527 research outputs found
Uniqueness of ground states for quasilinear elliptic equations in the exponential case
No full textThe boundary value problem and notation are as in previous paper of Pucci and Serrin [Indiana Univ. Math. J. 47 (1998), no. 2, 501-528]. A uniqueness theorem for positive solutions is now proved for nonlinearities f of exponential type. In order that the main theorem of the authors' paper can be applied, the only difficulty is verification of the hypothesis (F/f)′≥0. This is very delicate. Also, under the stronger regularity condition on the elliptic part, ground states with connected support are shown to be radially symmetric in the plane, and hence unique up to translations by the main theorem of Pucci and Serrin. Related results were obtained recently by Adimurthi [Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 5, 895-906]
Entire solutions for some critical equations in the Heisenberg group
No full textWe complete the study started in the paper [P. Pucci, L. Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal p-Laplacian equations
Concentration compactness results for systems in the Heisenberg group
No full textIn this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895-922] on some variants of the concentration-compactness principle in bounded PS domains Ω of the Heisenberg group [formula]. The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms
The impact of the mountain pass theory in nonlinear analysis: a mathematical survey
No full textThis article provides a survey on mountain pass theory. The mountain pass theory is a useful tool to find the critical points of functionals, and it plays a central role in analysis of nonlinear problems, especially nonlinear differential equations. Various versions of the mountain pass theorem have been investigated. We state the generalizations to nonsmooth functionals of three versions of the mountain pass theorem (the case of mountains of positive altitude), the Pucci-Serrin theorem (the case of mountains of zero altitude), and the Goussoub-Preiss theorem. Ekeland's variational principle is a central tool to prove these results on nonsmooth functionals. Finally, several relevant applications to semilinear elliptic partial differential equations are submitted
Geometric description of the mountain pass critical points
No full textIn this short commentary I present the main results contained in the famous paper [P. Pucci & J. Serrin, The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc. 299 (1987), no. 1, 115–132]. This paper deals with critical points obtained via the mountain pass lemma of A.
Ambrosetti and P.H. Rabinowitz. The structure of such critical points is studied with a remarkable precision and several general results are obtained, such as, for example, in the case of an infinite-dimensional problem, a general alternative between saddle points of mountain pass type or a set of local minima whose closure intersects at least two components of the set of saddle points. The questions which arose in the paper inspired a number of further research papers and the commentary lists the applications
A three critical points result in a bounded domain of a Banach space and applications
No full textUsing the bounded mountain pass lemma and the Ekeland variational principle we prove a bounded version of the Pucci-Serrin three critical points result in the intersection of a ball with a wedge in a Banach space. The localization constraints are overcome by boundary and invariance conditions. The result is applied to obtain multiple positive solutions for some semilinear problems
Existence of entire solutions for a class of variable exponent elliptic equations
No full textThe paper deals with the existence of entire solutions for a quasilinear equation (Eλ) in RN, depending on a real parameter λ, which involves a general variable exponent elliptic operator A in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value λ∗ > 0 with the property that (Eλ) admits nontrivial nonnegative entire solutions if and only if λ ≥ λ∗. Furthermore, under the further assumption that the potential of A is uniform convex, we give the existence of a second independent nontrivial nonnegative entire solution of (Eλ), when λ > λ∗. Our results extend the previous work of [G. Autuori and P. Pucci, Nonlinear Differential Equations Appl. NoDEA 20 (2013), 977–1009] from the case of constant exponents p, q and r to the case of variable exponents. More interesting, we weaken the condition max{2, p} 2. Hence
the results of this paper are new even in the canonical case p(·) ≡ 2
Energy-based localization and multiplicity of radially symmetric states for the stationary p-Laplace diffusion
No full textIt is investigated the role of the state-dependent source-term for the localization by means of the kinetic energy of radially symmetric states for the stationary p-Laplace diffusion. It is shown that the oscillatory behavior of the source-term, with respect to the state amplitude, yields multiple possible states, located in disjoint energy bands. The mathematical analysis makes use of critical point theory in conical shells and of a version of Pucci-Serrin three-critical point theorem for the intersection of a cone with a ball. A key ingredient is a Harnack type inequality in terms of the energetic norm
An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains
No full textWe consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying Ambrosetti-Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni, Lancelotti and Perera in 2010, we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p=2 and complements some recent results obtained in a recent paper of Autuori, Pucci and Varga in 2013
L'épopée des usciti dans le Centiloquio de Antonio Pucci
No full textNel proemio del Centiloquio., Antonio Pucci indica apertamente l'opera che è la fonte storica della sua narrazione : la Nuova Cronica di Giovanni Villani. L'autore esplicita anche il motivo della trasposizione in versi della cronaca del suo concittadino : il «diletto » che i lettori potranno trarre dalla sua opera faciliterà la memorizzazione del contenuto. Quest'intenzione si traduce in un lungo racconto della storia di Firenze, il cui titolo e la cui forma metrica -la terzina di endecasillabi - sono inoltre un omaggio, seppur indiretto, a Dante, illustre precursore e concittadino. La storia e le passioni fiorentine forniscono a Pucci la materia di una narrazione che assume toni epici. La costruzione di tale "voce" narrante costituisce l'oggetto di questo contributo.Dans le proemio du Centiloquio, Antonio Pucci affiche ouvertement la source qui sert de fondement historique à sa narration : la Nuova cronica de Giovanni Villani. Pucci justifie également la raison de sa transposition en vers : le «plaisir » (diletto) que les lecteurs pourront en tirer facilitera la mémorisation du contenu. Pucci livre ainsi un long récit sur l'histoire de Florence, qui, par son titre et par sa forme métrique -le tercet d'hendécasyllabes —, se veut un hommage explicite, bien qu'indirect, à Dante, son illustre prédécesseur et concitoyen. L'histoire et les passions florentines fournissent à Pucci la matière d'une narration à la tournure épique. C'est la construction de cette "voix" narrative qui fait l'objet de cette contribution.Gasparini Patrizia. L'épopée des usciti dans le Centiloquio de Antonio Pucci. In: Arzanà 16-17, 2013. Écritures de l’exil dans l’Italie médiévale, sous la direction de Anna Fontes Baratto et Marina Gagliano. pp. 87-112
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