Một số định lí kiểu liouville cho các bài toán elliptic và parabolic suy biến phi tuyến

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION VU THI HIEN ANH LIOUVILLE TYPE THEOREMS FOR NONLINEAR DEGENERATE ELLIPTIC AND PARABOLIC PROBLEMS DISSERTATION OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Speciality: Code: Differential and Integral Equations 46 01 03 Under the guidance of Associate Professor DUONG Anh Tuan Associate Professor PHAN Quoc Hung Ha Noi, 2023 DECLARATION I am the creator of this dissertation, which has been conducted at the Faculty of Mathematics and Informatics, Hanoi National University of Education, under the guidance and direction of Associate Professor DUONG Anh Tuan and Associate Professor PHAN Quoc Hung I hereby affirm that the results presented in this dissertation are truly provided and have not been included in any other dissertations or theses submitted to any other universities or institutions for a degree or diploma “I certify that I am the PhD student named below and that the information provided is correct” Full name: VU THI HIEN ANH Signed: Date: i ACKNOWLEDGMENT First and foremost, I want to extend my heartfelt gratitude to my mentors, Associate Professor DUONG Anh Tuan and Associate Professor PHAN Quoc Hung Their invaluable guidance, profound insights, and unwavering support throughout my tenure at Hanoi National University of Education are deeply appreciated The meticulous research ethics, dedicated work ethic, and sincere commitment to their students exhibited by both professors have not only inspired me but will leave a lasting impact on me I am grateful to Associcate Professor Tran Dinh Ke and other members of the weekly seminar at the Division of Mathematical Analysis, Faculty of Mathematics and Informatics, Hanoi National University of Education, for their discussions and valuable comments on my research results I am also grateful to my colleagues at the Division of Mathematics, Ha Long High school for Gifted students for their help and support during the time of my PhD research and study I am forever grateful to my parents for endless love and unconditional support they have been giving me Last but not least, I am indebted to my beloved husband, Mr Manh Hung who always stay beside me and always motivates me to try everyday None of this would have been possible without their continuous and unconditional love, kindness and comfort through my journey Furthermore, I would like to convey my deep appreciation to the VinIF Scholarship for being a significant source of motivation, encouragement, and support that has played a crucial role in helping me successfully complete my academic program The author ii Contents DECLARATION i ACKNOWLEDGMENT ii TABLE OF CONTENTS LIST OF SYMBOLS AND ACRONYMS INTRODUCTION Literature review and motivations Objectives 12 Scope of research 13 Methodology 14 The structure and results of the dissertation 15 Chapter PRELIMINARIES AND AUXILIARY RESULTS 16 PRELIMINARIES AND AUXILIARY RESULTS 16 The ∆λ -Laplace operator 16 1.1.1 Hypotheses on ∆λ and some properties 16 1.1.2 The ∆λ - functional setting 18 1.1.3 Examples of ∆λ - Laplacian 18 1.1 1.2 Some auxiliary results 20 1.2.1 Some inequalities 20 1.2.2 Stability and the variations of energy 22 1.2.3 Stable solutions for a class of elliptic systems 24 Chapter CLASSIFICATION OF SOLUTIONS FOR A DEGENERATE SYS- TEM INVOLVING THE ∆λ - LAPLACIAN 28 2.1 Problem setting and results 28 2.1.1 Problem formulation 28 2.1.2 Nonexistence results 29 2.2 Proof of nonexistence results 31 2.2.1 Nonexistence of positive super-solutions 31 2.2.2 Nonexistence of positive stable solutions 35 Chapter LIOUVILLE TYPE THEOREMS FOR KIRCHHOFF ELLIPTIC EQUA- TIONS INVOLVING ∆λ -LAPLACIAN 51 3.1 Problem setting and results 51 3.1.1 Problem formulation 51 3.1.2 Nonexistence of stable solutions 52 3.2 Proof of nonexistence of stable solutions 55 3.2.1 Nonexistence of stable solutions for negative exponent 56 3.2.2 Nonexistence of stable solutions for exponential nonlinearity 60 3.2.3 Nonexistence of stable solutions for polynomial nonlinearity Chapter LIOUVILLE TYPE THEOREMS FOR DEGENERATE PARABOLIC SYSTEMS WITH ADVECTION TERMS 4.1 63 68 Problem setting and results 68 4.1.1 Problem formulation 68 4.1.2 Nonexistence of positive super-solutions 69 4.2 Proof of nonexistence results 72 4.2.1 Nonexistence of positive super-solutions of the equation 72 4.2.2 Nonexistence of positive super-solutions of the system 76 CONCLUDING REMARKS 80 LIST OF PUBLICATIONS 82 References 82 LIST OF SYMBOLS AND ACRONYMS R the set of real numbers R+ the set of positive real numbers RN the N -dimensional Euclidean space C n (Ω) the space of continuous differentiable function space of order n in Ω ⊂ RN C ∞ (Ω) the space of infinitely differentiable functions in Ω ⊂ RN Cc∞ (Ω) the space of infinitely differentiable functions with compact support in Ω ⊂ RN ∆λ ∆λ − operator ∇λ Hλ1 (RN ) Gradient associated to ∆λ  R the space defined by u ∈ L (RN ) : RN |∇λ u|2 dx < ∞ Gα Grushin operator ∆ Laplace operator ∇ gradient vector div ≡ ▽ divergence X ,→ Y X is embedded in Y ut partial derivative of u in variable t A := B A is defined by B i.e id est (that is) p.5 page PDE Partial differential equation □ the proof is complete INTRODUCTION Literature review and motivations The classical Liouville theorem asserts that any harmonic function, which is bounded across the entire space, must remain constant This theorem was first stated by Liouville [39] for the special case of double periodic functions in a report in Comptes Rendus (Paris, 23/12/1944), Cauchy [9] gave the first proof of the above theorem The Liouville type theorem is the non-existence of solutions (non-trivial) over the whole space or half space In the past four decades, Liouville type results have been widely studied for both elliptic problems, see [7,16,28–30,58], and parabolic problems, see [5,42,43,52,54,59,69] In particular, Gidas and Spruck [29] established an optimal Liouville type theorem for the Lane-Emden equation −∆u = u p , p > in RN Gidas and Spruck proved that, for p < ps := N +2 N −2 , (1) every non-negative solution over the entire space of (1) must be identically zero They then used this result, combined with the rescaling technique (also known as the blow-up method), to prove priori estimates for the corresponding boundary value problem The result of Gidas and Spruck was later extended to the case of multiple harmonic operators in [63] Concerning the class of positive super-solutions, the existence and nonexistence results were also proved, (see e.g [2],) in which, the critical exponent is given by pc = N N −2 (pc = ∞ when N ≤ 2) It is noteworthy to highlight that the categorization of stable solutions to the Lane-Emden equation has recently been comprehensively solved [26], with the explicit computation of the critical exponent achieved For the Lane-Emden system   −∆u = v p  −∆v = uq in RN , (2) the Lane-Emden conjecture states that system (2) has no classical positive solution on the whole space if and only if p > 0, q > 0, pq > and N N + > N − p+1 q+1 Until now, this conjecture has been proved for the classical radial solutions [42, 58] For non-radial solutions in spaces with dimension N ≤ 2, this conjecture follows from a result of Mitidieri and Pohozaev [44] When N = 3, the conjecture was proved by Serrin and Zou [57] with the additional assumption that the the growth rate of solutions does not exceed polynomial degree at infinity This restriction was removed in the work of Polacik, Quittner and Souplet [52], so the conjecture holds when N = In 2009, Souplet [59] gave a proof for the case of dimension N = When N ≥ 5, the conjecture has only been demonstrated for a few isolated cases, see [9, 59] Along with results for elliptic equations, Liouville type theorems for parabolic equations u t − ∆u = u p , (3) have also been widely studied over the past twenty years The first result, also known as the Fujita result (see [50]), asserts that equation (3) has no nonnegative non-trivial solution on half-space RN × R+ if and only if < p ≤ + N2 Later on, the Liouville type theorem for equation (3) was proved in [52] with optimal condition p < ps = N +2 N −2 for nonnegative radial solutions For non- radial solutions, the Liouville type theorem has been proved with condition p < pB := N (N +2) (N −1)2 (see [7, 52]) It is easy to see that this condition is not opti- mal because pB < ps when N > In 2016, Quittner [54] proved the optimal Liouville type result for the case N = Some related results for nodal solutions have also been established by Bartsch, Polacik and Quittner in [5] Recently, Liouville type theorems for parabolic systems have attracted much attention from mathematicians However, due to the fact that the Liouville type theorem for equation (3) has not yet been fully resolved, similar results for parabolic systems are still very few Here we mention some important works in this direction: Escobedo and Herrero [24], Andreucci, Herrero and Velázquez [2], Merle and Zaag [43], Quittner [54], Phan and Souplet [49] Since the pioneering work of Gidas and Sprick [29] on the Liouville type theorem for the Lane-Emden equation, there have been many established extensions in different directions Here we refer to some important results like [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   −∆ u = v λ in RN , p ∈ R,  −∆ v = u p λ (4) where ∆λ is a strongly degenerate operator of the form   N X ∂ ∂ λi (x) , ∆λ = ∂ ∂xi i=1 x i and λi satisfies some of the conditions given in the Chapter Let us now consider the special case where λi = for i = 1, 2, , N Then, the problem (4) reduces to   −∆u = v  −∆v = u p in RN (5) Recently, the system (5) has been considerably studied by many mathematicians, see [2, 11, 12, 31, 34, 65] For p ≤ 1, the nonexistence of super-solutions of (5) was proved, see e.g [2] It is shown for p > that the existence of positive classical solutions of the problem (5) and of the biharmonic problem ∆2 u = u p (6)