Yuxin Ge

Sign Changing Tower of Bubbles for an Elliptic Problem at the Critical Exponent in Pierced Non-Symmetric Domains

We consider the problem in Ωϵ, u = 0 on ∂Ωϵ, where Ωϵ: = Ω \ {B(a, ϵ) ∪ B(b, ϵ)}, with Ω a bounded smooth domain in ℝ N , N ≥ 3, a ≠ b two points in Ω, and ϵ is a positive small parameter. As ϵ goes to zero, we construct sign changing solutions with multiple blow up both at a and at b.

An almost Schur theorem on 4-dimensional manifolds

In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

A few symmetry results for nonlinear elliptic PDE on noncompact manifolds

Abstract We prove some symmetry theorems for positive solutions of elliptic equations in some noncompact manifolds, which generalize and extend symmetry results known in the case of the euclidean space R n . The (variational) technique that we use relies on Sobolev inequalities available for manifolds together with the well known method of moving planes. In the particular case of the standard n-dimensional hyperbolic space H n we get the radial symmetry of positive solutions of the equation −Δ H n u=f(u) in H n , which tend to zero at infinity (or belong to the Sobolev space H 1 ( H n ) in some cases), under different hypotheses on the relationship between the behavior of the nonlinearity f in a neighborhood of zero and the summability properties of the solution. One of the main features of this work is to single out and study the connection between the geometric properties of the manifold considered and the growth conditions on the nonlinearity in order to have our symmetry results.

Regularity of optimal transport on compact, locally nearly spherical, manifolds

Abstract Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold M, we look for a smooth optimal transportation map G, pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge–Ampère type satisfied by the potential function u, such that G = exp(grad u). This approach boils down to proving an a priori upper bound on the Hessian of u, which was done on the flat torus by the first author. The recent local C 2 estimate of Ma–Trudinger–Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image G(m) of a generic point m ∈ M, uniformly away from the cut-locus of m; he checked a fourth-order inequality satisfied by the squared distance cost function, proving its so-called (strict) regularity. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in C 2 norm—specifying and proving a conjecture stated by Trudinger.

Symmetry Results for Positive Solutions of Some Elliptic Equations on Manifolds

We adapt the method of moving planes in order to obtainsymmetry results for positive solutions of some elliptic equations onmanifolds, in the case where our problem satisfies certain symmetryhypothesis. We also obtain monotonicity results using the slidingmethod.

Isoperimetric-type inequalities on constant curvature manifolds

Abstract. By exploiting optimal transport theory on Riemannian manifolds and adapting Gromovs proof of the isoperimetric inequality in the Euclidean space, we prove an isoperimetric-type inequality on simply connected constant curvature manifolds.

Brezis-Nirenberg Problem and Coron Problem for Polyharmonic Operators

where F (x, u) = ∫ u 0 f(x, t)dt. Our motivation for the problem (1) to (3) comes from the fact that it resembles some variational problems in geometry and physics where lack of compactness occurs. For example, when K = 1, it arises from the famous Yamabe’s problem and when K = 2, it is similar to a conformally covariant operator studied by Paneitz. For related problems, we infer [2], [4], [5], [15], [18] and the references therein.

COMPARISON, SYMMETRY AND MONOTONICITY RESULTS FOR SOME DEGENERATE ELLIPTIC OPERATORS IN CARNOT-CARATHÉODORY SPACES

This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.

Multiple solutions of compact

We prove here the multiplicity results for the solutions of compact

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