Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations

By Yanyan Li, Lei Zhang. 2003

276 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6dcb8a090afa878d7812033ea0677a389958a9cd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Prescribing scalar and boundary mean curvature on the three dimensional half sphere (2003, unknown): “−∆u = u in R+, u > 0; ∂u ∂ν = a u q on ∂R+, where 1 < p < n+2 n−2 , 1 < q < n n−2 and a ≤ 0, see [18] and [32].”
Conformally invariant fully nonlinear elliptic equations and isolated singularities (2005, unknown): “[18] or [13], that there exists λ0(x) ∈ (0, |x|) such that for all 0 < λ < λ0(x),”
Positive solutions of nonlinear problems involving the square root of the Laplacian (2009, methodology): “Zhang [18], [17] using the moving spheres method.”
A Harnack type inequality for the Yamabe equation in low dimensions (2004, methodology): “This implies, by a calculus lemma (see, e.g., Lemma 11.2 in [ 27 ]), that U is a constant, contradicting to (38).”
Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations (2005, unknown): “We refer to [11, 15, 21, 23, 26, 27] and the references therein for discussions of other Liouville theorems.”
On some conformally invariant fully nonlinear equations (2003, methodology): “On the other hand, works on the Yamabe equation by Caﬀarelli, Gidas and Spruck ([3]), Schoen ([22] and [23]), Li and Zhu ([19]), and Li and Zhang ([18]), have played an important role in our approach to the study of (5) as developed in [15]. together with Let g 1 = u 4 n − 2 g 0 be a conformal…”
Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle (2010, background): “For instance, it is known that − u = f u in n has no classical positive solutions provided s− n+2 n−2 f s is an increasing function on 0 ; see [30] and the references therein.”
SHARP HARDY-LITTLEWOOD-SOBOLEV INEQUALITY ON THE UPPER HALF SPACE (2013, background): “Under stronger assumptions (f ∈ C1(Rn+)), these lemmas were early proved by Li and Zhu [36], and Li and Zhang [35].”
Remark on some conformally invariant integral equations: the method of moving spheres (2003, methodology): “A P] 1 8 Ju l 2 00 3 Remark on some conformally invariant integral equations: the method of moving spheres YanYan Li Department of Mathematics Rutgers University 110 Frelinghuysen Rd. Piscataway, NJ 08854”
Local gradient estimates of solutions to some conformally invariant fully nonlinear equations (2006, background): “1 in [18] or [13], for any x in R, there exists λ0(x) > 0 such that”
Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities (2013, background): “ 1 +|y|2 n−2σ 2 , (15) upon some multiple, scaling and translation. On the other hand, we are going to show that wλ,x(y) ≤ w(y) ∀ λ > 0,x ∈ Rn, |y −x| ≥ λ. (16) By an elementary calculus lemma in [20], (16) implies that w ≡ constant. This contradicts to (15). Let us arbitrarily ﬁx x0 ∈ Rn and λ0 ”
On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, methodology): “8 uses the method of moving spheres and is inspired by [76], [75] and [25].”
On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe (2005, methodology): “An important step was taken by Zhang and the second author in [46], which gives a proof of Schoen's Harnack-type inequality for the Yamabe equation without using the Liouville-type theorem in [6].”
Compactness of solutions to the Yamabe problem (2004, unknown): “Analogous results, extensions, as well as different proofs of (12) can be found in [87], [90], [81], [22], [21], [74], [34], [33], [68], [16], [17], [36], [ 69 ], [64], [14], [78], [37], [65], [15], [88], [89], and others.”
On the critical p-Laplace equation (2022, unknown): “2), with p = 2 (see also [6] and [19]).”