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 Poho{\v z}aev-type identities."

}, keywords = {Liouville-type equations, Min{\textendash}max solutions, Non-existence results, Toda system}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2015.12.011}, url = {http://www.sciencedirect.com/science/article/pii/S0022123615004942}, author = {Luca Battaglia and Andrea Malchiodi} } @article {farina2016symmetry, title = {Symmetry properties of some solutions to some semilinear elliptic equations}, journal = {Annali della Scuola Normale Superiore di Pisa. Classe di scienze}, volume = {16}, number = {4}, year = {2016}, pages = {1209{\textendash}1234}, publisher = {Classe di Scienze}, author = {Farina, Alberto and Andrea Malchiodi and Matteo Rizzi} } @article {BATTAGLIA2015937, title = {A general existence result for the Toda system on compact surfaces}, journal = {Advances in Mathematics}, volume = {285}, year = {2015}, pages = {937 - 979}, abstract = {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{\textendash}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."

}, keywords = {Geometric PDEs, Min{\textendash}max schemes, Variational methods}, issn = {0001-8708}, doi = {https://doi.org/10.1016/j.aim.2015.07.036}, url = {http://www.sciencedirect.com/science/article/pii/S0001870815003072}, author = {Luca Battaglia and Aleks Jevnikar and Andrea Malchiodi and David Ruiz} } @article {jevnikar2015topological, title = {A topological join construction and the Toda system on compact surfaces of arbitrary genus}, journal = {Analysis \& PDE}, volume = {8}, number = {8}, year = {2015}, pages = {1963{\textendash}2027}, publisher = {Mathematical Sciences Publishers}, doi = {10.2140/apde.2015.8.1963}, author = {Aleks Jevnikar and Kallel, Sadok and Andrea Malchiodi} } @article {2014, title = {Conformal invariants from nodal sets. I. negative eigenvalues and curvature prescription}, number = {International Mathematics Research Notices;volume 2014; issue 9; pages 2356-2400;}, year = {2014}, publisher = {Oxford University Press}, abstract = {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{\textquoteright}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.}, doi = {10.1093/imrn/rns295}, url = {http://urania.sissa.it/xmlui/handle/1963/35128}, author = {Rod R. Gover and Yaiza Canzani and Dmitry Jakobson and Rapha{\"e}l Ponge and Andrea Malchiodi} } @article {2012, title = {Critical points of the Moser-Trudinger functional on a disk}, number = {Journal of the European Mathematical Society}, year = {2014}, note = {16 pages}, publisher = {European Mathematical Society}, abstract = {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.}, doi = {10.4171/JEMS/450}, url = {http://hdl.handle.net/1963/6560}, author = {Andrea Malchiodi and Luca Martinazzi} } @article {battaglia2013moser, title = {A Moser-Trudinger inequality for the singular Toda system}, journal = {Bull. Inst. Math. Acad. Sin.}, volume = {9}, number = {1}, year = {2014}, pages = {1{\textendash}23}, author = {Luca Battaglia and Andrea Malchiodi} } @article {2012, title = {An improved geometric inequality via vanishing moments, with applications to singular Liouville equations}, journal = {Communications in Mathematical Physics 322, nr.2 (2013): 415-452}, number = {arXiv:1206.0225;}, year = {2013}, publisher = {SISSA}, doi = {10.1007/s00220-013-1731-0}, url = {http://hdl.handle.net/1963/6561}, author = {Mauro Bardelloni and Andrea Malchiodi} } @article {2013, title = {A variational Analysis of the Toda System on Compact Surfaces}, journal = {Communications on Pure and Applied Mathematics, Volume 66, Issue 3, March 2013, Pages 332-371}, number = {arXiv:1105.3701;}, year = {2013}, note = {pre-peer version, to appear in Comm. Pure Applied Math}, publisher = {Wiley}, abstract = {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.}, doi = {10.1002/cpa.21433}, url = {http://hdl.handle.net/1963/6558}, author = {Andrea Malchiodi and David Ruiz} } @article {2012, title = {A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group.}, journal = {Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198}, year = {2012}, publisher = {SISSA}, abstract = {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}, doi = {10.1515/CRELLE.2011.159}, url = {http://hdl.handle.net/1963/6556}, author = {Andrea Malchiodi and Paul Yang and Jih-Hsin Cheng and JennFang Hwang} } @article {2012, title = {Non-uniqueness results for critical metrics of regularized determinants in four dimensions}, journal = {Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37}, number = {arXiv:1105.3762;}, year = {2012}, note = {35 pages, title changed, added determinant of half-torsion, references added. Comm. Math. Phys., to appear}, publisher = {Springer}, abstract = {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{\textquoteright}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.}, doi = {10.1007/s00220-012-1535-7}, url = {http://hdl.handle.net/1963/6559}, author = {Matthew Gursky and Andrea Malchiodi} } @article {2012, title = {Weighted barycentric sets and singular Liouville equations on compact surfaces}, journal = {Journal of Functional Analysis 262 (2012) 409-450}, number = {arXiv:1105.2363;}, year = {2012}, publisher = {Elsevier}, abstract = {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]).}, doi = {10.1016/j.jfa.2011.09.012}, url = {http://hdl.handle.net/1963/5218}, author = {Alessandro Carlotto and Andrea Malchiodi} } @article {2011, title = {Axial symmetry of some steady state solutions to nonlinear Schr{\"o}dinger equations}, journal = {Proc. Amer. Math. Soc. 139 (2011), 1023-1032}, number = {SISSA;75/2010/M}, year = {2011}, publisher = {American Mathematical Society}, abstract = {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.}, keywords = {Nonlinear Schr{\"o}dinger equation}, doi = {10.1090/S0002-9939-2010-10638-X}, url = {http://hdl.handle.net/1963/4100}, author = {Changfeng Gui and Andrea Malchiodi and Haoyuan Xu and Paul Yang} } @article {2011, title = {A class of existence results for the singular Liouville equation}, journal = {Comptes Rendus Mathematique 349 (2011) 161-166}, year = {2011}, publisher = {Elsevier}, abstract = {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{\textendash}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.}, doi = {10.1016/j.crma.2010.12.016}, url = {http://hdl.handle.net/1963/5793}, author = {Alessandro Carlotto and Andrea Malchiodi} } @article {2011, title = {Critical points of the Moser-Trudinger functional}, number = {SISSA;46/2011/M}, year = {2011}, institution = {SISSA}, keywords = {Moser-Trudinger inequality}, url = {http://hdl.handle.net/1963/4592}, author = {Francesca De Marchis and Andrea Malchiodi and Luca Martinazzi} } @article {2011, title = {New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces}, journal = {Geometric and Functional Analysis 21 (2011) 1196-1217}, number = {SISSA;74/2010/M}, year = {2011}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s00039-011-0134-7}, url = {http://hdl.handle.net/1963/4099}, author = {Andrea Malchiodi and David Ruiz} } @article {2011, title = {Supercritical conformal metrics on surfaces with conical singularities}, journal = {Int Math Res Notices (2011) 2011 (24): 5625-5643}, number = {SISSA;56/2010/M}, year = {2011}, publisher = {Oxford University Press}, abstract = {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.

}, doi = {10.1093/imrn/rnq285}, url = {http://hdl.handle.net/1963/4095}, author = {Mauro Bardelloni and Francesca De Marchis and Andrea Malchiodi} } @article {2010, title = {Concentration of solutions for some singularly perturbed mixed problems. Part I: existence results}, journal = {Arch. Ration. Mech. Anal. 196 (2010) 907-950}, number = {SISSA;02/2009/M}, year = {2010}, abstract = {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.}, doi = {10.1007/s00205-009-0259-0}, url = {http://hdl.handle.net/1963/3406}, author = {Jesus Garcia Azorero and Andrea Malchiodi and Luigi Montoro and Ireneo Peral} } @article {2010, title = {Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 37-56}, number = {SISSA;03/2009/M}, year = {2010}, abstract = {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.}, doi = {10.1016/j.anihpc.2009.06.005}, url = {http://hdl.handle.net/1963/3409}, author = {Jesus Garcia Azorero and Andrea Malchiodi and Luigi Montoro and Ireneo Peral} } @article {2009, title = {Some new entire solutions of semilinear elliptic equations on Rn}, journal = {Adv. Math. 221 (2009) 1843-1909}, year = {2009}, publisher = {Elsevier}, doi = {10.1016/j.aim.2009.03.012}, url = {http://hdl.handle.net/1963/3645}, author = {Andrea Malchiodi} } @article {2008, title = {Concentrating solutions of some singularly perturbed elliptic equations}, journal = {Front. Math. China 3 (2008) 239-252}, year = {2008}, abstract = {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.}, doi = {10.1007/s11464-008-0015-z}, url = {http://hdl.handle.net/1963/2657}, author = {Andrea Malchiodi} } @article {2008, title = {Entire solutions of autonomous equations on Rn with nontrivial asymptotics}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008) 65-72}, year = {2008}, abstract = {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.}, url = {http://hdl.handle.net/1963/2640}, author = {Andrea Malchiodi} } @article {2008, title = {Existence of conformal metrics with constant $Q$-curvature}, journal = {Ann. of Math. 168 (2008) 813-858}, number = {SISSA;70/2004/M}, year = {2008}, abstract = {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.}, url = {http://hdl.handle.net/1963/2308}, author = {Zindine Djadli and Andrea Malchiodi} } @article {2008, title = {Morse theory and a scalar field equation on compact surfaces}, journal = {Adv. Differential Equations 13 (2008) 1109-1129}, year = {2008}, publisher = {Khayyam Publishing}, url = {http://hdl.handle.net/1963/3531}, author = {Andrea Malchiodi} } @article {2008, title = {Topological methods for an elliptic equation with exponential nonlinearities}, journal = {Discrete Contin. Dyn. Syst. 21 (2008) 277-294}, year = {2008}, abstract = {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.}, doi = {10.3934/dcds.2008.21.277}, url = {http://hdl.handle.net/1963/2594}, author = {Andrea Malchiodi} } @article {2008, title = {Transition layer for the heterogeneous Allen-Cahn equation}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631}, number = {arXiv.org;math/0702878v1}, year = {2008}, abstract = {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 $j \\\\to + \\\\infty$.}, doi = {10.1016/j.anihpc.2007.03.008}, url = {http://hdl.handle.net/1963/2656}, author = {Fethi Mahmoudi and Andrea Malchiodi and Juncheng Wei} } @article {2007, title = {Boundary interface for the Allen-Cahn equation}, journal = {J. Fixed Point Theory Appl. 1 (2007) 305-336}, year = {2007}, doi = {10.1007/s11784-007-0016-7}, url = {http://hdl.handle.net/1963/2027}, author = {Andrea Malchiodi and Juncheng Wei} } @article {2007, title = {Boundary-clustered interfaces for the Allen{\textendash}Cahn equation}, journal = {Pacific Journal of Mathematics 229 (2007), No. 2, 447{\textendash}468}, year = {2007}, publisher = {Mathematical Sciences Publishers}, url = {http://hdl.handle.net/1963/5089}, author = {Andrea Malchiodi and Wei-Ming Ni and Juncheng Wei} } @article {2007, title = {Concentration on minimal submanifolds for a singularly perturbed Neumann problem}, journal = {Adv. Math. 209 (2007) 460-525}, number = {arXiv.org;math/0611558}, year = {2007}, abstract = {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 {\textrightarrow} 0.}, doi = {10.1007/s00039-005-0542-7}, url = {http://hdl.handle.net/1963/4866}, author = {Andrea Malchiodi} } @article {2005, title = {A fourth order uniformization theorem on some four manifolds with large total Q-curvature}, journal = {C. R. Acad. Sci. Paris, Ser. I 340 (2005) 341-346.}, year = {2005}, publisher = {Elsevier}, abstract = {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.}, doi = {10.1016/j.crma.2005.01.013}, url = {http://hdl.handle.net/1963/4868}, author = {Zindine Djadli and Andrea Malchiodi} } @article {2005, title = {Ground states of nonlinear Schroedinger equations with potentials vanishing at infinity}, journal = {J. Eur. Math. Soc. 7 (2005) 117-144}, number = {SISSA;16/2004/M}, year = {2005}, abstract = {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 $\\\\theta= (p+1)/(p-1)-1/2$.}, url = {http://hdl.handle.net/1963/2352}, author = {Antonio Ambrosetti and Veronica Felli and Andrea Malchiodi} } @article {2005, title = {Minimal surfaces in pseudohermitian geometry}, journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177.}, number = {arXiv:math/0401136;}, year = {2005}, publisher = {Scuola Normale Superiore}, abstract = {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.}, doi = {10.2422/2036-2145.2005.1.05}, url = {http://hdl.handle.net/1963/4579}, author = {Jih-Hsin Cheng and JennFang Hwang and Andrea Malchiodi and Paul Yang} } @article {2005, title = {Multiple clustered layer solutions for semilinear Neumann problems on a ball}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163}, year = {2005}, publisher = {Elsevier}, doi = {10.1016/j.anihpc.2004.05.003}, url = {http://hdl.handle.net/1963/3532}, author = {Andrea Malchiodi and Wei-Ming Ni and Juncheng Wei} } @article {2004, title = {Multidimensional boundary layers for a singularly perturbed Neumann problem}, journal = {Duke Math. J. 124 (2004) 105-143}, year = {2004}, publisher = {Duke University Press}, doi = {10.1215/S0012-7094-04-12414-5}, url = {http://hdl.handle.net/1963/2960}, author = {Andrea Malchiodi and Marcelo Montenegro} } @article {2004, title = {Singularity perturbed elliptic equations with symmetry: existence of solutions concetrating on spheres, Part II}, journal = {Indiana Univ. Math. J. 53 (2004) 297-392}, number = {SISSA;93/2002/M}, year = {2004}, publisher = {Indiana University Mathematics Journal}, doi = {10.1512/iumj.2004.53.2400}, url = {http://hdl.handle.net/1963/1663}, author = {Antonio Ambrosetti and Andrea Malchiodi and Wei-Ming Ni} } @article {2004, title = {Solutions concentrating at curves for some singularly perturbed elliptic problems}, journal = {C. R. Acad. Sci. Paris, Ser. I 338 (2004) 775-780}, year = {2004}, publisher = {Elsevier}, doi = {10.1016/j.crma.2004.03.023}, url = {http://hdl.handle.net/1963/4869}, author = {Andrea Malchiodi} } @article {2003, title = {Prescribing scalar and boundary mean curvature on the three dimensional half sphere}, journal = {J. Geom. Anal. 13 (2003) 255-289}, number = {SISSA;36/2001/M}, year = {2003}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/BF02930697}, url = {http://hdl.handle.net/1963/3086}, author = {Zindine Djadli and Andrea Malchiodi and Mohameden Ould Ahmedou} } @article {2003, title = {Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I}, journal = {Comm. Math. Phys. 235 (2003) no.3, 427-466}, number = {SISSA;63/2002/M}, year = {2003}, publisher = {Springer}, doi = {10.1007/s00220-003-0811-y}, url = {http://hdl.handle.net/1963/1633}, author = {Antonio Ambrosetti and Andrea Malchiodi and Wei-Ming Ni} } @article {2002, title = {Curvature theory of boundary phases: the two-dimensional case}, journal = {Interfaces Free Bound. 7 (2002) 345-370}, year = {2002}, publisher = {European Mathematical Society}, abstract = {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.}, url = {http://hdl.handle.net/1963/3537}, author = {Andrea Braides and Andrea Malchiodi} } @article {2002, title = {Prescribing a fourth oder conformal invariant on the standard sphere - Part I: a perturbation result}, journal = {Commun. Contemp. Math., 2002, 4, 375}, number = {SISSA;82/00/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1142/S0219199702000695}, url = {http://hdl.handle.net/1963/1539}, author = {Zindine Djadli and Mohameden Ould Ahmedou and Andrea Malchiodi} } @article {2002, title = {Prescribing a fourth oder conformal invariant on the standard sphere - Part II: blow up analysis and applications}, journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci., 2002, 1, 387}, number = {SISSA;83/00/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1540}, author = {Zindine Djadli and Andrea Malchiodi and Mohameden Ould Ahmedou} } @article {2002, title = {The scalar curvature problem on $S\\\\sp n$: an approach via Morse theory}, journal = {Calc. Var. Partial Differential Equations 14 (2002), no. 4, 429-445}, number = {SISSA;117/99/M}, year = {2002}, publisher = {Springer}, doi = {10.1007/s005260100110}, url = {http://hdl.handle.net/1963/1331}, author = {Andrea Malchiodi} } @article {2002, title = {Singular elliptic problems with critical growth}, journal = {Comm. Partial Differential Equations 27 (2002), no. 5-6, 847-876}, number = {SISSA;54/99/M}, year = {2002}, publisher = {Dekker}, doi = {10.1081/PDE-120004887}, url = {http://hdl.handle.net/1963/1268}, author = {Paolo Caldiroli and Andrea Malchiodi} } @article {2002, title = {Solutions concentrating on spheres to symmetric singularly perturbed problems}, journal = {C.R.Math.Acad.Sci. Paris 335 (2002),no.2,145-150}, number = {SISSA;23/2002/M}, year = {2002}, publisher = {SISSA Library}, abstract = {We discuss some existence results concerning problems (NLS) and (N), proving the existence of radial solutions concentrating on a sphere.}, doi = {10.1016/S1631-073X(02)02414-7}, url = {http://hdl.handle.net/1963/1594}, author = {Antonio Ambrosetti and Andrea Malchiodi and Wei-Ming Ni} } @article {2002, title = {On the Yamabe problem and the scalar curvature problems under boundary conditions}, journal = {Math. Ann., 2002, 322, 667}, number = {SISSA;52/00/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1007/s002080100267}, url = {http://hdl.handle.net/1963/1510}, author = {Antonio Ambrosetti and Li YanYan and Andrea Malchiodi} } @article {2001, title = {Adiabatic limits of closed orbits for some Newtonian systems in R-n}, journal = {Asymptotic Anal., 2001, 25, 149-181}, number = {SISSA;53/00/M}, year = {2001}, publisher = {SISSA Library}, abstract = {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.}, url = {http://hdl.handle.net/1963/1511}, author = {Andrea Malchiodi} } @article {2001, title = {Multiple positive solutions of some elliptic equations in \\\\bold R\\\\sp N}, journal = {Nonlinear Anal. 43 (2001) 159-172}, number = {SISSA;133/1998/M}, year = {2001}, publisher = {Elsevier}, doi = {10.1016/S0362-546X(99)00186-8}, url = {http://hdl.handle.net/1963/3094}, author = {Andrea Malchiodi} } @article {2001, title = {Multiplicity results for some nonlinear Schrodinger equations with potentials}, journal = {Arch. Ration. Mech. An., 2001, 159, 253}, number = {SISSA;108/00/AF}, year = {2001}, publisher = {SISSA Library}, doi = {10.1007/s002050100152}, url = {http://hdl.handle.net/1963/1564}, author = {Antonio Ambrosetti and Andrea Malchiodi and Simone Secchi} } @article {2001, title = {Non-compactness and multiplicity results for the Yamabe problem on Sn}, journal = {J. Funct. Anal. 180 (2001) 210-241}, number = {SISSA;130/99/M}, year = {2001}, publisher = {Elsevier}, doi = {10.1006/jfan.2000.3699}, url = {http://hdl.handle.net/1963/1345}, author = {Massimiliano Berti and Andrea Malchiodi} } @article {2001, title = {On the symmetric scalar curvature problem on S\\\\sp n}, journal = {J. Differential Equations 170 (2001) 228-245}, year = {2001}, publisher = {Elsevier}, abstract = {We discuss some existence results dealing with the scalar curvature problem on S\\\\sp n in the presence of various symmetries.}, doi = {10.1006/jdeq.2000.3816}, url = {http://hdl.handle.net/1963/3095}, author = {Antonio Ambrosetti and Andrea Malchiodi} } @mastersthesis {2000, title = {Existence and multiplicity results for some problems in Riemannian geometry}, year = {2000}, school = {SISSA}, keywords = {Yamabe problem}, url = {http://hdl.handle.net/1963/5948}, author = {Andrea Malchiodi} } @article {2000, title = {A note on the scalar curvature problem in the presence of symmetries}, journal = {Ricerche Mat. 49 (2000), suppl., 169-176}, number = {SISSA;150/99/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1365}, author = {Antonio Ambrosetti and Li YanYan and Andrea Malchiodi} } @article {2000, title = {Scalar curvature under boundary conditions}, journal = {Cr. Acad. Sci. I-Math, 2000, 330, 1013}, number = {SISSA;48/00/M}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0764-4442(00)00312-8}, url = {http://hdl.handle.net/1963/1506}, author = {Antonio Ambrosetti and Li YanYan and Andrea Malchiodi} } @article {1999, title = {A multiplicity result for the Yamabe problem on $S\\\\sp n$}, journal = {J. Funct. Anal. 168 (1999), no. 2, 529-561}, number = {SISSA;50/99/M}, year = {1999}, publisher = {Elsevier}, abstract = {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.}, doi = {10.1006/jfan.1999.3458}, url = {http://hdl.handle.net/1963/1264}, author = {Antonio Ambrosetti and Andrea Malchiodi} } @article {1999, title = {On the scalar curvature problem under symmetry}, number = {SISSA;73/99/M}, year = {1999}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1287}, author = {Antonio Ambrosetti and Andrea Malchiodi} }