1,721,008 research outputs found
An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime
Full text linkNew existence results for the mean field equation on compact surfaces via degree theory
Full text linkA mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects
Full text linkBlow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities
Full text linkOn the uniqueness and monotonicity of solutions of free boundary problems
No full textFor any smooth and bounded domain Ω⊂RN, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on Ω in a neat interval depending only by the best constant of the Sobolev embedding H01(Ω)↪L2p(Ω), [Formula presented] and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for p>1, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for p=1. The threshold, which is sharp for p=1, yields a new condition which guarantees that there is no free boundary inside Ω. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes
NEW UNIVERSAL ESTIMATES for FREE BOUNDARY PROBLEMS ARISING in PLASMA PHYSICS
No full textFor Ω ⊂ R2 a smooth and bounded domain, we derive a sharp universal energy estimate for non-negative solutions of free boundary problems on Ω arising in plasma physics. As a consequence, we are able to deduce new universal estimates for this class of problems. We first come up with a sharp positivity threshold which guarantees that there is no free boundary inside Ω or either, equivalently, with a sharp necessary condition for the existence of a free boundary in the interior of Ω. Then we derive an explicit bound for the L∞-norm of non-negative solutions and also obtain explicit estimates for the thresholds relative to other neat density boundary values. At least to our knowledge, these are the first explicit estimates of this sort in the superlinear case
Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains
Full text linkThe aim of this paper is to complete the program initiated in [51], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov–Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems
MULTIPLICITY RESULTS FOR THE MEAN FIELD EQUATION ON COMPACT SURFACES
Full text linkAbstract. We are concerned with the following class of equations with expo-nential nonlinearities on a compact surface Σ: −∆u = ρ1 h eu´ Σ h e u dV
Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria
Full text linkWe are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality
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