# Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.49 (2017)

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**CONTENTS / SOMMARIO**

Drábek Pavel, Kufner Alois

Hardy inequality, compact embeddings and properties of certain eigenvalue problems

Torres Pedro J.

The prescribed mean curvature problem with Neumann boundary conditions in FLRW spacetimes

Schmitt Klaus

On the existence of nontrivial solutions of differential equations subject to linear constraints

Orsina Luigi, Ponce Augusto C.

Flat solutions of the 1-Laplacian equation

Clapp Mónica, Rizzi Matteo

Benevieri Pierluigi, Calamai Alessandro, Furi Massimo, Pera Maria Patrizia

On general properties of n-th order retarded functional differential equations

Krisztin Tibor, Walther Hans-Otto

Smoothness issues in differential equations with state-dependent delay

Ponce Augusto C., Van Schaftingen Jean

Ekeland Ivar

Hamilton-Jacobi on the symplectic group

Došlá Zuzana, Marini Mauro, Matucci Serena

Positive decaying solutions to BVPs with mean curvature operator

Mavinga Nsoki, Nkashama Mubenga N.

Rynne Bryan P.

Caicedo Francisco, Castro Alfonso, Duque Rodrigo, Sanjuán Arturo

The semilinear wave equation with non-monotone nonlinearity: a review

Franca Matteo, Johnson Russell

Remarks on nonautonomous bifurcation theory

Gurban Daniela, Jebelean Petru

Positive radial solutions for systems with mean curvature operator in Minkowski space

Margheri Alessandro, Ortega Rafael, Rebelo Carlota

On a family of Kepler problems with linear dissipation

Antón Inmaculada, López-Gómez Julián

Principal eigenvalues of weighted periodic-parabolic problems

Bonanno Gabriele

Dirichlet problems without asymptotic conditions on the nonlinear term

Sovrano Elisa, Zanolin Fabio

A periodic problem for first order differential equations with locally coercive nonlinearities

Massamba Fortuné, Ntumba Patrice P.

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- PublicationA periodic problem for first order differential equations with locally coercive nonlinearities(EUT Edizioni Università di Trieste, 2017)
;Sovrano, ElisaZanolin, FabioIn this paper we study the periodic boundary value problem associated with a first order ODE of the form x' + g(t, x) = s where s is a real parameter and g is a continuous function, T-periodic in the variable t. We prove an Ambrosetti-Prodi type result in which the classical uniformity condition on g(t, x) at infinity is considerably relaxed. The Carathéodory case is also discussed.385 172 - PublicationBifurcation from infinity and multiplicity of solutions for nonlinear periodic boundary value problems(EUT Edizioni Università di Trieste, 2017)
;Mavinga, NsokiNkashama, Mubenga N.We are concerned with multiplicity and bifurcation results for solutions of nonlinear second order diff erential equations with general linear part and periodic boundary conditions. We impose asymptotic conditions on the nonlinearity and let the parameter vary. We then proceed to establish a priori estimates and prove multiplicity results (for large-norm solutions) when the parameter belongs to a (nontrivial) continuum of real numbers. Our results extend and complement those in the literature. The proofs are based on degree theory, continuation methods, and bifurcation from infinity techniques.369 370 - PublicationDirichlet problems without asymptotic conditions on the nonlinear term(EUT Edizioni Università di Trieste, 2017)Bonanno, GabrieleThis paper is devoted, with my great esteem, to Jean Mawhin. Jean Mawhin, who is for me a great teacher and a very good friend, is a fundamental reference for the research in nonlinear differential problems dealt both with topological and variational methods. Here, owing to this occasion in honor of Jean Mawhin, Dirichlet problems depending on a parameter are investigated, ensuring the existence of non-zero solutions without requiring asymptotic conditions neither at zero nor at infinity on the nonlinear term which, in addition, is not forced by subcritical or critical growth. The approach is based on a combination of variational and topological tools that in turn are developed by starting from a fundamental estimate.
344 166 - PublicationFlat solutions of the 1-Laplacian equation(EUT Edizioni Università di Trieste, 2017)
;Orsina, LuigiPonce, Augusto C.Abstract. For every f ∈ Ln(Ω) defined in an open bounded subset Ω of Rn, we prove that a solution u ∈ W01,1 (Ω) of the 1-Laplacian equation – div (∇u / |∇u|)= f in Ω satisfies ∇u = 0 on a set of positive Lebesgue measure. The same property holds if f ∈/ Ln(Ω) has small norm in the Marcinkiewicz space of weak–Ln functions or if u is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia’s truncation method.370 182 - PublicationGauge-measurable functions(EUT Edizioni Università di Trieste, 2017)
;Ponce, Augusto C.Van Schaftingen, JeanIn 1973, E. J. McShane introduced an alternative definition of the Lebesgue integral based on Riemann sums, where gauges are used to decide what tagged partitions are allowed. Such an approach does not require any preliminary knowledge of Measure Theory. We investigate in this paper a definition of measurable functions also based on gauges. Its relation to the gauge-integrable functions that satisfy McShane’s definition is obtained using elementary tools from Real Analysis. We show in particular a dominated integration property of gauge-measurable functions.394 325 - PublicationGlobal stability, or instability, of positive equilibria of p-Laplacian boundary value problems with p-convex nonlinearities(EUT Edizioni Università di Trieste, 2017)Rynne, Bryan P.We consider the parabolic, initial value problem vt = Δp(v) + λg(x, v)φp(v), in Ω x (0,∞), v = 0, in ∂Ω x (0,∞), (IVP) v = v0 > 0, in Ω x {0}, where Ω is a bounded domain in RN , for some integer N > 1, with smooth boundary ∂Ω, φp(s) := |s|p−1 sgn s , s ∈ R , and Δp denotes the p -Laplacian, with p > max{2,N} , v0 ∈ C0(Ω) , and λ > 0 . The function g : Ω x [0,∞) → (0,∞) is C0 and, for each x ∈ Ω , the function g(x, ·) : [0,∞) → (0,∞) is Lipschitz continuous and strictly increasing. Clearly, (IVP) has the trivial solution v ≡ 0 , for all λ > 0 . In addition, there exists 0 < λmin(g) < λmax(g) such that: • if λ ∈/ (λmin(g),λmax(g)) then (IVP) has no non-trivial, positive equilibrium; • there exists a closed, connected set of positive equilibria bifurcating from (λmax(g), 0) and ‘meeting infinity’ at λ = λmin(g) . We prove the following results on the positive solutions of (IVP): • if 0 < λ < λmin(g) then the trivial solution is globally asymptotically stable; • if λmin(g) < λ < λmax(g) then the trivial solution is locally asymptotically stable and all non-trivial, positive equilibria are unstable; • if λmax(g) < λ then any non-trivial solution blows up in finite time.
348 138 - PublicationHamilton-Jacobi on the symplectic group(EUT Edizioni Università di Trieste, 2017)Ekeland, IvarThe classical Hamilton-Jacobi-Bellman theory in the calculus of variations, which is associated with the Bolza problem, is extended to other kinds of boundary-value problems, such as periodicity. By using the dual action principle of Clarke and earlier results by the author, one can establish the analogue of HJB on the symplectic group and show that it has a solution.
273 164 - PublicationHardy inequality, compact embeddings and properties of certain eigenvalue problems(EUT Edizioni Università di Trieste, 2017)
;Drábek, PavelKufner, AloisWe point out the connection between the Hardy inequality, compact embedding of weighted function spaces and the properties of the spectra of certain eigenvalue problems. Necessary and sufficient conditions in terms of the Muckenhoupt function are formulated.469 358 - PublicationOn a family of Kepler problems with linear dissipation(EUT Edizioni Università di Trieste, 2017)
;Margheri, Alessandro ;Ortega, RafaelRebelo, CarlotaWe consider the dissipative Kepler problem for a family of dissipations that is linear in the velocity. Under mild assumptions on the drag coefficient, we show that its forward dynamics is qualitatively similar to the one obtained in [15] and [16] for a constant drag coefficient. In particular, we extend to this more general framework the existence of a continuous vector-valued first integral I obtained as the limit along the trajectories of the Runge-Lenz vector. We also establish the existence of asymptotically circular orbits, so improving the result about the range of I contained in [16].360 166 - PublicationOn general properties of n-th order retarded functional differential equations(2017)
;Benevieri, Pierluigi ;Calamai, Alessandro ;Furi, MassimoPera, Maria PatriziaConsider the second order RFDE (retarded functional differential equation) x’’ (t) = f(t, xt), where f is a continuous realvalued function defined on the Banach space R x C1([−r, 0],R). The weak assumption of continuity on f (due to the strong topology of C1([−r, 0],R)) makes not convenient to transform this equation into a first order RFDE of the type z’ (t) = g(t, zt). In fact, in this case, the associated R2-valued function g could be discontinuous (with the C0- topology) and, in addition, not necessarily defined on the whole space R x C([−r, 0],R2). Consequently, in spite of what happens for ODEs, the classical results regarding existence, uniqueness, and continuous dependence on data for first order RFDEs could not apply. Motivated by this obstruction, we provide results regarding general properties, such as existence, uniqueness, continuous dependence on data and continuation of solutions of RFDEs of the type x(n)(t) = f(t, xt), where f is an Rk-valued continuous function on the Banach space R x C(n−1)([−r, 0],Rk). Actually, for the sake of generality, our investigation will be carried out in the case of infinite delay.348 327 - PublicationOn sheaves of differential operators(EUT Edizioni Università di Trieste, 2017)
;Massamba, FortunéNtumba, Patrice P.Given a C∞ manifold X, denote by Cm / X the sheaf of m-times differentiable real-valued functions and by Dm,r / X the sheaf of differential operators of order ≤ m with coefficient functions of class C r . We prove that the natural morphism Dm−r,r / X → H omRX(Cm / X , Cr / X) is an isomorphism.312 144 - PublicationOn the existence of nontrivial solutions of differential equations subject to linear constraints(EUT Edizioni Università di Trieste, 2017)Schmitt, KlausThe purpose of this paper is to consider boundary value problems for second order ordinary diff erential equations where the solutions sought are subject to a host of linear constraints (such as multipoint constraints) and to present a unifying framework for studying such. We show how Leray-Schauder continuation techniques may be used to obtain existence results for nontrivial solutions of a variety of nonlinear second order diff erential equations. A typical example may be found in studies of the four-point boundary value problem for the diff erential equation y’’(t)+a(t)f(y(t)) = 0 on [0, 1], where the values of y at 0 and 1 are each some multiple of y(t) at two interior points of (0, 1). The techniques most often used in such studies have their origins in fixed point theory. By embedding such problems into parameter dependent ones, we show that detailed information may be obtained via global bifurcation theory. Of course, such techniques, as they are consequences of properties of the topological degree, are similar in nature.
329 314 - PublicationPositive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents(EUT Edizioni Università di Trieste, 2017)
;Clapp, MónicaRizzi, MatteoWe study the problem −Δv + λv = |u|p−2 u in Ω, u= 0 on ∂Ω, for λ ∈ R and supercritical exponents p, in domains of the form Ω := {(y, z) ∈ RN−m−1 x Rm+1 : (y, |z|) ∈ Θ}, where m ≥ 1, N − m ≥ 3, and Θ is a bounded domain in RN−m whose closure is contained in RN−m−1 x (0,∞). Under some symmetry assumptions on Θ, we show that this problem has infinitely many solutions for every λ in an interval which contains [0,∞) and p > 2 up to some number which is larger than the (m+1)st critical exponent 2∗N,m := 2(N−m)/N−m−2 . We also exhibit domains with a shrinking hole, in which there are a positive and a nodal solution which concentrate on a sphere, developing a single layer that blows up at an m dimensional sphere contained in the boundary of Ω, as the hole shrinks and p → 2∗N,m from above. The limit profile of the positive solution, in the transversal direction to the sphere of concentration, is a rescaling of the standard bubble, whereas that of the nodal solution is a rescaling of a nonradial sign-changing solution to the problem −Δu = |u|2∗n−2 u, u ∈ D1,2(Rn), where 2∗n := 2n n−2 is the critical exponent in dimension n.323 145 - PublicationPositive decaying solutions to BVPs with mean curvature operator(EUT Edizioni Università di Trieste, 2017)
;Došlá, Zuzana ;Marini, MauroMatucci, SerenaA boundary value problem on the whole half-closed interval [1,∞) , associated to diff erential equations with the Euclidean mean curvature operator or with the Minkowski mean curvature operator is here considered. By using a new approach, based on a linearization device and some properties of principal solutions of certain disconjugate second-order linear equations, the existence of global positive decaying solutions is examined.475 327 - PublicationPositive radial solutions for systems with mean curvature operator in Minkowski space(EUT Edizioni Università di Trieste, 2017)
;Gurban, DanielaJebelean, PetruWe are concerned with a Dirichlet system, involving the mean curvature operator in Minkowski space M(w) = div (∇w / 1−|∇w|2) in a ball in RN. Using topological degree arguments, critical point theory and lower and upper solutions method, we obtain non existence, existence and multiplicity of radial, positive solutions. The examples we provide involve Lane-Emden type nonlinearities in both sublinear and superlinear cases.468 225 - PublicationPrincipal eigenvalues of weighted periodic-parabolic problems(EUT Edizioni Università di Trieste, 2017)
;Antón, InmaculadaLópez-Gómez, JuliánBased on a recent characterization of the strong maximum principle, [3], this paper gives some periodic parabolic counterparts of some of the results of Chapters 8 and 9 of J. L´opez-G´omez [22]. Among them count some pivotal monotonicity properties of the principal eigenvalue σ[P+V,B,QT ], as well as its concavity with respect to the periodic potential V through a point-wise periodic-parabolic Donsker–Varadhan min-max characterization. Finally, based on these findings, this paper sharpens, substantially, some classical results of A. Beltramo and P. Hess [4], K. J. Brown and S. S. Lin [6], and P. Hess [14] on the existence and uniqueness of principal eigenvalues for weighted boundary value problems.318 267 - PublicationRemarks on nonautonomous bifurcation theory(EUT Edizioni Università di Trieste, 2017)
;Franca, MatteoJohnson, RussellWe study some elementary bifurcation patterns when the bifurcation parameter is subjected to fast oscillations.347 151 - PublicationRendiconti dell’Istituto di matematica dell’Università di Trieste. An International Journal of Mathematics. Vol. 49 (2017)(EUT Edizioni Università di Trieste, 2017)Università degli Studi di Trieste, Dipartimento di Matematica e InformaticaRendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in.
232 1017 - PublicationSmoothness issues in differential equations with state-dependent delay(EUT Edizioni Università di Trieste, 2017)
;Krisztin, TiborWalther, Hans-OttoFor differential equations with state-dependent delays a satisfactory theory is developed by the second author [6] on the solution manifold to guarantee C1 -smoothness for the solution operators. We present examples showing that better than C1 -smoothness cannot be expected in general for the solution manifold and for local stable manifolds at stationary points on the solution manifold. Then we propose a new approach to overcome the diffi culties caused by the lack of smoothness. The mollification technique is used to approximate the nonsmooth evaluation map with smooth maps. Several examples show that the mollified systems can have nicer smoothness properties than the original equation. Examples are also given where better smoothness than C1 can be obtained on the solution manifold.386 194 - PublicationThe prescribed mean curvature problem with Neumann boundary conditions in FLRW spacetimes(EUT Edizioni Università di Trieste, 2017)Torres, Pedro J.We provide sufficient conditions for the existence of solution of the radially symmetric prescribed curvature problem with Neumann boundary condition on a general Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime.
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