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Carlo MERCURI

Professore Associato
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2022 - New multiplicity results for critical p-Laplacian problems [Articolo su rivista]
Mercuri, C.; Perera, K.
abstract

We prove new multiplicity results for the Brézis-Nirenberg problem for the p-Laplacian. Our proofs are based on a new abstract critical point theorem involving the Z2-cohomological index that requires less compactness than the (PS) condition.

2021 - Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger–Poisson systems [Articolo su rivista]
Dutko, T.; Mercuri, C.; Tyler, T. M.
abstract

We study a nonlinear Schrödinger–Poisson system which reduces to the nonlinear and nonlocal PDE -Δu+u+λ2(1ω|x|N-2⋆ρu2)ρ(x)u=|u|q-1ux∈RN,where ω= (N- 2) | SN-1| , λ> 0 , q∈ (1 , 2 ∗- 1) , ρ: RN→ R is nonnegative, locally bounded, and possibly non-radial, N= 3 , 4 , 5 and 2 ∗= 2 N/ (N- 2) is the critical Sobolev exponent. In our setting ρ is allowed as particular scenarios, to either (1) vanish on a region and be finite at infinity, or (2) be large at infinity. We find least energy solutions in both cases, studying the vanishing case by means of a priori integral bounds on the Palais–Smale sequences and highlighting the role of certain positive universal constants for these bounds to hold. Within the Ljusternik–Schnirelman theory we show the existence of infinitely many distinct pairs of high energy solutions, having a min–max characterisation given by means of the Krasnoselskii genus. Our results cover a range of cases where major loss of compactness phenomena may occur, due to the possible unboundedness of the Palais–Smale sequences, and to the action of the group of translations.

2020 - Groundstate asymptotics for a class of singularly perturbed p-Laplacian problems in RN [Articolo su rivista]
Albalawi, W.; Mercuri, C.; Moroz, V.
abstract

We study the asymptotic behaviour of positive groundstate solutions to the quasilinear elliptic equation [Figure not available: see fulltext.]where 1 < p< N, p< q< l< + ∞ and ε> 0 is a small parameter. For ε→ 0 , we give a characterization of asymptotic regimes as a function of the parameters q, l and N. In particular, we show that the behaviour of the groundstates is sensitive to whether q is less than, equal to, or greater than the critical Sobolev exponent p∗:=pNN-p.

2020 - On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density [Articolo su rivista]
Mercuri, C.; Tyler, T. M.
abstract

In the spirit of the classical work of P.H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system {equation presented} under different assumptions on ρ: R3→ R+at infinity. Our results cover the range p ∈ (2, 3) where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a 'limiting problem' at infinity and of the possible unboundedness of the Palais-Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem {equation presented}, in various functional settings which are suitable for both variational and perturbation methods.

2019 - A Liouville theorem for the p-Laplacian and related questions [Articolo su rivista]
Farina, Alberto; Mercuri, Carlo; Willem, Michel
abstract

We prove several classification results for p-Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of sign-changing solutions to p-Laplacian equations on RN involving critical nonlinearities. Moreover, on radial domains we characterise the compactness of possibly sign-changing Palais-Smale sequences.

2019 - Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations [Articolo su rivista]
Mercuri, C.; Moreira Dos Santos, E.
abstract

We consider a class of weighted Emden-Fowler equations [EQUATION PRESENTED] posed on the unit ball B = B(0,1)⊂RN, N ≤ 1. We prove that symmetry breaking occurs for the groundstate solutions as the parameter α ← &inf;. The above problem reads as a possibly large perturbation of the classical Hénon equation. We consider a radial function Vα having a spherical shell of zeroes at |x| = R &insin; (0,1). For N ≤ 3, a quantitative condition on R for this phenomenon to occur is given by means of universal constants, such as the best constant for the subcritical Sobolev's embedding H10(B) ⊂ Lp+1(B). In the case N = 2 we highlight a similar phenomenon when R = R(α) is a function with a suitable decay. Moreover, combining energy estimates and Liouville type theorems we study some qualitative and quantitative properties of the groundstate solutions to (Pα) as α ← &inf;.

2018 - Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces [Articolo su rivista]
Bellazzini, J.; Ghimenti, M.; Mercuri, C.; Moroz, V.; Van Schaftingen, J.
abstract

We prove scaling invariant Gagliardo–Nirenberg type inequalities of the form (Formula Presented) involving fractional Sobolev norms with s > 0 and Coulomb type energies with 0 < α < d and q ≥ 1. We establish optimal ranges of parameters for the validity of such inequalities and discuss the existence of the optimizers. In the special case p = 2d/d-2s our results include a new refinement of the fractional Sobolev inequality by a Coulomb term. We also prove that if the radial symmetry is taken into account, then the ranges of validity of the inequalities could be extended and such a radial improvement is possible if and only if α > 1.

2016 - A regularity result for the p-laplacian near uniform ellipticity [Articolo su rivista]
Mercuri, C.; Riey, G.; Sciunzi, B.
abstract

We consider weak solutions to a class of Dirichlet boundary value problems involving the p-Laplace operator, and prove that the second weak derivatives are in Lq with q as large as it is desirable, provided p is suffciently close to p0 = 2. We show that this phenomenon is driven by the classical Calderón-Zygmund constant. As a byproduct of our analysis we show that C1,α regularity improves up to C1,1, when p is close enough to 2. This result we believe is particularly interesting in higher dimensions n>2, when optimal C1,α regularity is related to the optimal regularity of p-harmonic mappings, which is still open.

2016 - Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency [Articolo su rivista]
Mercuri, C.; Moroz, V.; Van Schaftingen, J.
abstract

We study the nonlocal Schrödinger–Poisson–Slater type equation -Δu+(Iα ∗ |u|p)|u|p-2u=|u|q-2u in RN,where N∈ N, p> 1 , q> 1 and Iα is the Riesz potential of order α∈ (0 , N). We introduce and study the Coulomb–Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.

2015 - On Coron's problem for the p-Laplacian [Articolo su rivista]
Mercuri, C.; Sciunzi, B.; Squassina, M.
abstract

We prove that the critical problem for the p-Laplacian operator admits a nontrivial solution in annular shaped domains with sufficiently small inner hole. This extends Coron's result [4] to a class of quasilinear problems.

2014 - On the pure critical exponent problem for the p-Laplacian [Articolo su rivista]
Mercuri, C.; Pacella, F.
abstract

In this paper we prove existence and multiplicity of positive and sign-changing solutions to the pure critical exponent problem for the p-Laplacian operator with Dirichlet boundary conditions on a bounded domain having nontrivial topology and discrete symmetry. Pioneering works related to the case p = 2 are Brezis and Nirenberg (Comm Pure Appl Math 36, 437-477, 1983), Coron (C R Acad Sci Paris Sr I Math 299, 209-212, 1984), and Bahri and Coron (Comm. Pure Appl. Math. 41, 253-294, 1988). A global compactness analysis is given for the Palais-Smale sequences in the presence of symmetries. © 2013 Springer-Verlag Berlin Heidelberg.

2013 - Global compactness for a class of quasi-linear elliptic problems [Articolo su rivista]
Mercuri, C.; Squassina, M.
abstract

We prove a global compactness result for Palais-Smale sequences associated with a class of quasi-linear elliptic equations on exterior domains. © 2012 Springer-Verlag.

2012 - Concentration on circles for nonlinear schrödinger-poisson systems with unbounded potentials vanishing at infinity [Articolo su rivista]
Bonheure, D.; Di Cosmo, J.; Mercuri, C.
abstract

The present paper is devoted to weighted nonlinear Schrödinger-Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle. © 2012 World Scientific Publishing Company.

2011 - Embedding theorems and existence results for nonlinear Schrödinger-Poisson systems with unbounded and vanishing potentials [Articolo su rivista]
Bonheure, D.; Mercuri, C.
abstract

Motivated by existence results for positive solutions of non-autonomous nonlinear Schrödinger-Poisson systems with potentials possibly unbounded or vanishing at infinity, we prove embedding theorems for weighted Sobolev spaces. We both consider a general framework and spaces of radially symmetric functions when assuming radial symmetry of the potentials. © 2011 Elsevier Inc.

2010 - A global compactness result for the P-laplacian involving critical nonlinearities [Articolo su rivista]
Mercuri, C.; Willem, M.
abstract

We prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.

2009 - Foliations of small tubes in Riemannian manifolds by capillary minimal discs [Articolo su rivista]
Fall, M. M.; Mercuri, C.
abstract

Letting Γ be an embedded curve in a Riemannian manifold M, we prove the existence of minimal disc-type surfaces centered at Γ inside the surface of revolution of M around Γ, having small radius, and intersecting it with constant angles. In particular we obtain that small tubular neighborhoods can be foliated by minimal discs. © 2008 Elsevier Ltd. All rights reserved.

2009 - Minimal disc-type surfaces embedded in a perturbed cylinder [Articolo su rivista]
Fall, M. M.; Mercuri, C.
abstract

In the present note, we deal with small perturbations of an infinite cylinder in three-dimensional Euclidian space. We find minimal disc-type surfaces embedded in the cylinder and intersecting its boundary perpendicularly. The existence and localization of those minimal discs is a consequence of a non-degeneracy condition for the critical points of a functional related to the oscillations of the cylinder from the flat configuration.

2008 - Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity [Articolo su rivista]
Mercuri, C.
abstract

We deal with a weighted nonlinear Schrodinger-Poisson system, allowing the potentials to vanish at infinity.