A sup + inf Inequality for Some Nonlinear Elliptic Equations Involving Exponential Nonlinearities

Brezis, Haı̈m; Li, Yanyan; Shafrir, Itaï

doi:10.1006/jfan.1993.1094

Cited by 71 publications

(65 citation statements)

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“…Under (14) and (15), the following Harnack type inequality is proved by Brezis, Li and Shafrir in [9] through the method of moving planes: Every solution of (16) satisfies, on any compact subset K of ,…”

Section: Corollary 03 Let {ξ N } Be the Subsequence In Theorem 02 mentioning

confidence: 99%

“…It is raised as an open question in [9] whether the above Harnack type inequality still holds when replacing ∇V n L ∞ ( ) by V n C α ( ) (0 < α < 1). The answer is affirmative due to some recent work of Chen and Lin [19].…”

Section: Corollary 03 Let {ξ N } Be the Subsequence In Theorem 02 mentioning

confidence: 99%

“…Many more applications of the moving plane method have been given by various authors. The method of moving planes was used to obtain some Harnack type inequalities by Schoen in [36], subsequently by Brezis, Li, and Shafrir in [9], and by Chen and Lin in [18]. Our proof of Theorem 0.3 requires some new ingredients.…”

Section: Corollary 03 Let {ξ N } Be the Subsequence In Theorem 02 mentioning

confidence: 99%

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Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

273

271

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Abstract:A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern-Simons Higgs model and have been studied recently by various authors. IntroductionLet (M, g) be a compact Riemann surface without boundary, V be a positive function on M , W be a function with M W dv g = 1. Throughout the paper dv g denotes the volume element of g, g denotes the Laplace Beltrami operator with respect to g. For λ ∈ R, we seek a solution ofClearly M W dv g = 1 is a necessary condition for (E u ) λ to have a solution. If we set ξ = u − log M V e u dv g for a solution of (E u ) λ , then ξ satisfies

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Section: Corollary 03 Let {ξ N } Be the Subsequence In Theorem 02 mentioning

confidence: 99%

Section: Corollary 03 Let {ξ N } Be the Subsequence In Theorem 02 mentioning

confidence: 99%

Section: Corollary 03 Let {ξ N } Be the Subsequence In Theorem 02 mentioning

confidence: 99%

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Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

273

271

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“…In particular, the Toda system (1) arises in the study of the non-Abelian non-relativistic ChernSimons theory with gauge group SU (3). See, for instance, the books [9], [25] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, it holds that ∇ ln h(x j ) + ∇ x H(x j , x j ) + i =j ∇ x G(x i , x j ) = 0, j = 1, ..., m (3) where H(x, y) = G(x, y) − 1 2π…”

Section: Introductionmentioning

confidence: 99%