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Literature review and motivations
The classical Liouville theorem states that any harmonic function that is bounded throughout the entire space must be constant Initially proposed by Liouville in 1944 for double periodic functions, the theorem was later proven by Cauchy This theorem, often referred to as a Liouville-type result, indicates the non-existence of non-trivial solutions in both the entire space and half-space Over the past forty years, Liouville-type results have been extensively explored in the context of elliptic and parabolic problems.
In particular, Gidas and Spruck[29]established an optimal Liouville type the- orem for the Lane-Emden equation
Gidas and Spruck demonstrated that for \( p < p_s := N + 2N - 2 \), every non-negative solution in the entire space of the equation must be identically zero They utilized this finding, along with the rescaling technique, to establish a priori estimates for the related boundary value problem Subsequently, their results were expanded to encompass multiple harmonic operators.
Recent studies have established both the existence and nonexistence of positive super-solutions for the Lane-Emden equation, identifying a critical exponent defined as \( p_c = \frac{N}{N-2} \) (with \( p_c = \infty \) when \( N \leq 2 \)) Furthermore, the classification of stable solutions has been thoroughly addressed, culminating in an explicit calculation of the critical exponent.
For the Lane-Emden system
−∆v=u q inR N , (2) the Lane-Emden conjecture states that system (2) has no classical positive so- lution on the whole space if and only if p>0,q>0,pq>1 and
The conjecture has been validated for classical radial solutions, while for non-radial solutions in dimensions N ≤ 2, it is supported by the findings of Mitidieri and Pohozaev In three dimensions, Serrin and Zou proved the conjecture under the condition that the growth rate of solutions does not exceed polynomial degree at infinity, a limitation later lifted by Polacik, Quittner, and Souplet, confirming the conjecture for N = 3 In 2009, Souplet extended the proof to dimension N = 4 However, for dimensions N ≥ 5, the conjecture has only been established in a few isolated instances.
Liouville type theorems for parabolic equations, specifically \( u_t - \Delta u = u^p \), have been extensively researched over the past two decades The foundational Fujita result indicates that there are no nonnegative non-trivial solutions in the half-space \( \mathbb{R}^N \times \mathbb{R}^+ \) if \( 1 < p \leq 1 + \frac{N}{2} \) Subsequent studies established an optimal condition for nonnegative radial solutions, proving the Liouville type theorem for \( p < p_s = \frac{N + N - 2}{2} \) For non-radial solutions, the theorem holds under the condition \( p < p_B = \frac{N(N + (N - 1))}{2} \), although this condition is suboptimal since \( p_B < p_s \) for \( N > 1 \) In 2016, Quittner achieved the optimal Liouville type result for the case \( N = 2 \), while Bartsch, Polacik, and Quittner also contributed related findings for nodal solutions.
Liouville type theorems for parabolic systems have garnered significant interest among mathematicians recently Despite this, the Liouville type theorem for equation (3) remains unresolved, leading to a scarcity of similar results in the context of parabolic systems Notable contributions in this area include research by Escobedo and Herrero, as well as Andreucci, Herrero, and Velázquez.
[2], Merle and Zaag[43], Quittner [54], Phan and Souplet[49].
Since the foundational research by Gidas and Sprick on the Liouville type theorem related to the Lane-Emden equation, numerous significant extensions have been developed in various areas Notable results in this field include advancements that enhance our understanding of the theorem's implications and applications.
[11, 26, 62, 63] for classically stable solutions, [2, 58] for p-Laplace operator and[15, 46, 47, 68] for the Grushin operator.
Coming back to the our thesis, the first subject is to study the following equation
−∆ λ v=u p in R N , p∈R, (4) where∆ λ is a strongly degenerate operator of the form
, andλ i satisfies some of the conditions given in the Chapter 1.
Let us now consider the special case whereλ i =1 fori =1, 2, ,N Then, the problem (4) reduces to
Recent studies by mathematicians have focused on system (5), highlighting significant findings in the field For values of p less than or equal to 1, it has been demonstrated that super-solutions do not exist, as noted in reference [2] Additionally, for p greater than 1, research confirms the existence of positive classical solutions for both the problem (5) and the biharmonic problem.
∆ 2 u=u p (6) are equivalent, see [31, 63, 65] The nonexistence of positive solutions of (6) is proved in the cases N ≤4 and p>1 WhenN ≥5, this result holds true for
1 < p < N+ N − 4 4 , see also [63, 64] The classification of stable positive solutions were studied in [11, 31, 65] In particular, it was shown in [65] that (6) has no stable positive solution with N ≤8, in [11] with N ≤ 10 and in [31] with
N ≤12, (see also[34]) The condition N ≤12 is optimal due to the existence of stable radial solutions whenN ≥13 and p≥p J L , see[35].
In recent years, there has been a significant focus on the study of elliptic equations and systems that involve degenerate operators, particularly the Grushin operator This growing interest highlights the importance of understanding the properties and implications of degenerate operators in mathematical analysis.
The Grushin operator serves as a prime example of the ∆λ-Laplacian, with significant properties detailed in reference [36] Monticelli demonstrated the nonexistence of nontrivial nonnegative classical solutions for the equation involving the Grushin operator, denoted as -Gαu up in RN Following this, Yu established Liouville-type theorems for nonnegative weak solutions of the same equation Recent studies have further confirmed nonexistence results for stable solutions of elliptic equations and systems that incorporate the Grushin operator or the ∆λ-Laplacian.
To date, there has been no research addressing the system involving degenerate operators, primarily due to the absence of a spherical mean formula for the sub-elliptic operator ∆λ, which complicates the application of ODE techniques Drawing inspiration from findings related to the Laplace operator, it is conjectured that the system has no positive super-solutions for p ≤ 1 This thesis aims to resolve this conjecture and examines the nonexistence of stable solutions when p > 1.
The second subject of this thesis is to consider the following equation
−M(∥∇ λ u∥ 2 )∆ λ u=w(x)f(u) in R N , (7) where M is a continuously differentiable monotone function,
R N |∇ λ u| 2 dx and w is a weight function.
Let us first recall that in the caseλ i =1, M =1, andw=1, the problem (7) becomes
−∆u= f(u)inR N (8) When f(u)is of the form|u| p− 1 u, e u or−u −p , p>1, the classification of stable solutions to (8) has been established in[17, 22, 26, 27, 41].
We now consider the case M =1 and w=1, (7) becomes
The nonexistence of positive solutions to the equation −∆ λ u= f(u) in R N has been established, particularly involving the Grushin operator, as demonstrated in [68] through the application of the moving plane method For insights on the nonexistence of stable solutions to this equation, readers are directed to references [19] and [56].
In the case M(t) =a+bt, a>0,b≥0, the problem (7) reduces to
The equation \((a+b\| \nabla \lambda u \|^2) \Delta \lambda u = w(x)f(u)\) in \(\mathbb{R}^N\) has been extensively studied, particularly through the pioneering work of Lions, who introduced a functional analysis approach to this problem For recent advancements related to this equation involving the Laplace operator, readers are directed to several key papers, including references [3, 4, 25, 38, 48, 66].
Recently, based on Farina’s approach [26] and the technique in[13], Yun- feng Wei and collaborators[66]assume that the weightw(x)satisfies a power lower bound near infinity, that is,
(W) "The function w ∈ C(R N ) is non-negative and there exist constants α >
In the study presented in [66], the authors have proven the nonexistence of stable solutions for the equation (10), which includes the Laplace operator, as articulated below.
Theorem A.([66])"Letλ i =1, i=1, 2, ,N.Suppose that the weight w satisfies the condition (W) and one of the following conditions are satisfied
Then, there is no stable solution of (10)with f(u) =e u "
The results introduced in[66]have been expanded and generalized in[67] to include Kirchhoff equations that incorporate Grushin operators.
To the best of our knowledge, there have been no established results in- dicating the nonexistence of solutions for (7) in general cases involving the
∆ λ -operator This thesis aims to address this gap by investigating the nonexis- tence of stable solutions for (7), with a specific focus on examining f(u)in the forms of−u − p ,e u , andu p
The third subject in this thesis is concerned with the following parabolic equations u t −∆ λ u+aã ∇ λ u=u p inR N ìR, (11) and the parabolic system
, and a is a smooth vector field satisfying some growth condition at infinity.
The elliptic counterpart of (11) is given by
−∆ λ u+aã ∇ λ u=u p in R N , (13) and of (12) is given by
In a special case,a =0 and λ i =1, (13) becomes the celebrated Lane-Emden equation
Ifλ i =1 and a̸=0, the equation (13) becomes
Some results for this equation have been mentioned above.
Recent years have seen significant attention on this equation, with notable references including [6, 13, 14, 32, 33] Research has focused on positive stable solutions, particularly the Liouville type results for p > 1, as discussed in [13, 14] Additionally, the nonexistence of positive super-solutions in exterior domains has been established in [6, 32] A key theorem was also proved in [32].
Theorem B.(Hara[32])"Assume that sup x ∈R N
Then the equation(14)has no positive super-solutions provided p≤ N−2 N "
The optimal nonexistence range is defined as p ≤ N - N/2 with a = 0 Additionally, it has been highlighted in reference [32] that Theorem B is also precise for the class of positive weak super-solutions in exterior domains Recently, significant developments regarding Theorem B have emerged.
B has been generalized in[18]where the advection terma belongs to a larger class of vector fields.
We next consider the parabolic equation (11) in the case λ i =1 and a =0, i.e. u t −∆u=u p in R N ×R (16)
The nonexistence of positive solutions is conjectured to be true in the range
1< p < N+ N − 2 2 So far, this conjecture has been only proved for dimension N ≤
2, see e.g., [7, 54] and also [5, 51, 53, 55] Very recently, the classification of positive super-solutions to (16) has been completely solved in[21] where the nonexistence result is proved in the range of the exponent p∈(−∞, 1)∪
Theorem C (Duong-Phan[21]) "The equation(16)has no positive super-solutions if and only if p 0 defined above."
In this subsection, we present some examples of∆ λ -Laplace.
G α :=∆ x +|x| 2α ∆ y , whereα≥0,N α :=N 1 +(1+α)N 2 is called the homogeneous dimension to the
Example 1.1.2 "We putR N as follows
,x (i) ∈R N i ,i=1, ,k, the point ofR N Let∆ (i) be the classical Laplace operator in R N i
Given a muti-indexα= (α 1, ,α r− 1)α j ≥1,j=1, ,r−1, we put
1 (i) ,i =1, ,r A group of dilation is given by δ t :R N −→R N , δ t x (1) , ,x (r)
(1.9) withε 1=1 andε i =α i − 1 ε i − 1+1,i=2, ,k In particular, ifα 1= =α k − 1 1, the operator (1.8) and the dilation (1.9) become, respectively
Example 1.1.3 We write R N = R N 1 ìR N 2 Suppose that à : R N 1 −→ R is continuous, of class C 1 and strictly positive outside the coordinate axis In addition, we suppose thatà t x (1)
, forα >0, x (1) ∈R N 1 and t >0. Then, if we putλ= 1 ( 1 ) ,à1 ( 2 )
This operator satisfies(H1) with respect to the dilation δ t :R N −→R N , δ t x ( 1 ) ,x ( 2 )
The class of the operators in (1.10) obviously containsG α in Example 1.1.1, as well as see[61]
4|x (1) |, then the operators∆ λ in (1.10) is of the form
Some auxiliary results
In this section, we present some auxiliary results.
Lemma 1.1 "(Young’s inequality) Let a,b be non-negative real numbers, p,q be positive real numbers, greater than 1 such that 1 p +1 q =1 Then a b≤ a p p + b q q
Equality holds if and only if a p = b q ”
Lemma 1.2 "(Hửlder’s inequality) Let f ∈ L p (Ω) and g ∈ L q (Ω) be integral functions over the domainΩ⊂R N and p,q>1satisfying 1 p + 1 q =1.Then
Lemma 1.3 (Lemma about polynomials) Let p>1be a constant and γ 0:v t 2p p+1+ v u t 2p p+1− v t 2p p+1.
Then there exists only a number t max that is a solution of P in[2γ 0,+∞) Fur- thermore, t max > 5(p p+ − 1 1)
Proof The proof can be found in [31] Here we present the details for the completeness We have
From the definition ofγ 0, we get γ 2 0
Moreover, for all p>1,t ≥2γ 0, one has
≥48(p+1) 2p p+1−64p2p>0 in [2γ 0,+∞) with γ 2 0 ≥ 2p p+1 Then, we get that P is a convex function in
Since lim s →∞ P(t) =∞and P(2γ 0) 0 When we substitute this type of solution into problem (1.18) and let t → ∞, we obtain
(1.19) where we denote u = u k,l and v = v k,l to emphasize the dependence of the solution(u,v)on(k,l), and the matrix M from the linearization has entries a(x) = f u (x,u k,l ,v k,l ), b(x) = f v (x,u k,l ,v k,l ), c(x) = g u (x,u k,l ,v k,l ), d(x) = g v (x,u k,l ,v k,l ).
The linearized problem (1.19) has a positive eigenvalueηand a positive eigen- function (ϕ,ψ) associated with it We can define a general concept based on the ideas explained above.
A solution (u,v) of the equation is deemed stable when the first eigenvalue η 1 from the corresponding eigenvalue problem is positive and is associated with a positive eigenfunction Notably, η 1 is the sole eigenvalue that possesses a positive eigenfunction.
Inspired by the above analysis, one defines the stability of solution of theLane-Emden system as follows.
Recall that the Lane-Emden system is of form
A positive solution(u,v)of (1.20) is called stable if only if there exists a pair of functions(ξ,η)∈C ∞ (R N )such that
Now, with a stable positive solution (u,v) of (1.20), we have the following stable inequality evaluation.
Lemma 1.4 Let(u,v)be a stable positive solution of (1.20)Then,
Proof Suppose(u,v) is a stable positive solution of (1.20) and there are two functionξ,χ∈C 2 (R) such that
We next show that for any test-functionφ and ψ∈C c 2 (R N ), there holds
Combining the above two expressions, we get
2ặ pq v p − 1 u q − 1 φψ≤ χ ξ pv p− 1 φ 2 + ξ χ qu q− 1 ψ 2 (1.23) Inequalities (1.22) and (1.23) lead to
Chooseφ=ψ, the proof is complete.
This chapter explores the classification of solutions for a degenerate system characterized by the ∆ λ -Laplacian It begins by demonstrating that positive super-solutions do not exist when the exponent is less than or equal to 1 Additionally, it presents a Liouville-type theorem applicable to stable positive solutions within this context.
Main content of this chapter is written based on paper [P1] in the List of publications.
Problem setting and results
In this chapter, we are concerned with the following system
−∆ λ v=u p inR N , with u>0, v>0, (2.1) where p is a real number and∆ λ is of the form
, (2.2) where x = (x 1 , ,x N ) ∈ R N and the functions λ i :R N → R satisfy the con- ditions (H1), (H2), ( ˜H3)in Chapter 1 Notice that, in the case λ i =1, (2.1) becomes
The Lane-Emden system, a significant focus of research in various fields, particularly in physics and geometry, serves as a foundational framework for understanding complex phenomena Specifically, when N = 3, the Lane-Emden equation is essential for analyzing stellar structures in astrophysics, as referenced in studies [26, 29, 57, 59] Additionally, for N = 3 and p = (N + 2) / (N - 2), this equation plays a vital role in addressing conformal geometry issues, such as the prescribed scalar curvature problem.
In this chapter, we investigate some Liouville properties for positive super- solutions and stable positive solutions of the system (2.1).
First, we give the definition of positive super-solutions
Definition 2.1.A positive super-solution(u,v)of (2.1) means(u,v)∈C 2 (R N )×
The first result is the following.
Theorem 2.1 Under the hypotheses (H1) and (H2), the system (2.1) has no positive super-solution provided p≤1.
Notice that, in the caseλ i =1, (2.1) becomes
For p≤1, the nonexistence of super-solutions of (2.3) was proved in[2].
Recall that the Grushin operator is a typical example of∆ λ operator and is given by
Then we have the following corollary.
−G α v =u p inR N =R N 1 ×R N 2 has no positive super-solution provided p≤1.
According to Theorem 2.1, we focus exclusively on the case where the exponent p is greater than 1 In this context, we will categorize stable positive solutions to equation (2.1) Before presenting our second result, it is important to revisit the definition of stability, which is influenced by sources [11, 20, 45].
Definition 2.2 A positive solution (u,v) of (2.1) means (u,v) ∈ C 2 (R N )×
Definition 2.3 A positive solution(u,v)∈C 2 (R N )×C 2 (R N )of (2.1) is called stable if there exist two positive functionsξandηbelongingC 2 (R N )such that
−∆ λ η=pu p− 1 ξ. The second result is as follows.
Theorem 2.2 Assume that (H1), (H2) and( ˜ H3) hold and p >1 Let t max be the largest root of the polynomial
Then, the problem (2.1)has no stable positive solutions provided that the homo- geneous dimension Q satisfies
Remark 2.1 This result is a natural extension of[31, Theorem 1.1]for the case of Laplace operator On the other hand, as shown in[31], the right hand side of (2.4) is always larger than 12, then the system (2.1) has no stable positive solutions providedQ≤12.
A direct consequence of Theorem 2.2 is the following.
Corollary 2.2 Assume that p> 1.Let t max be given in Theorem 2.2 Then, the problem
−G α v=u p in R N has no stable positive solution provided that the homogeneous dimension N α N 1 + (1+α)N 2 associated to the Grushin operator satisfies
LIOUVILLE TYPE THEOREMS FOR KIRCHHOFF ELLIPTIC EQUA-
Problem formulation
In this chapter, our focus lies on the following equations
The nonlinearity f(u) is of the form u p ,e u or −u − p and M(ã):R + → R + is a monotone function such that γ:=sup t ≥ 0 §t M ′ (t)
The weight functionwis nonnegative and satisfies w(x) =O |x| α λ as|x| λ → ∞ for someα≥0.
Here∆ λ is of the form
, (3.3) where x = (x 1 , ,x N )∈R N and the functionsλ i :R N →Rsatisfy the condi- tions(H1)-(H2)in Chapter 1.
Recall that Q denotes the homogeneous dimension of R N concerning the dilation group{δ t } t> 0, given by
Q=ε 1+ .+ε N , where ε i represents the homogeneous degree of λ i concerning the dilation group{δ t } t>0 (refer to Chapter 1).
Whenλ i =1 and M =a+bt, we can see that (3.1) becomes the following
∆u= f(u) inR N , (3.4) which has been extensively studied in the last decade.
The equation (3.4) is related to the stationary analogue of the Kirchhoff equation u t t − a+b
∆u= f(x,u) withΩ⊂R N bounded domain, which was proposed by Kirchhoff in 1883 as an extension of the classical D’Alembert’s wave equation ρu t t −
The equation |u x | 2 d x u 2 x x = f(x,u) describes the free vibrations of elastic strings, incorporating Kirchhoff’s model, which factors in the length changes of the string due to transverse vibrations In this context, L represents the string's length, h denotes the cross-sectional area, E is the Young's modulus of the material, ρ indicates the mass density, and P 0 signifies the initial tension.
In this chapter, we study some Liouville properties for stable solutions of the equation (3.1).
Nonexistence of stable solutions
This section focuses on studying the nonexistence of positive stable solutions of the equation (3.1) in the case of negative exponents, exponential nonlinear- ity, or polynomial nonlinearity.
Now we recall condition (W) which is given in Introduction.
(W) "The function w ∈ C(R N ) is non-negative and there exist constants α >
The definitions of a solution and a stable solution in (3.1) are as follows
Definition 3.2 A solutionuof (3.1) is stable if
The first result is concerned with the case of negative exponents.
Theorem 3.1 Let Q>2, p>0and f(u) =−u −p Suppose in addition that(W) and(3.2)hold Then, the equation(3.1)has no positive stable solutions provided that
The second result is the case of exponential nonlinearity.
Theorem 3.2 Let f(u) =e u Suppose that(W)and(3.2)holds and
Then, the equation(3.1)has no stable solutions.
The third result is concerned with the case of polynomial nonlinearity.
Theorem 3.3 Let Q > 2, p> 1+2γ and f(u) = u p Suppose in addition that (W) and (3.2) hold Then, the equation (3.1) has no positive stable solutions provided that
In the caseλ i (x) =1 for all 1≤i ≤N and Kirchhoff function M(t) =a+bt witha>0,b≥0, we investigate the Kirchhoff equations
We have the following corollaries
Corollary 3.1 "Assume that(W)and one the following conditions are satisfied
Then, there is no stable solution of (3.7)."
Corollary 3.2 "Assume that(W)and one of the following conditions are satisfied
Then there is no nontrivial stable solution of (3.8)."
These results are direct consequences of Theorem 3.2 and Theorem 3.3 and has been demonstrated in detail in[66].
Especially, in the case M(t) = a+ bt+c t 2 with a,b,c >0, it is no hard to seeγ=2 Then, we obtain following corollaries.
Corollary 3.3 Under the hypothesis of Theorem 3.1 and M(t) = a+bt+c t 2 with a,b,c>0,the equation(3.1)has no positive stable solution provided that
Corollary 3.4 Under the hypothesis of Theorem 3.2, M(t) = a+bt+c t 2 with a,b,c >0,
5(α+2), the equation(3.1)has no stable solution.
Corollary 3.5 Under the hypothesis of Theorem 3.3 and M(t) = a+bt+c t 2 with a,b,c>0,the equation(3.1)has no positive stable solutions provided that
LIOUVILLE TYPE THEOREMS FOR DEGENERATE PARABOLIC
Problem setting and results
In this article, we explore the conditions under which a continuously differentiable vector field \( a \) and a real number \( p \) lead to the nonexistence of positive super-solutions for the parabolic equation \( u_t - \Delta_\lambda u + a \cdot \nabla_\lambda u = u^p \) in \( \mathbb{R}^N \times \mathbb{R} \) Additionally, we examine the implications of these findings on the associated parabolic system.
u t −∆ λ u+aã ∇ λ u=v p v t −∆ λ v+aã ∇ λ v=u q in R N ×R, (4.2) where p,q∈R N and ∆ λ is of the form
Herex = (x 1 , ,x N )∈R N and the functionsλ i :R N →Rsatisfy the conditions
The elliptic counterpart of (4.1) is given by
−∆ λ u+aã ∇ λ u=u p in R N (4.4) Whenλ i =1 and a̸=0, (4.4) becomes
The equation −∆u + a · ∇u = u^p in ℝ^N involves a smooth, divergence-free vector field 'a' with the condition |a(x)| ≤ 1 + |εx|, where ε is sufficiently small In a previous study, Cowan classified stable positive solutions to this equation using a combination of the test function method and the generalized Hardy inequality Notably, this equation is relevant in the context of elliptic systems with gradient terms, which are significant in electrochemical models and various applications in fluid dynamics.
In this chapter, we prove some Liouville properties of positive super-solutions of (4.1) and (4.2).
4.1.2 Nonexistence of positive super-solutions
First, we give the definition of positive super-solutions
Definition 4.1 A positive super-solution of (4.1) is defined as a positive func- tionu∈C 2,1 (R N ×R)that satisfies, point-wise in R N ×R, u t −∆ λ u+aã ∇ λ u≥u p
Motivated by recent advancements in the study of elliptic and parabolic equations of Lane-Emden type [13, 18, 20, 21], we establish a Liouville type theorem for the class of positive super-solutions of (4.1).
Theorem 4.1 Suppose that the assumptions (H1) and (H2) hold Assume in addition that∇ λ ãa=0and there exists a constantθ 1andmax¦ (p+1) pq− 1, (q+1) pq− 1 ©
Remark that we also exclude the case p,q>0,pq=1 since in this case the system has a positive super-solutionu= pβ 1 e pβt ,v=e βt withβ =p − q q +1.
In the case where a = 0, the relationship ∆ λ ≡ ∆ reveals that our theorem extends previous findings in [21] Notably, our results are broader than those presented in [18], which indicate that the function is independent of the critical exponent In contrast, we have established that the function does indeed influence the critical exponent.
In the case∆ λ ≡G α , we get the result as follow
Corollary 4.2 Under the hypothesis of Corollary 4.2, the problem (4.2)has no positive super-solution if(p,q)satisfies one of the following condition
(iii) p,q>0,pq>1andmax¦ p+ 1 pq−1, pq−1 q+ 1 ©
Proof of nonexistence results
4.2.1 Nonexistence of positive super-solutions of the equation
To prove Theorem 4.1, we employ the rescaled test-function method for the case
The method becomes more complex due to the presence of the degenerate operator ∆ λ, necessitating the selection of suitable rescaled test functions Notably, when p = 1, a positive solution of the equation (4.1) can be expressed as u = e^t Additionally, for the case where p < 1, we establish a maximum principle akin to those utilized in previous studies [10, 20, 21] to tackle issues related to degenerate operators and advection processes.
The rest will be used to present the details of the proof.
Proof The proof of Theorem 4.1 is divided into two cases.
Assume that u is a positive super-solution of (4.1) with p < 1 Using a change of variable v= 1 u >0 and combining with some computation, one gets
B r ={(x,t)∈R N ×R;|x i |0, define ψ r (x,t) =ψ m x 1 r ε 1 , x 2 r ε 2 , , x N r ε N , t r 2
, where mis a positive parameter Suppose, to the contrary, that uis a positive super-solution of (4.1) Using the weak form of (4.1) with the test function ψ r (x,t), we obtain
By using the fact that∇ λ ãa=0 and an integration by parts, we arrive at
As in the first case, given the assumptions on the functionsλ i , we derive
In addition, the behavior of the advection terms implies that
Combining these estimates with (4.13) , we deduce that
Subsequently, by applying the Hửlder inequality to the right-hand side of (4.14), we obtain
We now choose the parameter m= p− 2p 1 or equivalently (m − m 2)p = 1 Thus, we deduce from (4.15) that
It follows from (4.16) and (4.14) that
When p < Q+ 2 − min Q+2 ( 2,1 −θ) , the exponent in the right hand side of (4.14) is negative Thus, by letting r→ ∞in (4.17), we deduce that
Asuis a positive function, this leads to a contradiction.
We finally consider the borderline case p= Q+2 − min(2,1 Q + 2 −θ ) Then (4.17) yields
R N ×R u p dx dt0 The proof is complete.
4.2.2 Nonexistence of positive super-solutions of the system
To demonstrate Theorem 4.2, we implement reductions that convert system (4.2) into the scalar equation (4.1), subsequently applying Theorem 4.1 to achieve the desired outcome This methodology is inspired by the work of Duong and Phan[21] We will initially examine the cases where p=0 or q=0 The resulting systems, u_t - ∆_λ u + a∇_λ u = 1 or v_t - ∆_λ v + a∇_λ v = 1, do not possess positive super-solutions, as established by Theorem 4.1.
Next, it is sufficient to consider the case where p̸=0 and q̸=0 and p≥q.
We now deal with the case of negative exponents.
Lemma 4.1 Under the assumption of Theorem 4.1, the system(4.2)has no pos- itive super-solutions provided p1 It is easy to verify that sm
From (4.18) and (4.19) we get w t −∆ λ w+aã ∇ λ w≥C w −s which has no positive super-solutions according to Theorem 4.1.
The next lemma completes the proof of Theorem 4.2.
Lemma 4.2 Suppose that the hypothesis of Theorem 4.1 holds If p and q satisfy one of the following conditions
(iii) p,q>0,pq>1andmax¦ 2 ( p + 1 ) pq−1 , 2 pq−1 ( q + 1 ) ©
>N −2; then the system(4.2)has no positive super-solutions.
Proof We also prove this lemma by contradiction Suppose that(u,v)is a posi- tive super-solution of (4.2) We introduce an auxiliary functionw=u κ v k where κ,k>0 andκ+k=1 A simple computation yields
Applying the Young inequality to the right hand side of (4.21) and using the fact that(1−κ)(1−k) =κk, one obtains
∇ λ w=κ∇ λ uu κ− 1 v k +k∇ λ v v k− 1 u κ (4.23) and w t =αu t u κ− 1 v k +kv t v k − 1 u κ Therefore, combining this, (4.22) and (4.23), one has w t −∆ λ w+aã ∇ λ w≥κ(u t −∆ λ u+aã ∇ λ u)u κ− 1 v k
Since(u,v)is a super-solution of (4.2), the estimate (4.24) yields w t −∆ λ w+aã ∇ λ w≥κu κ− 1 v p+k +kv k − 1 u q+κ
Again, the Young inequality implies that κv p u +ku q v ≥C v p u m 1 u q v m m −1
, (4.26) where C >0 independent ofu,v and m>1 will be chosen as below It results from (4.25) and (4.26) that w t −∆ λ w+aã ∇ λ w≥Cu κ− m 1 +q m−1 m v k− m−1 m + m p (4.27)
We select the constant \( m \) defined by the equation \( m = 1 + \kappa p + k k q + \kappa \) This leads to the relationship \( \kappa - m \frac{1 + q}{m(m - 1)} k - m - \frac{m}{1 + m} p = \kappa k \) Notably, \( m \) is greater than 1 if \( p \) and \( q \) meet conditions (ii) or (iii) Conversely, if \( p \) and \( q \) fulfill condition (i), we can choose \( \kappa \) slightly less than 1 and \( k \) slightly greater than 0, ensuring that \( m \) remains greater than 1.
Plugging (4.28) into (4.27), there holds w t −∆ λ w+aã ∇ λ w≥C w 1+ pq −1 κ( p +1)+ k ( q +1) (4.29)
However, if p and q satisfy (i) or (ii) or (iii), then we chooseκ close to 1 such that
Q+2−min(2, 1−θ).According to Theorem 4.1, (4.29) has no positive super-solutions We obtain a contradiction and finish the proof.
This chapter investigates the absence of positive super-solutions for a degenerate parabolic equation that includes advection terms The study employs the test-function method and introduces a maximum principle to address challenges associated with advection and degenerate operators.
We demonstrate the nonexistence of a positive super-solution for a degenerate parabolic system with advection terms by employing specific reductions that transform the system into a scalar equation Our findings yield significant results in understanding this mathematical framework.
• The nonexistence of positive super-solution for a degenerate parabolic equation with advection terms in the case
Q+2−min(2, 1−θ) and, in the case of p 1, N ≥ 3 and the advection term a(x) is a smooth vector field.
4 Establish some Liouville type theorems for some other elliptic or parabolic equation/systems involving the∆ λ - Laplacian such as the Ginzburg-Landau system
(P1) Anh Tuan Duong, Trung Hieu Giang, Phuong Le and Thi Hien Anh Vu
(2021), "Classification results for a sub-elliptic system involving the∆ λ - Laplacian",Mathematical Methods in the Applied Sciences, volume 44, issue
5, pp 3615-3629, https://doi.org/10.1002/mma.6968 (SCIE-Q2).
(P2) Thi Thu Huong Nguyen, Dao Trong Quyet and Thi Hien Anh Vu (2023),
"Liouville type theorems for Kirchhoff elliptic equations involving ∆ λ - operators",Topological Methods in Nonlinear Analysis, https://doi.org/10.12775/TMNA.2022.071 (SCIE-Q2).
(P3) Vu Trong Luong, Duc Hiep Pham and Thi Hien Anh Vu (2020), "Liouville type theorems for degenerate parabolic systems with advection terms",
Journal of Elliptic and Parabolic Equations, issue 6, pp 871–882, https://doi.org/10.1007/s41808-020-00086-6 (Scopus-Q2).
[1] Anh, Cung The and My, Bui Kim (2016), “Liouville type theorems for ellip- tic inequalities involving the ∆ λ -Laplace operator”,Complex Var Elliptic Equ., 7, 1002–1013.
[2] Armstrong, S N., and Sirakov, B (2011), “Nonexistence of positive su- persolutions of elliptic equations via the maximum principle”, Comm. Partial Differential Equations, 11, 2011–2047.
[3] Autuori, G., and Pucci, P (2010), “Kirchhoff systems with nonlinear source and boundary damping terms”, Commun Pure Appl Anal., 9,
[4] Autuori, G., and Pucci, P and Salvatori, M C (2009), “Asymptotic stabil- ity for nonlinear Kirchhoff systems”,Nonlinear Anal Real World Appl., 2,
[5] Bartsch, T., and Polácik, P., and Quittner, P (2009), “Liouville type the- orems and asymptotic behavior of nodal radial solutions of semilinear heat equations”,J Eur Math Soc (JEMS), 1, 219–247.
[6] Berestycki, H., and Hamel, F., and Rossi, L (2007), “Liouville type results for semilinear elliptic equations in unbounded domains”,Ann Mat Pura Appl., 3, 469–507.
[7] Bidaut-Véron, M-F (1998), “Initial blow-up for the solutions of a semi- linear parabolic equation with source term”,Équations aux dérivées par- tielles et applications, 189–198.
[8] Capuzzo Dolcetta, I., and Cutri, A (1997), “On the Liouville property for sublaplacians”, Annali della Scuola Normale Superiore di Pisa - Classe diScienze, 25, 239–2567.
[9] Cauchy, A (1844), “Mémoires sur les fonctions complémentaires”,C R. Acad Sci Paris, 19, 1377–1384.
[10] Cheng, Z., and Huang, G., and Li, C (2016), “On the Hardy-Littlewood- Sobolev type systems”,Commun Pure Appl Anal., 6, 2059–2074.
[11] Cowan, C (2013), “Liouville theorems for stable Lane-Emden systems with biharmonic problems”,Commun Pure Appl Anal., 8, 2357–2371.
[12] Cowan, C (2014), “A Liouville theorem for a fourth order Hénon equa- tion”,Adv Nonlinear Stud., 3, 767–776.
[13] Cowan, C (2016), “Stability of entire solutions to supercritical elliptic problems involving advection”, Nonlinear Anal., 1, 1–11.
[14] Cowan, C., and Fazly, M (2012), “On stable entire solutions of semi- linear elliptic equations with weights”, Proc Amer Math Soc., 6, 2003–
[15] D’Ambrosio, L., and Lucente, S (2003), “Nonlinear Liouville theorems for Grushin and Tricomi operators”,J Differential Equations, 2, 511–541.
[16] de Figueiredo, D G., and Felmer, P L (1994), “A Liouville-type theorem for elliptic systems”,Ann Scuola Norm Sup Pisa Cl Sci (4), 3, 387–397.
[17] Du, Y., and Guo, Z (2009), “Positive solutions of an elliptic equation with negative exponent: stability and critical power”,J Differential Equations,
[18] Duong, Anh Tuan (2019), “On the classification of positive supersolutions of elliptic systems involving the advection terms”, J Math Anal Appl.,
[19] Duong, Anh Tuan and Nguyen, Nhu Thang (2017), “Liouville type the- orems for elliptic equations involving Grushin operator and advection”,
Electron J Differential Equations, Paper No 108, 11.
[20] Duong, Anh Tuan and Phan, Quoc Hung (2019), “Liouville type theorem for nonlinear elliptic system involving Grushin operator”,J Math Anal. Appl., 2, 785–801.
[21] Duong, Anh Tuan and Phan, Quoc Hung (2019), “Optimal Liouville type theorems for a system of parabolic inequalities”,Communications in Con- temporary Mathematics, 1950043.
[22] Dupaigne, L., and Farina, A (2010), “Stable solutions of−∆u= f(u)in
[23] Dupaigne, L (2011), “Stable solutions of elliptic partial differential equa- tions”,Chapman & Hall/CRC, Boca Raton, Chapman & Hall/CRC Mono- graphs and Surveys in Pure and Applied Mathematics, FL, Vol 143.
[24] Escobedo, M., and Herrero, M A (1991), “Boundedness and blow up for a semilinear reaction-diffusion system”, J Differential Equations, 1,
[25] Fan, H., and Liu, X (2015), “Positive and negative solutions for a class of Kirchhoff type problems on unbounded domain”,Nonlinear Anal., 186–
[26] Farina, A (2007), “On the classification of solutions of the Lane-Emden equation on unbounded domains of R N ”, J Math Pures Appl (9), 5,
[27] Farina, A (2007), “Stable solutions of −∆u = e u on R N ”, C R Math. Acad Sci Paris, 2, 63–66.
[28] Felmer, P., and Quaas, A (2011), “Fundamental solutions and Liouville type theorems for nonlinear integral operators”, Adv Math., 3, 2712–
[29] Gidas, B., and Spruck, J (1981), “Global and local behavior of positive solutions of nonlinear elliptic equations”, Comm Pure Appl Math., 4,
[30] Guo, Y., and Liu, J (2008), “Liouville type theorems for positive solutions of elliptic system inR N ”,Comm Partial Differential Equations, 1-3, 263–
[31] Hajlaoui, H., and Harrabi, A., and Ye, D (2014), “On stable solutions of the biharmonic problem with polynomial growth”,Pacific J Math., 1,
[32] Hara, T (2017), “Liouville theorems for supersolutions of semilinear el- liptic equations with drift terms in exterior domains”, J Math Anal. Appl., 1, 601–618.
[33] Hu, L-G (2018), “Liouville type theorems for stable solutions of the weighted elliptic system with the advection term: p ≥ ϑ >1”, NoDEA Nonlinear Differential Equations Appl., 1, Art 7, 30.
[34] Hu, L-G., and Zeng, J (2016), “Liouville type theorems for stable solu- tions of the weighted elliptic system”,J Math Anal Appl., 2, 882–901.
[35] Karageorgis, P (2009), “Stability and intersection properties of solutions to the nonlinear biharmonic equation”,Nonlinearity, 7, 1653–1661.
[36] Kogoj, A E., and Lanconelli, E (2012), “On semilinear∆ λ -Laplace equa- tion”,Nonlinear Anal., 12, 4637–4649.
[37] Kogoj, A E., and Lanconelli, E (2018), “Linear and semilinear problems involving ∆ λ -Laplacians”, In Proceedings of the International Conference
“Two nonlinear days in Urbino 2018", vol 25 of Electron J Differ Equ.
Conf., Texas State Univ.–San Marcos, Dept Math., San Marcos, TX, 167– 178.
[38] Le, Phuong and Huynh, Nhat Vy and Ho, Vu (2019), “Classification results for Kirchhoff equations inR N ”,Complex Var Elliptic Equ., 7, 1146–1157.
[40] Luyen, Duong Trong and Tri, Nguyen Minh (2015), “Existence of solu- tions to boundary-value problems for similinear ∆ γ differential equa- tions”,Math Notes, 1-2, 73–84.
[41] Ma, L., and Wei, J C (2008), “Properties of positive solutions to an el- liptic equation with negative exponent”,J Funct Anal., 4, 1058–1087.
[42] Merle, F., and Zaag, H (1998), “Optimal estimates for blowup rate and behavior for nonlinear heat equations”,Comm Pure Appl Math., 2, 139–
[43] Merle, F., and Zaag, H (2000), “A Liouville theorem for vector-valued nonlinear heat equations and applications”,Math Ann., 1, 103–137.
[44] Mitidieri, E., and Pohozaev, S I (2001), “A priori estimates and the ab- sence of solutions of nonlinear partial differential equations and inequal- ities”,Tr Mat Inst Steklova, 3, 1–384.
[45] Montenegro, M (2005), “Minimal solutions for a class of elliptic sys- tems”,Bull London Math Soc., 3, 405–416.
[46] Montenegro, M (2010), “Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators”, J Eur Math. Soc (JEMS), 3, 611–654.
[47] Morbidelli, D (2009), “Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics”, Israel J Math., 379–
[48] Nie, J., and Wu, X (2012), “Existence and multiplicity of non-trivial so- lutions for Schrửdinger-Kirchhoff-type equations with radial potential”,
[49] Phan, Quoc Hung and Souplet, P (2016), “A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems”,
[50] Pinsky, R G (1997), “Existence and nonexistence of global solutions for u t =∆u+a(x)u p inR d ”,J Differential Equations, 1, 152–177.
[51] Poláˇcik, P., and Quittner, P (2006), “A Liouville-type theorem and the de- cay of radial solutions of a semilinear heat equation”,Nonlinear Anal., 8,
[52] Poláˇcik, P., and Quittner, P., and Souplet, P (2007), “Singularity and decay estimates in superlinear problems via Liouville-type theorems I Elliptic equations and systems”,Duke Math J., 3, 555–579.
[53] Poláˇcik, P., and Quittner, P., and Souplet, P (2007), “Singularity and de- cay estimates in superlinear problems via Liouville-type theorems II. Parabolic equations”,Indiana Univ Math J., 2, 879–908.
[54] Quittner, P (2016), “Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure”,Math Ann., 1-2, 269–292.
[55] Quittner, P., and Souplet, P (2007), “Superlinear parabolic problems”,
Birkhọuser Advanced Texts: Basler Lehrbỹcher [Birkhọuser Advanced Texts: Basel Textbooks], Blow-up, global existence and steady states, xii+584.
[56] Rahal, B (2018), “Liouville type theorems with finite Morse index for semilinear ∆ λ -Laplace operators”, NoDEA Nonlinear Differential Equa- tions Appl., 3, Art 21, 19.
[57] Serrin, J., and Zou, H (1996), “Non-existence of positive solutions of Lane-Emden systems”,Differential Integral Equations, 4, 635–653.
[58] Serrin, J., and Zou, H (1996), “Cauchy-Liouville and universal bound- edness theorems for quasilinear elliptic equations and inequalities”,Acta Math., 1, 79–142.
[59] Souplet, P (2009), “The proof of the Lane-Emden conjecture in four space dimensions”,Adv Math., 5, 1409–1427.