Dimitri Mugnai

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Verified

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• PhD in Mathematics
• Professor (Full) at University of Tuscia

We study the null controllability for a degenerate/singular wave equation with drift in non divergence form. In particular, considering a control localized on the non degenerate boundary point, we provide some conditions for the boundary controllability via energy methods and boundary observability.

We prove the existence of entire, radial, and signed bounded solutions for a quasilinear elliptic equation in {{\mathbb{R}}}^{N} driven by a Leray-Lions operator of the (p, q)-type. For this, we need an extension of related results by Boccardo-Murat-Puel and a variational approach in intersections of Banach spaces introduced by Candela-Palmieri.

We study the stabilization of degenerate 1-D wave equations in non divergence form with drift. The degeneracy takes place in one boundary point and the stabilization is obtained by a nonlinear damping in the nondegeneracy one.

We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the as...

We study an elliptic quasilinear fractional problem with fractional Neumann boundary conditions, proving an existence and multiplicity result without assuming the classical Ambrosetti–Rabinowitz condition. Improving previous results, we also provide the weak formulation of solutions without regularity assumptions and we provide an example, even in...

We study a quasilinear fractional problem with fractional Neumann boundary conditions. Improving previous results, we also provide the weak formulation of solutions without regularity assumptions and we provide and example, even in the linear case, for which such a regularity cannot indeed be assumed.

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions f...

We study a nonlinear, nonlocal Dirichlet problem driven by the fractional p-Laplacian, involving a \((p-1)\)-sublinear reaction. By means of a weak comparison principle we prove uniqueness of the solution. Also, comparing the problem to ’asymptotic’ weighted eigenvalue problems for the same operator, we prove a necessary and sufficient condition fo...

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions f...

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions f...

We study the null controllability for a degenerate/singular wave equation with drift in non divergence form. In particular, considering a control localized on the non degenerate boundary point, we provide some conditions for the boundary controllability via energy methods and boundary observability.

In this paper we prove the existence of signed bounded solutions for a quasilinear elliptic equation in \({\mathbb {R}}^N\) driven by a Leray–Lions operator of (p, q)–type in presence of unbounded potentials. A direct approach seems to be a hard task, and for this reason we will study approximating problems in bounded domains, whose resolutions nee...

We prove the existence and multiplicity of weak solutions for a mixed local-nonlocal problem at resonance. In particular, we consider a not necessarily positive operator which appears in models describing the propagation of flames. A careful adaptation of well known variational methods is required to deal with the possible existence of negative eig...

We prove the existence of a weak solution for boundary value problems driven by a mixed local–nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive definite.

In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used...

We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the as...

In this paper, we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e. [Formula: see text] Our main result is resemblant to the celebrated work by Brezis–Oswald [Remarks on sublinear elliptic equat...

We prove the existence of a weak solution for boundary value problems driven by a mixed local--nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive definite.

We study a nonlinear, nonlocal Dirichlet problem driven by the fractional p-Laplacian, involving a (p-1)-sublinear reaction. By means of a weak comparison principle we prove uniqueness of the solution. Also, comparing the problem to 'asymptotic' weighted eigenvalue problems for the same operator, we prove a necessary and sufficient condition for th...

We consider some problems governed by the sum of a Laplacian and of a fractional Laplacian in presence of so–called (α,β)-Neumann conditions in an essentially linear context. Indeed, first we show an existence result for asymptotically linear elliptic problems, and then we establish some qualitative properties for solutions of the associated homoge...

In this note we complete the study started in Biagi et al. (2021) providing a full characterization of the existence of a unique positive weak solution of a p-sub-li-near Dirichlet boundary value problem driven by a mixed local-nonlocal operator.

We consider a degenerate wave equation with drift in presence of a leading operator which is not in divergence form. We provide some conditions for the boundary controllability of the associated Cauchy problem.

This paper is a corrigendum of one hypothesis introduced in Mem. Amer. Math. Soc. 242 (2016), no. 1146, and used again in J. Differential Equations 260 (2016), pp. 1314–1371 and Adv. Nonlinear Anal. 6 (2017), pp. 61–84]. We give here the corrected proofs of the concerned results, improving most of them.

We consider nonlocal problems in which the leading operator contains a sign-changing weight which can be unbounded. We begin studying the existence and the properties of the first eigenvalue. Then we study a nonlinear problem in which the nonlinearity does not satisfy the usual Ambrosetti-Rabinowitz condition. Finally, we study a problem with gener...

We give some fundamental definitions and some Hardy-type inequalities with boundary or interior degeneracy. We also show the equivalence between null controllability and observability inequality.

We consider parabolic problems in divergence and non divergence form with interior degeneracy and singularity given by general functions, showing well posedness and null controllability.

We consider non degenerate singular parabolic problems, giving some existence or non existence results, which depend on the value of the parameter of the singular term. Null controllability results are presented, as well.

We consider parabolic problems in divergence form with boundary degeneracy and power singularity, showing well posedness and null controllability.

We show Carleman estimates for parabolic problems in divergence or non divergence form with degeneracy at the boundary or in the interior of the space domain. By them we obtain observability inequalities, proving that the problems are null controllable.

In this paper we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e., $$\mathcal{L}_{p,s} = -\Delta_p + (-\Delta)^s_p.$$ Our main result is resemblant to the celebrated work by Brezis-Oswald [10]....

We consider nonlinear problems governed by the fractional p-Laplacian in presence of nonlocal Neumann boundary conditions and we show three different existence results: the first two theorems deal with a p-superlinear term, the last one with a source having p-linear growth. For the p-superlinear case we face two main difficulties. First: the p-supe...

We first prove that solutions of fractional p-Laplacian problems with nonlocal Neumann boundary conditions are bounded and then we apply such a result to study some resonant problems by means of variational tools and Morse theory.

Nowadays, energy represents the most important resource; however, we need to face several energy-related rising issues, one main concern is how energy is consumed. In particular, how we can stimulate consumers on a specific behaviour. In this work, we present a model facing energy allocation and payment. Thus, we start with the explanation of the f...

We consider nonlinear problems governed by the fractional $p-$Laplacian in presence of nonlocal Neumann boundary conditions. We face two problems. First: the $p-$superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the lin...

We consider nonlocal problems in which the leading operator contains a sign-changing weight which can be unbounded. We begin studying the existence and the properties of the first eigenvalue. Then we study a nonlinear problem in which the nonlinearity doesn't satisfy the usual Ambrosetti-Rabinowitz condition. Finally, we study a problem with genera...

We consider a boundary value problem driven by the p-fractional Laplacian with nonlocal Robin boundary conditions and we provide necessary and sufficient conditions which ensure the existence of a unique positive (weak) solution. The results proved in this paper can be considered a first step towards a complete generalization of the classical resul...

We deal with fractional generalized logistic problems in presence of a signed and unbounded weight. We describe the first eigenpair of the underlying operators and we show a bifurcation result for positive solutions, which are proved to be unique. A symmetry result is established under suitable geometric constraints.

We show the existence of nontrivial solutions for a class of highly quasilinear problems in which the governing operators depend on the unknown function. By using a suitable variational setting and a weak version of the Cerami-Palais-Smale condition, we establish the desired result without assuming that the nonlinear source satisfies the Ambrosetti...

We develop some properties of the p−Neumann derivative for the fractional p−Laplacian in bounded domains with general p > 1. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear probl...

In this paper, we consider fractional Choquard equations with confining potentials. First, we show that they admit a positive ground state and infinitely many bound states. Then we prove the existence of two signed solutions when a superlinear and subcritical perturbation is added; in this case, the main feature is that such a perturbation does not...

We study the existence of radially symmetric solutions for a nonlinear planar Schrödinger-Poisson system in presence of a superlinear reaction term which doesn’t satisfy the Ambrosetti-Rabinowitz condition. The system is re-written as a nonlinear Hartree equation with a logarithmic convolution term, and the existence of a positive and a negative so...

We consider a system for a generalized Schnakenberg model, showing Turing pattern formation for a wide class of nonlinearities.

We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear p...

We consider a nonlinear elliptic equation with Robin boundary condition driven by the p−Laplacian and with a reaction term which depends also on the gradient. By using a topological approach based on the Leray-Schauder alternative principle, we show the existence of a smooth solution. © American Institute of Mathematical Sciences. All rights reserv...

We prove a null controllability result for a parabolic problem with Neumann boundary conditions. We consider non smooth coefficients in presence of a strongly singular potential and a strongly degenerate coefficient, both vanishing at an interior point. This paper concludes the study of the Neumann case.

We prove a null controllability result for a parabolic Dirichlet problem with non smooth coefficients in presence of strongly singular potentials and a coefficient degenerating at an interior point. We cover the case of weights falling out the class of Muckenhoupt functions, so that no Hardy-type inequality is available; for instance, we can consid...

In this paper we show existence and multiplicity results for a linearly perturbed elliptic problem driven by nonlocal operators, whose prototype is the fractional Laplacian. More precisely, when the perturbation parameter is close to one of the eigenvalues of the leading operator, the existence of three nontrivial solutions is proved.

In this work, we face a payment estimation problem that involves a community of users and an energy distributor (or producer). Our aim is to compute payments for every user in the community according to the single user's consumption, the community's consumption and the available energy. The proposed scheme influences the community in consuming in a...

We prove the multiplicity result in [12] under more general assumptions. More precisely, we prove the existence of three nontrivial solutions for a nonlocal problem when a parameter approaches one of the eigenvalues of the leading operator, without assuming the Ambrosetti-Rabinowitz condition.

We establish Hardy - Poincar\'e and Carleman estimates for non-smooth degenerate/singular parabolic operators in divergence form with Neumann boundary conditions. The degeneracy and the singularity occur both in the interior of the spatial domain. We apply these inequalities to deduce well-posedness and null controllability for the associated evolu...

In this work, we face a payment estimation problem that involves a community of users and an energy distributor (or producer). Our aim is to compute a payment function in relation to single user's consumption and available energy. We want to influence the community in consuming in a virtuous way. In order to reach this goal, our payment function di...

We study a partial differential inclusion, driven by the p-Laplacian operator, involving a p-superlinear nonsmooth potential, and subject to Neumann boundary conditions. By means of nonsmooth critical point theory, we prove the existence of at least two constant sign solutions (one positive, the other negative). Then, by applying the nonsmooth Mors...

We consider existence and multiplicity results for a semilinear problem driven by the square root of the negative Laplacian in presence of a nonlinear term which is the difference of two powers. In the case of convex–concave powers, a precise description of the problem at the threshold value of a given parameter is established through variational m...

In this paper, first we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sublevels of the associated functional, which per...

We consider a parametric nonlinear Robin problem driven by the p-Laplacian plus an indefinite potential and a Carathéodory reaction which is (p-1)- superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the ex...

We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on observability results through Carleman estimates for the associated adjoint problem. The novelties of the present paper are two. First, the coefficient of the leading operator only belongs to a Sobolev space. Second, the degeneracy point is allowe...

We consider a class of pseudo-relativistic Hartree equations in presence of general nonlinearities not satisfying the Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory, we show the existence of two non trivial signed solutions, one positive and one negative.

We consider a class of pseudo-relativistic Hartree equations in presence of general nonlinearities not satisfying the Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory, we show the existence of two non trivial signed solutions, one positive and one negative.

In this paper we consider a nonlinear elliptic problem driven by a nonhomogeneous differential operator with Robin boundary conditions. We produce conditions on the reaction term near

We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. First, we prove an existence theorem, and then, under stronger conditions on the reaction, we prove a multiplicity theorem producing three nontrivial solutions. The...

We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.

We show the existence of a nontrivial ground state solution for a class of nonlinear pseudo-relativistic systems in the entire space.

The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus \Omega\,, \end{array} \right. $$ where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator, $\lamb...

We establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non smooth coeffi...

We study the existence of cylindrically symmetric electro-magneto-static solitary waves for a system of a nonlinear Klein-Gordon equation coupled with Maxwell's equations in presence of a positive mass and of a nonnegative nonlinear potential. Nonexistence results are provided as well.

We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applie...

First, we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sub-levels of the associated functional, which permits to apply...

We consider a nonlinear elliptic equation driven by the sum of a p-Laplacian and a q-Laplacian, where 1<q≤2≤p<∞ with a (p-1)-superlinear Carathéodory reaction term which doesn’t satisfy the usual Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the pro...

We consider an elliptic problem driven by the square root of the negative Laplacian in the presence of a general logistic function having an indefinite weight. We prove a bifurcation result for the associated Dirichlet problem via regularity estimates of independent interest for when the weight belongs only to certain Lebesgue spaces.

We consider a diffusive p-logistic equation in the whole of R-N with absorption and an indefinite weight. Using variational and truncation techniques, we prove a bifurcation theorem and describe completely the bifurcation point. In the semilinear case p = 2, under an additional hypothesis on the absorption term, we show that the positive solution i...

In this paper we prove a general result giving the maximum and the antimaximum principles in a unitary way for linear operators of the form L+λIL+λI, provided that 00 is an eigenvalue of LL with associated constant eigenfunctions. To this purpose, we introduce a new notion of “quasi”–uniform maximum principle, named kk–uniform maximum principle: it...

We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on Carleman estimates for the associated adjoint problem. The novelty of interior degeneracy does not let us adapt previous Carleman estimates to our situation. As an application, observability inequalities are established.

In this paper, we face the question of describing the incremental motion of pre-stressed isotropic homogeneous compressible viscoelastic materials of differential type. We obtain a set of linear evolution equations which generalizes the previous mathematical description of the problem. Well-posedeness of the associated Cauchy problems and dissipati...

We consider quasilinear degenerate variational inequalities with pointwise constraint on the values of the solutions. The limit problem as the domain becomes unbounded in some directions is exhibited.

We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential. First we develop the spectral properties of such differential operators. Subsequently, using these spectral properties and variational methods based on critical point theory, truncation techniques and Morse theory, we prove existence and multiplicity theo...

We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on controllability results through Carleman estimates for the associated adjoint problem. The novelty of the present paper is that the degeneracy is also in the interior of the control region, so that no previous result can be adapted to this situati...

We prove several existence and non existence results of solitary waves for a class of nonlinear pseudo-relativistic Hartree equations with general nonlinearities. We use variational methods and some new variational identities involving the half Laplacian.

We show the incompleteness of a usually used version of the generalized Ambrosetti–Rabinowitz condition in superlinear problems, also used in the paper cited in the title, and we propose a complete one.

We study the existence of radially symmetric solitary waves for a nonlinear Schrödinger-Poisson system. In contrast to all previous results, we consider the presence of a positive potential, of interest in physical applications.

We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.

In this paper we consider linear operators of the form L+ λI between suitable functions spaces, when 0 is an eigenvalue of L with constant associated eigenfunctions.We introduce a new notion of “quasi”-uniform maximum principle, named k-uniform maximum principle, which holds for ? belonging to certain neighborhoods of 0 depending on k eR\epsilon\,...

We study the existence of radially symmetric solitary waves for a system of a nonlinear Klein–Gordon equation coupled with Maxwell's equation in presence of a positive mass. The nonlinear potential appearing in the system is assumed to be positive and with more than quadratical growth at infinity.

We prove some maximum principle results for weak solutions of elliptic inequalities, possibly inhomogeneous, on complete Riemannian manifolds.

We establish some maximum and comparison principles for weak distributional solutions of anisotropic elliptic inequalities in divergence form, both in the homogeneous and non-homogeneous cases. The main prototypes we have in mind are inequalities involving the p(⋅)-Laplace operator and the generalized mean curvature operator.

We show that a semilinear Dirichlet problem in bounded domains of \mathbbR2{\mathbb{R}}^2 in presence of subcritical exponential nonlinearities has four nontrivial solutions near resonance.

Inspired by a biological model on genetic repression proposed by P. Jacob and J. Monod, we introduce a new class of delay equations with nonautonomous past and nonlinear delay operator. With the aid of some new techniques from functional analysis we prove that these equations, which cover the biological model, are well–posed.

We consider two classes of semilinear wave equations with nonnegative damping which may be of type "on-o" or integrally positive. In both cases we give a sucient condition for the asymptotic stability of the solutions. In the case of integrally positive damping we show that such a condition is also necessary.

In this note we show that reversed variational inequalities cannot be studied in a general abstract framework as it happens for classical variational inequalities with Stampacchia's Lemma. Indeed, we provide two different situ- ations for reversed variational inequalities which are of the same type from an abstract point of view, but which behave q...

We prove maximum and comparison principles for weak distributional solutions of quasilinear, possibly singular or degenerate, elliptic differential inequalities in divergence form on complete Riemannian manifolds. A new definition of ellipticity for nonlinear operators on Riemannian manifolds is introduced, covering the standard important examples....

We study local bifurcation from an eigenvalue with multiplicity greater than one for a class of semilinear elliptic equations. In particular, we obtain the exact number of bifurcation branches of non trivial solutions at every eigenvalue of a square and at the second eigenvalue of a cube. We also compute the Morse index of the solutions in those br...

We prove the existence of three distinct nontrivial solutions for a class of semilinear elliptic variational inequalities involving a superlinear nonlinearity. The approach is variational and is based on linking and ∇-theorems. Some nonstandard adaptations are required to overcome the lack of the Palais–Smale condition.

We study local bifurcation from an eigenvalue with multiplicity greater than one for a class of semilinear elliptic equations. We evaluate the exact number of bifurcation branches of non trivial solutions and we compute the Morse index of the solutions in those branches.