Francesca De Marchis

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

We consider the semilinear Lane-Emden problem in a smooth bounded domain of the plane. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions as the exponent p of the nonlinearity goes to infinity. Among other results we show, under some symmetry assumptions on the domain, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p goes to infinity, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in the plane.

On the Ambjorn-Olesen electroweak condensates

We obtain sufficient conditions for the existence of the Ambjorn-Olesen [“On electroweak magnetism,” Nucl. Phys. B315, 606–614 (1989)10.1016/0550-3213(89)90004-7] electroweak N-vortices in case N ⩾ 1 and therefore generalize earlier results [D. Bartolucci and G. Tarantello, “Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory,” Commun. Math. Phys. 229, 3–47 (2002)10.1007/s002200200664; J. Spruck and Y. Yang, “On multivortices in the electroweak theory I: Existence of periodic solutions,” Commun. Math. Phys. 144, 1–16 (1992)10.1007/BF02099188] which handled the cases N ∈ {1, 2, 3, 4}. The variational argument provided here has its own independent interest as it generalizes the one adopted by Ding et al. [“Existence results for mean field equations,” Ann. Inst. Henri Poincare, Anal. Non Lineaire 16, 653–666 (1999)10.1016/S0294-1449(99)80031-6] to obtain solutions for Liouville-type equations on closed 2-manifolds. In fact, we obtain at once...

Supercritical Mean Field Equations on Convex Domains and the Onsager’s Statistical Description of Two-Dimensional Turbulence

We are motivated by the study of the Microcanonical Variational Principle within Onsager’s description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and “thin” enough domains in the supercritical (with respect to the Moser–Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains. We then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the Mean Field Equation on thin domains whose energy is uniformly bounded from above. Finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and, combining it with all the results obtained so far, we solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is shown to be concave.

Exact Morse index computation for nodal radial solutions of Lane–Emden problems

We consider the semilinear Lane–Emden problem where B is the unit ball of

Morse index and sign-changing bubble towers for Lane–Emden problems

\mathbb {R}^N

Prescribed Gauss curvature problem on singular surfaces

RN,

Generic multiplicity for a scalar field equation on compact surfaces

N\ge 2