Search

In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality \begin{equation*} {\Lambda} \left(\int_{\mathbb R^n}\frac{|u(x)|^{p}}{|x|^{{\beta} {p}}}\,dx\right)^{\frac{2}{p}}\leq \int_{\mathbb R^n}\int_{\mathbb R^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2\gamma}|x|^{{\alpha}}|y|^{{\alpha}}}\,dy\,dx \end{equation*} where $\gamma\in(0,1)$, $n\geq 2$, and $\alpha,\beta\in\mathbb R$ satisfy \begin{equation*} \alpha\leq \beta\leq \alpha+\gamma, \ -2\gamma

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points., Comment: 37 pages

We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted $L^\infty$ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an \emph{Ansatz} for the Yamabe problem with a prescribed singular set of maximal dimension $(n-2)/2$, for which, using the same machinery, we provide an alternative construction to the one given by Pacard. His linear theory uses $L^p$-theory on manifolds, while our approach studies the equations in the ambient space and is therefore suitable for generalization to nonlocal problems. In a forthcoming paper, we will prove analogous results in the fractional setting., 24 pages

We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an infinite-dimensional Lyapunov–Schmidt reduction method, that reduces the problem to an (infinite-dimensional) Toda-type system. The main technical part is the estimate of the interactions between different bubbles in the bubble towers.

We consider the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which we establish the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional-order ordinary differential equation (ODE). Thus, our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, we first provide a rigorous construction of radial fast-decaying solutions by a blowup argument and a bifurcation method. Then, second, we use conformal geometry to rewrite this nonlocal ODE, giving a hint of what a nonlocal phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrodinger equation with a Hardy-type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a 2-dimensional kernel as in the second-order case., A. DelaTorre’s research is supported by Swiss National Science Foundation, projects nr. PP00P2-144669 and PP00P2-170588/1 while she was a postdoc under the supervision of Luca Martinazzi, and by the Spanish government grant MTM2014-52402-C3-1-P. M. Fontelos is supported by the Spanish government grant MTM2017- 89423-P. M.d.M. Gonz´alez is supported by Spanish government grants MTM2014-52402-C3-1-P and MTM2017-85757-P, and the BBVA foundation grant for Researchers and Cultural Creators, 2016. The research of H. Chan and J. Wei is supported by NSERC of Canada.

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space [Formula: see text], where [Formula: see text] is any bounded, smooth, open subset of [Formula: see text], [Formula: see text]. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

We consider the fractional mean-field equation on the interval [Formula: see text] [Formula: see text] subject to Dirichlet boundary conditions, and prove that existence holds if and only if [Formula: see text]. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of nonlocal type.

We consider radial solutions with an isolated singularity for a semilinear equation involving the fractional Laplacian. In conformal geometry, this is equivalent to the study of singular metrics with constant fractional curvature (singular fractional Yamabe problem). Our main ideas are: first, to set up the problem into a natural geometric framework; and second, to reduce the problem to a non-local ODE for which we are able to perform some kind of phase portrait study.

We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity $(-\Delta)^\gamma w = c_{n, \gamma} w^{\frac{n+2\gamma}{n-2\gamma}}, w>0 \ \mbox{in} \ \mathbb{R}^n \backslash \{0\}$ We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates., Comment: 21 pages