We consider the SU(3) singular Toda system on a compact surface (Σ,g)−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑m=1Mα1m(δpm−1)−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑m=1Mα2m(δpm−1), where hi are smooth positive functions on Σ, ρi∈R+, pm∈Σ and αim>−1. We give both existence and non-existence results under some conditions on the parameters ρi and αim. Existence results are obtained using variational methods, which involve a geometric inequality of new type; non-existence results are obtained using blow-up analysis and localized Pohožaev-type identities."

VL - 270 UR - http://www.sciencedirect.com/science/article/pii/S0022123615004942 ER - TY - JOUR T1 - Symmetry properties of some solutions to some semilinear elliptic equations JF - Annali della Scuola Normale Superiore di Pisa. Classe di scienze Y1 - 2016 A1 - Farina, Alberto A1 - Andrea Malchiodi A1 - Matteo Rizzi PB - Classe di Scienze VL - 16 ER - TY - JOUR T1 - A general existence result for the Toda system on compact surfaces JF - Advances in Mathematics Y1 - 2015 A1 - Luca Battaglia A1 - Aleks Jevnikar A1 - Andrea Malchiodi A1 - David Ruiz KW - Geometric PDEs KW - Min–max schemes KW - Variational methods AB -In this paper we consider the following Toda system of equations on a compact surface:−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−Δu1=−4π∑j=1mα1,j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−Δu2=−4π∑j=1mα2,j(δpj−1), which is motivated by the study of models in non-abelian Chern–Simons theory. Here h1,h2 are smooth positive functions, ρ1,ρ2 two positive parameters, pi points of the surface and α1,i,α2,j non-negative numbers. We prove a general existence result using variational methods. The same analysis applies to the following mean field equation−Δu=ρ1(heu∫ΣheudVg−1)−ρ2(he−u∫Σhe−udVg−1), which arises in fluid dynamics."

VL - 285 UR - http://www.sciencedirect.com/science/article/pii/S0001870815003072 ER - TY - JOUR T1 - A topological join construction and the Toda system on compact surfaces of arbitrary genus JF - Analysis & PDE Y1 - 2015 A1 - Aleks Jevnikar A1 - Kallel, Sadok A1 - Andrea Malchiodi PB - Mathematical Sciences Publishers VL - 8 ER - TY - JOUR T1 - Conformal invariants from nodal sets. I. negative eigenvalues and curvature prescription Y1 - 2014 A1 - Rod R. Gover A1 - Yaiza Canzani A1 - Dmitry Jakobson A1 - Raphaël Ponge A1 - Andrea Malchiodi AB - In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the Graham, Jenne, Mason, and Sparling (GJMS) operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension n≥3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n≥3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in Appendix, the second named author and Andrea Malchiodi study the Q-curvature prescription problems for noncritical Q-curvatures. PB - Oxford University Press UR - http://urania.sissa.it/xmlui/handle/1963/35128 U1 - 35366 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Critical points of the Moser-Trudinger functional on a disk Y1 - 2014 A1 - Andrea Malchiodi A1 - Luca Martinazzi AB - On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional $$E(u)=\int_{B_1}(e^{u^2}-1)dx, u\in H^1_0(B_1)$$ and its restrictions to $M_\Lambda:=\{u \in H^1_0(B_1):\|u\|^2_{H^1_0}=\Lambda\}$ for $\Lambda>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{\Lambda_k}}$ (for some $\Lambda_k>0$) blows up as $k\to\infty$, then $\Lambda_k\to 4\pi$, and $u_k\to 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\loc}(\bar B_1\setminus\{0\})$. Using this we also prove that when $\Lambda$ is large enough, then $E|_{M_\Lambda}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe. PB - European Mathematical Society UR - http://hdl.handle.net/1963/6560 N1 - 16 pages U1 - 6487 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - A Moser-Trudinger inequality for the singular Toda system JF - Bull. Inst. Math. Acad. Sin. Y1 - 2014 A1 - Luca Battaglia A1 - Andrea Malchiodi VL - 9 ER - TY - JOUR T1 - An improved geometric inequality via vanishing moments, with applications to singular Liouville equations JF - Communications in Mathematical Physics 322, nr.2 (2013): 415-452 Y1 - 2013 A1 - Mauro Bardelloni A1 - Andrea Malchiodi PB - SISSA UR - http://hdl.handle.net/1963/6561 U1 - 6486 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - A variational Analysis of the Toda System on Compact Surfaces JF - Communications on Pure and Applied Mathematics, Volume 66, Issue 3, March 2013, Pages 332-371 Y1 - 2013 A1 - Andrea Malchiodi A1 - David Ruiz AB - In this paper we consider the Toda system of equations on a compact surface. We will give existence results by using variational methods in a non coercive case. A key tool in our analysis is a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u_1, u_2. PB - Wiley UR - http://hdl.handle.net/1963/6558 N1 - pre-peer version, to appear in Comm. Pure Applied Math U1 - 6489 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group. JF - Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198 Y1 - 2012 A1 - Andrea Malchiodi A1 - Paul Yang A1 - Jih-Hsin Cheng A1 - JennFang Hwang AB - In this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 1 PB - SISSA UR - http://hdl.handle.net/1963/6556 U1 - 6490 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Non-uniqueness results for critical metrics of regularized determinants in four dimensions JF - Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37 Y1 - 2012 A1 - Matthew Gursky A1 - Andrea Malchiodi AB - The regularized determinant of the Paneitz operator arises in quantum gravity (see Connes 1994, IV.4.$\gamma$). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in Branson (1996). A similar formula holds for Cheeger's half-torsion, which plays a role in self-dual field theory (see Juhl, 2009), and is defined in terms of regularized determinants of the Hodge laplacian on $p$-forms ($p < n/2$). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in Chang and Yang (1995), Branson, Chang, and Yang (1992), and Gursky (1997). We also study entire solutions of the Euler-Lagrange equation of $\log \det P$ and the half-torsion $\tau_h$ on $\mathbb{R}^4 \setminus {0}$, and show the existence of two families of periodic solutions. One of these families includes Delaunay-type solutions. PB - Springer UR - http://hdl.handle.net/1963/6559 N1 - 35 pages, title changed, added determinant of half-torsion, references added. Comm. Math. Phys., to appear U1 - 6488 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Weighted barycentric sets and singular Liouville equations on compact surfaces JF - Journal of Functional Analysis 262 (2012) 409-450 Y1 - 2012 A1 - Alessandro Carlotto A1 - Andrea Malchiodi AB - Given a closed two dimensional manifold, we prove a general existence result\\r\\nfor a class of elliptic PDEs with exponential nonlinearities and negative Dirac\\r\\ndeltas on the right-hand side, extending a theory recently obtained for the\\r\\nregular case. This is done by global methods: since the associated Euler\\r\\nfunctional is in general unbounded from below, we need to define a new model\\r\\nspace, generalizing the so-called space of formal barycenters and\\r\\ncharacterizing (up to homotopy equivalence) its very low sublevels. As a\\r\\nresult, the analytic problem is reduced to a topological one concerning the\\r\\ncontractibility of this model space. To this aim, we prove a new functional\\r\\ninequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after\\r\\n[6] and [8]). This study is motivated by abelian Chern- Simons theory in\\r\\nself-dual regime, or from the problem of prescribing the Gaussian curvature in\\r\\npresence of conical singularities (hence generalizing a problem raised by\\r\\nKazdan and Warner in [26]). PB - Elsevier UR - http://hdl.handle.net/1963/5218 U1 - 5040 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Axial symmetry of some steady state solutions to nonlinear Schrödinger equations JF - Proc. Amer. Math. Soc. 139 (2011), 1023-1032 Y1 - 2011 A1 - Changfeng Gui A1 - Andrea Malchiodi A1 - Haoyuan Xu A1 - Paul Yang KW - Nonlinear Schrödinger equation AB - In this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space. PB - American Mathematical Society UR - http://hdl.handle.net/1963/4100 U1 - 304 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A class of existence results for the singular Liouville equation JF - Comptes Rendus Mathematique 349 (2011) 161-166 Y1 - 2011 A1 - Alessandro Carlotto A1 - Andrea Malchiodi AB - We consider a class of elliptic PDEs on closed surfaces with exponential nonlinearities and Dirac deltas on the right-hand side. The study arises from abelian Chern–Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities. A general existence result is proved using global variational methods: the analytic problem is reduced to a topological problem concerning the contractibility of a model space, the so-called space of formal barycenters, characterizing the very low sublevels of a suitable functional. PB - Elsevier UR - http://hdl.handle.net/1963/5793 U1 - 5648 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - RPRT T1 - Critical points of the Moser-Trudinger functional Y1 - 2011 A1 - Francesca De Marchis A1 - Andrea Malchiodi A1 - Luca Martinazzi KW - Moser-Trudinger inequality PB - SISSA UR - http://hdl.handle.net/1963/4592 U1 - 4353 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces JF - Geometric and Functional Analysis 21 (2011) 1196-1217 Y1 - 2011 A1 - Andrea Malchiodi A1 - David Ruiz AB - We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results. PB - Springer UR - http://hdl.handle.net/1963/4099 U1 - 305 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Supercritical conformal metrics on surfaces with conical singularities JF - Int Math Res Notices (2011) 2011 (24): 5625-5643 Y1 - 2011 A1 - Mauro Bardelloni A1 - Francesca De Marchis A1 - Andrea Malchiodi AB -We study the problem of prescribing the Gaussian curvature on surfaces with conical singularities in supercritical regimes. Using a Morse-theoretical approach we prove a general existence theorem on surfaces with positive genus, with a generic multiplicity result.

PB - Oxford University Press UR - http://hdl.handle.net/1963/4095 U1 - 309 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions JF - Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 37-56 Y1 - 2010 A1 - Jesus Garcia Azorero A1 - Andrea Malchiodi A1 - Luigi Montoro A1 - Ireneo Peral AB - In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in. UR - http://hdl.handle.net/1963/3409 U1 - 926 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Concentration of solutions for some singularly perturbed mixed problems. Part I: existence results JF - Arch. Ration. Mech. Anal. 196 (2010) 907-950 Y1 - 2010 A1 - Jesus Garcia Azorero A1 - Andrea Malchiodi A1 - Luigi Montoro A1 - Ireneo Peral AB - In this paper we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero. UR - http://hdl.handle.net/1963/3406 U1 - 927 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some new entire solutions of semilinear elliptic equations on Rn JF - Adv. Math. 221 (2009) 1843-1909 Y1 - 2009 A1 - Andrea Malchiodi PB - Elsevier UR - http://hdl.handle.net/1963/3645 U1 - 659 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Concentrating solutions of some singularly perturbed elliptic equations JF - Front. Math. China 3 (2008) 239-252 Y1 - 2008 A1 - Andrea Malchiodi AB - We study singularly perturbed elliptic equations arising from models in physics or biology, and investigate the asymptotic behavior of some special solutions. We also discuss some connections with problems arising in differential geometry. UR - http://hdl.handle.net/1963/2657 U1 - 1466 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Entire solutions of autonomous equations on Rn with nontrivial asymptotics JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008) 65-72 Y1 - 2008 A1 - Andrea Malchiodi AB - We prove existence of a new type of solutions for the semilinear equation $- \\\\D u + u = u^p$ on $\\\\R^n$, with $1 < p < \\\\frac{n+2}{n-2}$. These solutions are positive, bounded, decay exponentially to zero away from three half-lines with a common origin, and at infinity are asymptotically periodic. UR - http://hdl.handle.net/1963/2640 U1 - 1483 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Existence of conformal metrics with constant $Q$-curvature JF - Ann. of Math. 168 (2008) 813-858 Y1 - 2008 A1 - Zindine Djadli A1 - Andrea Malchiodi AB - Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author. UR - http://hdl.handle.net/1963/2308 U1 - 1708 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Morse theory and a scalar field equation on compact surfaces JF - Adv. Differential Equations 13 (2008) 1109-1129 Y1 - 2008 A1 - Andrea Malchiodi PB - Khayyam Publishing UR - http://hdl.handle.net/1963/3531 U1 - 733 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Topological methods for an elliptic equation with exponential nonlinearities JF - Discrete Contin. Dyn. Syst. 21 (2008) 277-294 Y1 - 2008 A1 - Andrea Malchiodi AB - We consider a class of variational equations with exponential nonlinearities on compact surfaces. From considerations involving the Moser-Trudinger inequality, we characterize some sublevels of the Euler-Lagrange functional in terms of the topology of the surface and of the data of the equation. This is used together with a min-max argument to obtain existence results. UR - http://hdl.handle.net/1963/2594 U1 - 1528 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Transition layer for the heterogeneous Allen-Cahn equation JF - Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631 Y1 - 2008 A1 - Fethi Mahmoudi A1 - Andrea Malchiodi A1 - Juncheng Wei AB - We consider the equation $\\\\e^{2}\\\\Delta u=(u-a(x))(u^2-1)$ in $\\\\Omega$, $\\\\frac{\\\\partial u}{\\\\partial \\\\nu} =0$ on $\\\\partial \\\\Omega$, where $\\\\Omega$ is a smooth and bounded domain in $\\\\R^n$, $\\\\nu$ the outer unit normal to $\\\\pa\\\\Omega$, and $a$ a smooth function satisfying $-10} and {a<0}. Assuming $\\\\nabla a \\\\neq 0$ on $K$ and $a\\\\ne 0$ on $\\\\partial \\\\Omega$, we show that there exists a sequence $\\\\e_j \\\\to 0$ such that the above equation has a solution $u_{\\\\e_j}$ which converges uniformly to $\\\\pm 1$ on the compact sets of $\\\\O_{\\\\pm}$ as $j \\\\to + \\\\infty$. UR - http://hdl.handle.net/1963/2656 U1 - 1467 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Boundary interface for the Allen-Cahn equation JF - J. Fixed Point Theory Appl. 1 (2007) 305-336 Y1 - 2007 A1 - Andrea Malchiodi A1 - Juncheng Wei UR - http://hdl.handle.net/1963/2027 U1 - 2169 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Boundary-clustered interfaces for the Allen–Cahn equation JF - Pacific Journal of Mathematics 229 (2007), No. 2, 447–468 Y1 - 2007 A1 - Andrea Malchiodi A1 - Wei-Ming Ni A1 - Juncheng Wei PB - Mathematical Sciences Publishers UR - http://hdl.handle.net/1963/5089 U1 - 4905 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Concentration on minimal submanifolds for a singularly perturbed Neumann problem JF - Adv. Math. 209 (2007) 460-525 Y1 - 2007 A1 - Fethi Mahmoudi A1 - Andrea Malchiodi AB - We consider the equation $- \\\\e^2 \\\\D u + u= u^p$ in $\\\\Omega \\\\subseteq \\\\R^N$, where $\\\\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\partial \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, ..., N-2\\\\}$. We impose Neumann boundary conditions, assuming $11 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0. PB - Springer UR - http://hdl.handle.net/1963/4866 U1 - 4645 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - A fourth order uniformization theorem on some four manifolds with large total Q-curvature JF - C. R. Acad. Sci. Paris, Ser. I 340 (2005) 341-346. Y1 - 2005 A1 - Zindine Djadli A1 - Andrea Malchiodi AB - Given a four-dimensional manifold (M,g), we study the existence of a conformal metric for which the Q-curvature, associated to a conformally invariant fourth-order operator (the Paneitz operator), is constant. Using a topological argument, we obtain a new result in cases which were still open. PB - Elsevier UR - http://hdl.handle.net/1963/4868 U1 - 4649 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Ground states of nonlinear Schroedinger equations with potentials vanishing at infinity JF - J. Eur. Math. Soc. 7 (2005) 117-144 Y1 - 2005 A1 - Antonio Ambrosetti A1 - Veronica Felli A1 - Andrea Malchiodi AB - We deal with a class on nonlinear Schr\\\\\\\"odinger equations \\\\eqref{eq:1} with potentials $V(x)\\\\sim |x|^{-\\\\a}$, $0<\\\\a<2$, and $K(x)\\\\sim |x|^{-\\\\b}$, $\\\\b>0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\\\\e}$ belonging to $W^{1,2}(\\\\Rn)$ is proved under the assumption that $p$ satisfies \\\\eqref{eq:p}. Furthermore, it is shown that $v_{\\\\e}$ are {\\\\em spikes} concentrating at a minimum of ${\\\\cal A}=V^{\\\\theta}K^{-2/(p-1)}$, where $\\\\theta= (p+1)/(p-1)-1/2$. UR - http://hdl.handle.net/1963/2352 U1 - 1664 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Minimal surfaces in pseudohermitian geometry JF - Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177. Y1 - 2005 A1 - Jih-Hsin Cheng A1 - JennFang Hwang A1 - Andrea Malchiodi A1 - Paul Yang AB - We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold. PB - Scuola Normale Superiore UR - http://hdl.handle.net/1963/4579 U1 - 4347 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Multiple clustered layer solutions for semilinear Neumann problems on a ball JF - Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163 Y1 - 2005 A1 - Andrea Malchiodi A1 - Wei-Ming Ni A1 - Juncheng Wei PB - Elsevier UR - http://hdl.handle.net/1963/3532 U1 - 732 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Multidimensional boundary layers for a singularly perturbed Neumann problem JF - Duke Math. J. 124 (2004) 105-143 Y1 - 2004 A1 - Andrea Malchiodi A1 - Marcelo Montenegro PB - Duke University Press UR - http://hdl.handle.net/1963/2960 U1 - 1740 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Singularity perturbed elliptic equations with symmetry: existence of solutions concetrating on spheres, Part II JF - Indiana Univ. Math. J. 53 (2004) 297-392 Y1 - 2004 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi A1 - Wei-Ming Ni PB - Indiana University Mathematics Journal UR - http://hdl.handle.net/1963/1663 U1 - 2455 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Solutions concentrating at curves for some singularly perturbed elliptic problems JF - C. R. Acad. Sci. Paris, Ser. I 338 (2004) 775-780 Y1 - 2004 A1 - Andrea Malchiodi PB - Elsevier UR - http://hdl.handle.net/1963/4869 U1 - 4647 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Prescribing scalar and boundary mean curvature on the three dimensional half sphere JF - J. Geom. Anal. 13 (2003) 255-289 Y1 - 2003 A1 - Zindine Djadli A1 - Andrea Malchiodi A1 - Mohameden Ould Ahmedou AB - We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results. PB - Springer UR - http://hdl.handle.net/1963/3086 U1 - 1247 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I JF - Comm. Math. Phys. 235 (2003) no.3, 427-466 Y1 - 2003 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi A1 - Wei-Ming Ni PB - Springer UR - http://hdl.handle.net/1963/1633 U1 - 2485 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Curvature theory of boundary phases: the two-dimensional case JF - Interfaces Free Bound. 7 (2002) 345-370 Y1 - 2002 A1 - Andrea Braides A1 - Andrea Malchiodi AB - We describe the behaviour of minimum problems involving non-convex surface integrals in 2D, singularly perturbed by a curvature term. We show that their limit is described by functionals which take into account energies concentrated on vertices of polygons. Non-locality and non-compactness effects are highlighted. PB - European Mathematical Society UR - http://hdl.handle.net/1963/3537 U1 - 1164 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Prescribing a fourth oder conformal invariant on the standard sphere - Part I: a perturbation result JF - Commun. Contemp. Math., 2002, 4, 375 Y1 - 2002 A1 - Zindine Djadli A1 - Mohameden Ould Ahmedou A1 - Andrea Malchiodi PB - SISSA Library UR - http://hdl.handle.net/1963/1539 U1 - 2624 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Prescribing a fourth oder conformal invariant on the standard sphere - Part II: blow up analysis and applications JF - Ann. Sc. Norm. Super. Pisa Cl. Sci., 2002, 1, 387 Y1 - 2002 A1 - Zindine Djadli A1 - Andrea Malchiodi A1 - Mohameden Ould Ahmedou PB - SISSA Library UR - http://hdl.handle.net/1963/1540 U1 - 2623 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The scalar curvature problem on $S\\\\sp n$: an approach via Morse theory JF - Calc. Var. Partial Differential Equations 14 (2002), no. 4, 429-445 Y1 - 2002 A1 - Andrea Malchiodi PB - Springer UR - http://hdl.handle.net/1963/1331 U1 - 3124 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Singular elliptic problems with critical growth JF - Comm. Partial Differential Equations 27 (2002), no. 5-6, 847-876 Y1 - 2002 A1 - Paolo Caldiroli A1 - Andrea Malchiodi PB - Dekker UR - http://hdl.handle.net/1963/1268 U1 - 3187 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Solutions concentrating on spheres to symmetric singularly perturbed problems JF - C.R.Math.Acad.Sci. Paris 335 (2002),no.2,145-150 Y1 - 2002 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi A1 - Wei-Ming Ni AB - We discuss some existence results concerning problems (NLS) and (N), proving the existence of radial solutions concentrating on a sphere. PB - SISSA Library UR - http://hdl.handle.net/1963/1594 U1 - 2524 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Yamabe problem and the scalar curvature problems under boundary conditions JF - Math. Ann., 2002, 322, 667 Y1 - 2002 A1 - Antonio Ambrosetti A1 - Li YanYan A1 - Andrea Malchiodi PB - SISSA Library UR - http://hdl.handle.net/1963/1510 U1 - 2653 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Adiabatic limits of closed orbits for some Newtonian systems in R-n JF - Asymptotic Anal., 2001, 25, 149-181 Y1 - 2001 A1 - Andrea Malchiodi AB - We deal with a Newtonian system like x\\\'\\\' + V\\\'(x) = 0. We suppose that V: \\\\R^n \\\\to \\\\R possesses an (n-1)-dimensional compact manifold M of critical points, and we prove the existence of arbitrarity slow periodic orbits. When the period tends to infinity these orbits, rescaled in time, converge to some closed geodesics on M. PB - SISSA Library UR - http://hdl.handle.net/1963/1511 U1 - 2652 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Multiple positive solutions of some elliptic equations in \\\\bold R\\\\sp N JF - Nonlinear Anal. 43 (2001) 159-172 Y1 - 2001 A1 - Andrea Malchiodi PB - Elsevier UR - http://hdl.handle.net/1963/3094 U1 - 1239 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Multiplicity results for some nonlinear Schrodinger equations with potentials JF - Arch. Ration. Mech. An., 2001, 159, 253 Y1 - 2001 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi A1 - Simone Secchi PB - SISSA Library UR - http://hdl.handle.net/1963/1564 U1 - 2554 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Non-compactness and multiplicity results for the Yamabe problem on Sn JF - J. Funct. Anal. 180 (2001) 210-241 Y1 - 2001 A1 - Massimiliano Berti A1 - Andrea Malchiodi PB - Elsevier UR - http://hdl.handle.net/1963/1345 U1 - 3110 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the symmetric scalar curvature problem on S\\\\sp n JF - J. Differential Equations 170 (2001) 228-245 Y1 - 2001 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi AB - We discuss some existence results dealing with the scalar curvature problem on S\\\\sp n in the presence of various symmetries. PB - Elsevier UR - http://hdl.handle.net/1963/3095 U1 - 1238 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - THES T1 - Existence and multiplicity results for some problems in Riemannian geometry Y1 - 2000 A1 - Andrea Malchiodi KW - Yamabe problem PB - SISSA UR - http://hdl.handle.net/1963/5948 U1 - 5808 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - A note on the scalar curvature problem in the presence of symmetries JF - Ricerche Mat. 49 (2000), suppl., 169-176 Y1 - 2000 A1 - Antonio Ambrosetti A1 - Li YanYan A1 - Andrea Malchiodi PB - SISSA Library UR - http://hdl.handle.net/1963/1365 U1 - 3090 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Scalar curvature under boundary conditions JF - Cr. Acad. Sci. I-Math, 2000, 330, 1013 Y1 - 2000 A1 - Antonio Ambrosetti A1 - Li YanYan A1 - Andrea Malchiodi PB - SISSA Library UR - http://hdl.handle.net/1963/1506 U1 - 2657 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A multiplicity result for the Yamabe problem on $S\\\\sp n$ JF - J. Funct. Anal. 168 (1999), no. 2, 529-561 Y1 - 1999 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi AB - We prove a multiplicity result for the Yamabe problem on the manifold (S, g), where g is a perturbation of the standard metric g0 of Sn. Solutions are found by variational methods via an abstract perturbation result. PB - Elsevier UR - http://hdl.handle.net/1963/1264 U1 - 3191 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the scalar curvature problem under symmetry Y1 - 1999 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi PB - SISSA Library UR - http://hdl.handle.net/1963/1287 U1 - 3168 U2 - Mathematics U3 - Functional Analysis and Applications ER -