1,721,010 research outputs found
Harnack inequalities and quantization properties for the Liouville equation
No full textWe consider a quasilinear equation involving the n-Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known that the exponential nonlinearity converges, up to a subsequence, to a sum of Dirac measures. By performing a precise local asymptotic analysis we complete such a result by showing that the corresponding Dirac masses are quantized as multiples of a given one, related to the mass of limiting profiles after rescaling according to the classification result obtained by the first author in Esposito (Ann. Inst. H. Poincare Anal. Non Lineaire 35(3), 781-801, 2018). A fundamental tool is provided here by some Harnack inequality of "sup+inf" type, a question of independent interest that we prove in the quasilinear context through a new and simple blow-up approach
Interior estimates for some semilinear elliptic problem with critical nonlinearity
No full textWe study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary
Compactness of a nonlinear eigenvalue problem with a singular nonlinearity
No full textWe study the Dirichlet boundary value problem
on a bounded domain Ω ⊂ R^N. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a by-product, we give an uniqueness result for λ close to 0 and λ* in the class of all solutions with finite Morse index, λ* being the extremal value associated to the nonlinear eigenvalue problem
Perturbations of Paneitz-Branson operators on
No full textWe prove the existence of solutions on the standard unit sphere for the equation , small, and , where is the fourth order conformally invariant Paneitz-Branson operator. We will approach this problem via a finite dimensional reduction which lead us to consider the "stable" critical points of the "Melnikov function":
in the case a more subtle analysis will be carried out by means of a Morse relation for functions on manifolds with boundary which are quite degenerate on the boundary
Uniqueness and multiplicity for perturbations of the Yamabe problem on S^n
No full textMotivated by an uniqueness result for linear perturbations with constant coefficients of the conformal laplacian on the sphere, we investigate, via a finite dimensional reduction, more general perturbations of the conformal laplacian, exhibiting cases in which uniqueness fail
On some conjectures proposed by Haïm Brezis
No full textDruet (Ann. Inst. H. Poincaré Anal. Non Linèaire 19(2) (2002) 125) solved two conjectures proposed by Haim Brezis (Comm. Pure Appl. Math. 39 (1986) 17) about “low”-dimension
phenomena for some elliptic problem with critical Sobolev exponent. In Druet (Ann. Inst.
H. Poincaré Anal. Non Linèaire 19(2) (2002) 125), the proof of one of the two conjectures is reduced to an asymptotic analysis which is carried over with very general techniques involving
pointwise estimates. We propose here a different and simpler approach in the blow-up analysis based on integral estimates and on a careful expansion of the energy functional
Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation
No full textFor the Neumann sinh-Gordon equation on the unit ball B ⊂R^2
in , on we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ± are positive parameters. It answers partially an open problem formulated in Jost et al. [Calc Var Partial Diff Equ 31(2):263–276]
Blowing-up solutions for the Yamabe equation
No full textLet (M,g) be a smooth, compact Riemannian manifold of dimension N \geq 3. We consider the almost critical problem
(P_\epsilon)
-\Delta_g u+ {N-2\over 4(N-1)} Scal_g u= u^{{N+2\over N-2}+\epsilon } in} M,
u>0 in M,
where \Delta_g denotes the Laplace-Beltrami operator, Scal_g is the scalar curvature of g
and \epsilon \in R is a small parameter.
It is known that problem (P_\epsilon) does not have any blowing-up solutions when \epsilon \to 0^-, at least for N \leq 24 or in the locally conformally flat case, and this is not true anymore when \epsilon \to 0^+. Indeed, we prove that, if N \geq 7 and the manifold is not locally conformally flat, then problem (P_\epsilon) does have a family of solutions which blow-up at a maximum point of the function
\xi \to |Weyl_g(\xi)|_g
as \epsilon \to 0^+. Here Weyl_g denotes the Weyl curvature tensor of g
Uniqueness of solutions for an elliptic equation modeling MEMS
No full textWe show among other things, that for small voltage, the stable solution of the basic
nonlinear eigenvalue problem modelling a simple electrostatic MEMS is actually the unique solution,
provided the domain is star-shaped and the dimension is larger or equal than 3. In two dimensions,
we need the domain to be either strictly convex or symmetric. The case of a power permittivity
profile is also considered. Our results, which use an approach developed by Schaaf [13], extend and
simplify recent results by Guo and Wei [7], [8]
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