Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities

By J. Serrin, Heng Zou. 2002

406 citations recorded by the source corpus.

1. In t roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2. Liouville theorems I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3. Liouville theorems II . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4. The Harnack inequal i ty . . . . . . . . . . . . . . . . . . . . . . . . . 104 5. Universal a priori es t imates . . . . . . . . . . . . . . . . . . . . . . . 110 Chapte r II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6. A general integral inequali ty I . . . . . . . . . . . . . . . . . . . . . 114 7. A general integral inequali ty II . . . . . . . . . . . . . . . . . . . . . 124 8. Regular i ty theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9. Historical note: Canchy and Liouville, a quest ion of priori ty . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Subfields: Applied Mathematics, Computational Theory and Mathematics

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Citations in the corpus

Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy–Littlewood–Sobolev system of integral equations (2012, background): “The argument on the Lane–Emden conjecture can be seen in [20,22–25].”
Liouville theorems for the Navier–Stokes equations and applications (2007, background): “Analogous ideas were introduced in the study of many other classes of equations, such as semilinear elliptic and parabolic equations [10], [12], [13], [25], [26], [ 29 ], [30], the Navier‐Stokes equations [4], [7] and dispersive equations [18], [31], to name a few.”
Singular quasilinear and Hessian equations and inequalities (2009, background): “Quasilinear equations (1.1) in the special case 1 have been studied extensively over the past 20 years (see [5]‐[7], [8], [9], [ 30 ], [36]).”
Sharp criteria of Liouville type for some nonlinear systems (2013, positive): “Serrin and Zou [49] proved the existence if (5.2) is true.”
On the critical p-Laplace equation (2022, methodology): “1 in [27], where a truncation of |∇u| is introduced in order to deal with the critical set of u, and then pass to the limit as the relevant parameter ε tends to 0 to conclude.”
Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle (2010, background): “For more nonexistence results for positive solutions of quasilinear inequalities with pure power right-hand sides, we refer to Serrin and Zou [40], Liskevich et al. [32].”
Towards a Unified Approach to Nonexistence of Solutions for a Class of Differential Inequalities (2004, background): “For additional results the reader may refer to [4],[24],[25], [26],[38],[39],[44] and [45].”
Quasilinear and Hessian equations of Lane-Emden type (2005, methodology): “The existence and regularity theory, local and global estimates of sub- and super-solutions, the Wiener criterion, and Harnack inequalities associated with the p-Laplacian, as well as more general quasilinear operators, can be found in [HKM], [IM], [KM2], [M1], [MZ], [S1], [S2], [ SZ ], [TW4] where ”
Local and global properties of solutions of quasilinear Hamilton-Jacobi equations (2014, background): “86) for m > mc := N(p−1) N−p (see [38] for partial but very deep results).”
Monotonicity and one-dimensional symmetry for solutions of −Δpu = f(u) in half-spaces (2011, methodology): “1 holds true (accordingly to [28]) if f is subcritical w.”
Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations (2005, neutral): “We refer to [11, 15, 21, 23, 26, 27] and the references therein for discussions of other Liouville theorems.”
A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities (2010, methodology): “A(t) = |t |p−2t , nonexistence of positive solutions (not necessarily radial!) in an exterior domain has been recently proved in [5,58].”
A priori estimates and existence for quasi-linear elliptic equations (2008, background): “The weighted W 2,2 regularity (3) is established in [ 8 ],”
Liouville Theorems for Fractional Parabolic Equations (2021, background): “For the elliptic equations involving either local or nonlocal operators, there have been numerous articles that dedicated to the study of Liouville type theorems, such as [5, 15, 16, 18, 19, 25, 30, 33, 36] and so on.”
Basic Complex Analysis: A Comprehensive Course in Analysis, Part 2A (2015, neutral): “The historical record [370, 406, 519] provides support for naming the theorem after Liouville.”
Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations (2004, background): “It was pointed out to us that Serrin and Zou, in their celebrated paper [28], state in the case 1omo2 the solution m belongs to the Sobolev space W 1;2 loc ðOÞ; among other regularity results for solutions of quasilinear elliptic equations.”