{2m\lambda}{\lambda-1}}(x), \quad x\in {\rm{supp}}\left(\Delta^m\zeta_\tau\right). \end{equation} | (2.25) |
Finally, by (2.21) and (2.25), we obtain
\begin{aligned} \int_{\mathbb{R}^N\backslash B_1} |x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\, dx & = \int_{\tau < |x| < 2\tau} |x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\, dx \\ &\leq C\tau^{\frac{-2m\lambda-\mu}{\lambda-1}} \int_{\tau < |x| < 2\tau} \xi_\tau^{k-\frac{2m\lambda}{\lambda-1}}(x)\, dx\\ &\leq C\tau^{\frac{-2m\lambda-\mu}{\lambda-1}} \int_{r = \tau}^{2\tau} r^{N-1}\, dr\\ & = C \tau^{N-\frac{\mu+2m\lambda}{\lambda-1}}, \end{aligned} |
which proves (2.20).
Lemma 2.7. We have
J(\lambda, \mu, \varphi)\leq CT \tau^{N-\frac{\mu+2m\lambda}{\lambda-1}}. |
Proof. By (2.7) and (2.8), we have
J(\lambda, \mu, \varphi) = \left(\int_0^\infty G_T(t)\, dt\right) \left(\int_{\mathbb{R}^N\backslash B_1} |x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\, dx\right). |
Then, using Lemmas 2.4 and 2.6, we obtain the desired estimate.
Lemma 2.8. We have
\begin{equation} \int_{\mathbb{R}^N\backslash B_1}|x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau(x)\, dx\leq C \left(\tau^{N-\frac{\mu}{\lambda-1}}+\ln \tau\right). \end{equation} | (2.26) |
Proof. By (2.1)–(2.4), we have
\begin{aligned} \int_{\mathbb{R}^N\backslash B_1}|x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau(x)\, dx& = \int_{1 < |x| < 2\tau}|x|^{\frac{-\mu}{\lambda-1}}\left(1-|x|^{2-N}\right)\xi^\kappa\left(\frac{|x|}{\tau}\right) \, dx\\ &\leq \int_{1 < |x| < 2\tau}|x|^{\frac{-\mu}{\lambda-1}}\, dx\\ & = C \int_{r = 1}^{2\tau} r^{N-1-\frac{\mu}{\lambda-1}}\, dr\\ &\leq \left\{\begin{array}{llll} C \tau^{N-\frac{\mu}{\lambda-1}} &\mbox{if}& N-\frac{\mu}{\lambda-1} > 0, \\ [4pt] C \ln \tau &\mbox{if}& N-\frac{\mu}{\lambda-1} = 0, \\ [4pt] C &\mbox{if}& N-\frac{\mu}{\lambda-1} < 0 \end{array} \right.\\ &\leq C \left(\tau^{N-\frac{\mu}{\lambda-1}}+\ln \tau\right), \end{aligned} |
which proves (2.26).
Lemma 2.9. We have
K(\lambda, \mu, \varphi)\leq C T^{1-\frac{2\lambda}{\lambda-1}}\left(\tau^{N-\frac{\mu}{\lambda-1}}+\ln \tau\right). |
Proof. By (2.7) and (2.9), we have
K(\lambda, \mu, \varphi) = \left(\int_{0}^\infty G_T^{\frac{-1}{\lambda-1}}\left|\frac{d^2G_T}{dt^2}\right|^{\frac{\lambda}{\lambda-1}}\, dt\right)\left(\int_{\mathbb{R}^N\backslash B_1}|x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau(x)\, dx\right). |
Then, using Lemmas 2.5 and 2.7, we obtain the desired estimate.
Lemma 2.10. We have
\begin{equation} \int_{\mathbb{R}^2\backslash B_1} |x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\, dx\leq C \tau^{2-\frac{2m\lambda+\mu}{\lambda-1}}\ln \tau. \end{equation} | (2.27) |
Proof. Proceeding as in the proof of Lemma 2.6, we obtain
{\rm{supp}}\left(\Delta^m\zeta_\tau\right)\subset \left\{x\in \mathbb{R}^2: \tau\leq |x|\leq 2\tau\right\} |
and
|\Delta ^m\zeta_\tau(x)| \leq C \tau^{-2m}\ln \tau\, \xi_\tau^{k-2m}(x), \quad x\in {\rm{supp}}\left(\Delta^m\zeta_\tau\right). |
The above estimate yields
|x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\leq C \tau^{\frac{-2m\lambda-\mu}{\lambda-1}}\ln \tau\, \xi_\tau^{k-\frac{2m\lambda}{\lambda-1}}(x), \quad x\in {\rm{supp}}\left(\Delta^m\zeta_\tau\right). |
Then, it holds that
\begin{aligned} \int_{\mathbb{R}^2\backslash B_1} |x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau^{\frac{-1}{\lambda-1}}|(-\Delta)^m\zeta_\tau|^{\frac{\lambda}{\lambda-1}}\, dx&\leq C \tau^{\frac{-2m\lambda-\mu}{\lambda-1}}\ln \tau\int_{\tau < |x| < 2\tau}\xi_\tau^{k-\frac{2m\lambda}{\lambda-1}}(x)\, dx\\ &\leq C \tau^{\frac{-2m\lambda-\mu}{\lambda-1}}\ln \tau\int_{r = \tau}^{2\tau} r\, dr\\ &\leq C \tau^{2-\frac{2m\lambda+\mu}{\lambda-1}}\ln \tau, \end{aligned} |
which proves (2.27).
Using (2.7)-(2.8), Lemma 2.4, and Lemma 2.10, we obtain the following estimate of J(\lambda, \mu, \varphi) .
Lemma 2.11. We have
J(\lambda, \mu, \varphi)\leq CT \tau^{2-\frac{2m\lambda+\mu}{\lambda-1}}\ln \tau. |
Lemma 2.12. We have
\begin{equation} \int_{\mathbb{R}^2\backslash B_1}|x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau(x)\, dx\leq C \ln \tau\, \left(\tau^{2-\frac{\mu}{\lambda-1}}+\ln \tau\right). \end{equation} | (2.28) |
Proof. By (2.1)–(2.4), we have
\begin{aligned} \int_{\mathbb{R}^2\backslash B_1}|x|^{\frac{-\mu}{\lambda-1}}\zeta_\tau(x)\, dx& = \int_{1 < |x| < 2\tau}|x|^{\frac{-\mu}{\lambda-1}}\ln |x|\, \xi^\kappa\left(\frac{|x|}{\tau}\right) \, dx\\ &\leq \int_{1 < |x| < 2\tau}|x|^{\frac{-\mu}{\lambda-1}}\ln |x|\, dx\\ & = C \int_{r = 1}^{2\tau} r^{1-\frac{\mu}{\lambda-1}}\ln r\, dr\\ &\leq \left\{\begin{array}{llll} C \tau^{2-\frac{\mu}{\lambda-1}} \ln \tau &\mbox{if}& 2-\frac{\mu}{\lambda-1} > 0, \\ [4pt] C (\ln \tau )^2 &\mbox{if}& 2-\frac{\mu}{\lambda-1} = 0, \\ [4pt] C \ln \tau &\mbox{if}& 2-\frac{\mu}{\lambda-1} < 0 \end{array} \right.\\ &\leq C \ln \tau\, \left(\tau^{2-\frac{\mu}{\lambda-1}}+\ln \tau\right), \end{aligned} |
which proves (2.28).
Using (2.7), (2.9), Lemma 2.5, and Lemma 2.12, we obtain the following estimate of K(\lambda, \mu, \varphi) .
Lemma 2.13. We have
K(\lambda, \mu, \varphi)\leq C T^{1-\frac{2\lambda}{\lambda-1}}\ln \tau\, \left(\tau^{2-\frac{\mu}{\lambda-1}}+\ln \tau\right). |
This section is devoted to the proofs of Theorems 1.1 and 1.2.
By Remark 1.1, (1.10) is equivalent to (1.11). Without restriction of the generality, we assume that
\begin{equation} N-2m < \alpha, \quad I_{f_{m-1}} > 0. \end{equation} | (3.1) |
Indeed, exchanging the roles of (I_{f_{m-1}}, a, p) and (I_{g_{m-1}}, b, q) , the case
N-2m < \beta, \quad I_{g_{m-1}} > 0 |
reduces to (3.1).
We use the contradiction argument. Namely, let us suppose that (u, v) is a weak solution to (1.1)-(1.2) (in the sense of Definition 1.2). For k, T, \tau\gg 1 , let \varphi be the admissible test function defined by (2.7). Then, by Lemma 2.3, we have
\begin{equation} I_{f_{m-1}}^{\frac{pq-1}{p}}\leq CT^{-\frac{pq-1}{p}}\left(\left[J(p, a, \varphi)\right]^{\frac{p-1}{p}}+\left[K(p, a, \varphi)\right]^{\frac{p-1}{p}}\right) \left(\left[J(q, b, \varphi)\right]^{\frac{q-1}{q}}+\left[K(q, b, \varphi)\right]^{\frac{q-1}{q}}\right)^{q}. \end{equation} | (3.2) |
Making use of Lemmas 2.7 and 2.12, we obtain that for all N\geq 2 ,
\begin{equation} J(\lambda, \mu, \varphi)\leq CT \tau^{N-\frac{\mu+2m\lambda}{\lambda-1}} \ln \tau, \quad \lambda > 1, \, \mu\geq -2m. \end{equation} | (3.3) |
Similarly, by Lemmas 2.9 and 2.13, we obtain that for all N\geq 2 ,
\begin{equation} K(\lambda, \mu, \varphi)\leq C T^{1-\frac{2\lambda}{\lambda-1}}\left(\tau^{N-\frac{\mu}{\lambda-1}}+\ln \tau\right)\ln \tau, \quad \lambda > 1, \, \mu\geq -2m. \end{equation} | (3.4) |
In particular, for (\lambda, \mu) = (p, a) , we obtain by (3.3) and (3.4) that
\begin{equation} \begin{aligned} &\left[J(p, a, \varphi)\right]^{\frac{p-1}{p}}+\left[K(p, a, \varphi)\right]^{\frac{p-1}{p}}\\ &\leq C\left[T^{\frac{p-1}{p}} \tau^{\left(N-\frac{a+2mp}{p-1}\right)\frac{p-1}{p}} (\ln \tau)^{\frac{p-1}{p}}+ T^{\left(1-\frac{2p}{p-1}\right)\frac{p-1}{p}}\left(\tau^{N-\frac{a}{p-1}}+\ln \tau\right)^{\frac{p-1}{p}}(\ln \tau)^\frac{p-1}{p}\right]\\ & = C T^{\frac{p-1}{p}} \tau^{\left(N-\frac{a+2mp}{p-1}\right)\frac{p-1}{p}} (\ln \tau)^{\frac{p-1}{p}}\left[1+T^{-2}\left(\tau^{\frac{2mp}{p-1}}+\tau^{-\left(N-\frac{a+2mp}{p-1}\right)}\ln\tau\right)^{\frac{p-1}{p}}\right]. \end{aligned} \end{equation} | (3.5) |
Furthermore, taking T = \tau^\theta , where
\begin{equation} \theta > \max\left\{m, \left(\frac{a+2mp}{p-1}-N\right)\frac{p-1}{p}\right\}, \end{equation} | (3.6) |
we obtain
1+T^{-2}\left(\tau^{\frac{2mp}{p-1}}+\tau^{-\left(N-\frac{a+2mp}{p-1}\right)}\ln\tau\right)^{\frac{p-1}{p}}\leq C. |
Then, from (3.5), we deduce that
\begin{equation} \left[J(p, a, \varphi)\right]^{\frac{p-1}{p}}+\left[K(p, a, \varphi)\right]^{\frac{p-1}{p}}\leq C \left[\tau^{\theta+ N-\frac{a+2mp}{p-1}}\ln\tau\right]^{\frac{p-1}{p}}. \end{equation} | (3.7) |
Similarly, for
\begin{equation} \theta > \max\left\{m, \left(\frac{b+2mq}{q-1}-N\right)\frac{q-1}{q}\right\}, \end{equation} | (3.8) |
we obtain
\begin{equation} \left(\left[J(q, b, \varphi)\right]^{\frac{q-1}{q}}+\left[K(q, b, \varphi)\right]^{\frac{q-1}{q}}\right)^{q} \leq C \left[\tau^{\theta+ N-\frac{b+2mq}{q-1}}\ln\tau\right]^{q-1}. \end{equation} | (3.9) |
Thus, for T = \tau^\theta , where \theta satisfies (3.6) and (3.8), we obtain by (3.2), (3.7), and (3.9) that
I_{f_{m-1}}^{\frac{pq-1}{p}}\leq C\tau^{-\frac{\theta(pq-1)}{p}}\left[\tau^{\theta+ N-\frac{a+2mp}{p-1}}\ln\tau\right]^{\frac{p-1}{p}}\left[\tau^{\theta+ N-\frac{b+2mq}{q-1}}\ln\tau\right]^{q-1}, |
that is,
\begin{equation} I_{f_{m-1}}^{\frac{pq-1}{p}}\leq C \tau^\delta (\ln \tau)^{\frac{pq-1}{p}}, \end{equation} | (3.10) |
where
\begin{aligned} \delta & = \frac{pq-1}{p}\left[N-\frac{(b+2mq)p+a+2mp}{pq-1}\right]\\ & = \frac{pq-1}{p} \left(N-2m-\alpha\right). \end{aligned} |
Since N-2m < \alpha , we have \delta < 0 . Then, passing to the limit as \tau\to \infty in (3.10), we reach a contradiction with I_{f_{m-1}} > 0 . This completes the proof of Theorem 1.1.
Let us introduce the family of polynomial functions \left\{P_i\right\}_{0\leq i\leq m} , where
P_i(z) = \left\{\begin{array}{llll} 1 &\mbox{if}& i = 0, \\[10pt] \prod\limits_{j = 0}^{i-1} (z+2j)\prod\limits_{j = 1}^i (N-2j-z)&\mbox{if}& i = 1, \cdots, m. \end{array} \right. |
From (1.14), we deduce that
N-2j > \max\left\{\alpha, \beta\right\}, \quad j = 1, \cdots, m. |
Furthermore, because a, b\geq -2m and (a, b)\neq (-2m, -2m) , we have \alpha, \beta > 0 . Then,
\begin{equation} P_i(z) > 0, \quad i = 0, 1, \cdots, m, \quad z\in \{\alpha, \beta\}. \end{equation} | (3.11) |
For all
\begin{equation} 0 < \varepsilon\leq \min\left\{[P_m(\alpha)]^{\frac{1}{p-1}}, [P_m(\beta)]^{\frac{1}{q-1}}\right\}, \end{equation} | (3.12) |
we consider functions of the forms
\begin{equation} u_\varepsilon(x) = \varepsilon |x|^{-\alpha}, \quad x\in \mathbb{R}^N\backslash B_1 \end{equation} | (3.13) |
and
\begin{equation} v_\varepsilon(x) = \varepsilon |x|^{-\beta}, \quad x\in \mathbb{R}^N\backslash B_1. \end{equation} | (3.14) |
Since u_\varepsilon and v_\varepsilon are radial functions, elementary calculations show that
\begin{equation} (-\Delta)^i u_\varepsilon(x) = \varepsilon P_i(\alpha)|x|^{-\alpha-2i}, \quad i = 0, 1, \cdots, m, \quad x\in\mathbb{R}^N\backslash B_1 \end{equation} | (3.15) |
and
\begin{equation} (-\Delta)^i v_\varepsilon(x) = \varepsilon P_i(\beta)|x|^{-\beta-2i}, \quad i = 0, 1, \cdots, m, \quad x\in \mathbb{R}^N\backslash B_1. \end{equation} | (3.16) |
Taking i = m in (3.15), using (3.11)–(3.14), we obtain
\begin{aligned} (-\Delta)^m u_\varepsilon(x)& = \varepsilon P_m(\alpha)|x|^{-\alpha-2m}\\ & = |x|^a \varepsilon^p |x|^{-\beta p} \left(\varepsilon^{1-p}P_m(\alpha) |x|^{-\alpha-2m-a+\beta p}\right)\\ &\geq |x|^a v_\varepsilon^{p}(x)|x|^{-\alpha-2m-a+\beta p}. \end{aligned} |
On the other hand, by (1.12) and (1.13), one can show that
-\alpha-2m-a+\beta p = 0. |
Then, we obtain
\begin{equation} (-\Delta)^m u_\varepsilon(x)\geq |x|^a v_\varepsilon^{p}(x), \quad x\in \mathbb{R}^N\backslash B_1. \end{equation} | (3.17) |
Similarly, taking m = i in (3.16), using (3.11)–(3.14), we obtain
\begin{aligned} (-\Delta)^m v_\varepsilon(x)& = \varepsilon P_m(\beta)|x|^{-\beta-2m}\\ & = |x|^b \varepsilon^q |x|^{-\alpha q} \left(\varepsilon^{1-q}P_m(\beta) |x|^{-\beta-2m-b+\alpha q}\right)\\ &\geq |x|^b u_\varepsilon^{q}(x)|x|^{-\beta-2m-b+\alpha q}. \end{aligned} |
Using that
-\beta-2m-b+\alpha q = 0, |
we obtain
\begin{equation} (-\Delta)^m v_\varepsilon(x)\geq |x|^b u_\varepsilon^{q}(x), \quad x\in \mathbb{R}^N\backslash B_1. \end{equation} | (3.18) |
Furthermore, by (3.11) and (3.15), for all i = 0, \cdots, m-1 , we have
\begin{equation} (-\Delta)^i u_\varepsilon(x) = \varepsilon P_i(\alpha) > 0, \quad x\in \partial B_1. \end{equation} | (3.19) |
Similarly, by (3.11) and (3.16), for all i = 0, \cdots, m-1 , we have
\begin{equation} (-\Delta)^i v_\varepsilon(x) = \varepsilon P_i(\beta) > 0, \quad x\in \partial B_1. \end{equation} | (3.20) |
Finally, (3.17)–(3.20) show that for all \varepsilon satisfying (3.12), the pair of functions (u_\varepsilon, v_\varepsilon) given by (3.13) and (3.14) is a stationary solution to (1.1)-(1.2) with f_i\equiv \varepsilon P_i(\alpha) and g_i\equiv \varepsilon P_i(\beta) for all i = 0, \cdots, m-1 . The proof of Theorem 1.2 is then completed.
The system of polyharmonic wave inequalities (1.1) under the inhomogeneous Navier-type boundary conditions (1.2) was investigated. First, we established a nonexistence criterium for the nonexistence of weak solutions (see Theorem 1.1). Namely, under condition (1.10), we proved that (1.1)-(1.2) possesses no weak solution, provided I_{f_{m-1}}, I_{g_{m-1}}\geq 0 and (I_{f_{m-1}}, I_{g_{m-1}})\neq (0, 0) . Next, we proved the sharpness of the obtained criterium (1.10) by showing that under condition (1.14), (1.1)-(1.2) possesses weak solutions (stationary solutions) for some f_i, g_i\in L^1(\partial B_1) ( i = 0, \cdots, m-1 ) with I_{f_{m-1}}, I_{g_{m-1}} > 0 (see Theorem 1.2). From Theorem 1.1, we deduced an optimal criterium for the nonexistence of weak solutions to the corresponding stationary polyharmonic system (1.15) under the Navier-type boundary conditions (1.16) (see Corollary 1.1).
In this study, the critical case N\geq 2m+1 ,
N = \max\left\{{\rm{sign}}(I_{f_{m-1}})\times \frac{2mp(q+1)+pb+a}{pq-1}, {\rm{sign}}(I_{g_{m-1}})\times \frac{2mq(p+1)+qa+b}{pq-1}\right\} |
is not investigated. It would be interesting to know whether there is existence or nonexistence of weak solutions in this case.
Manal Alfulaij: validation, investigation, writing review and editing; Mohamed Jleli: Conceptualization, methodology, investigation and formal analysis; Bessem Samet: Conceptualization, methodology, validation and investigation. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Mohamed Jleli is supported by Researchers Supporting Project number (RSP2024R57), King Saud University, Riyadh, Saudi Arabia.
The authors declare no conflicts of interest.
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