In this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrödinger equations, $$-\varepsilon^2\Delta u+V(x)u\mp\varepsilon^{2+\gamma}u\Delta u^2=h(u),\ \ x\in \mathbb{R}^N, $$ where \(N\geqslant3, \varepsilon > 0, V(x)\) is a positive external potential,h is a real function with subcritical or critical growth

We are concerned with the qualitative analysis of solutions for three classes of nonlinear problems driven by the magnetic Laplace operator. We are mainly interested in the perturbation effects created by two reaction terms with different structure. Two equations are studied on bounded domains (under Dirichlet boundary condition) while the third problem is on the entire Euclidean space. Our main results

We provide an overview of technics that lead to an Euclidean upper bound on the volume of geodesic balls.

We are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain \({\Omega} {\subset} \mathbb{R}^{d}, d \geq 1\). When \(\Omega = [0, 1]\), such graphs include the standard Toeplitz graphs and, for \(\Omega = [0, 1]^{d}\), the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices

This study focuses on the existence and concentration of ground state solutions for a class of fractional Kirchhoff–Schrödinger equations. We first study the problem $$\left\{ \begin{array}{ll} M ([u]^{2}_{s} + \int_{\mathbb{R}^{N}} V(x)u^{2}) ((-{\Delta})^{s}u + V (x)u) = \bar{c}u + f(u)\, {\rm in}\,\, \mathbb{R}^N,\\ u > 0, u\, {\in} \, {H}^{s} (\mathbb{R}^N),\end{array} \right.$$ where \(s \in (0

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.

This paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on \(R^n\) and to a family of Hamiltonian systems involving double well potentials will also be discussed.

We study the motion of discrete interfaces driven by ferromagnetic interactions on the two-dimensional triangular lattice by coupling the Almgren, Taylor and Wang minimizing movements approach and a discrete-to-continuum analysis, as introduced by Braides, Gelli and Novaga in the pioneering case of the square lattice. We examine the motion of origin-symmetric convex “Wulfflike” hexagons, i.e. origin-symmetric

We are interested in quasilinear problems as follows: $$ \left\{ \begin{array}{ll} -\Delta u -u \Delta (u^2)= -\lambda |u|^{q-2}u+|u|^{22^*-2}u+\mu g(x,u), \quad \mathrm{in}~ \Omega ,\\ u=0,\quad \mathrm{on}~ \partial \Omega , \end{array}\right. $$(p) where \(\Omega \subset \mathbb {R}^N \)is a bounded domain with regular boundary \(\partial \Omega , N\ge 3, \lambda , \mu > 0,1

We study the existence of positive supersolutions of nonlocal equations of type \((-\Delta)^{s} u + |\Delta u|^{q} = \lambda f(u)\) posed in exterior domains where the datum f can be comparable with \(u^p\) near the origin. We prove that the existence of bounded supersolutions depends on the values of p, q and s.

Let \(\mathbb{D}\) be the two-dimensional real algebra generated by 1 and by a hyperbolic unit k such that \(k^{2} = 1\). This algebra is often referred to as the algebra of hyperbolic numbers. A function \(f : \mathbb{D} \rightarrow \mathbb{D}\) is called \(\mathbb{D}\)-holomorphic in a domain \(\Omega \subset \mathbb{D}\) if it admits derivative in the sense that \({\rm lim}_{h\rightarrow{0}}\fr

Regularization by noise for certain classes of fluid dynamic equations, a theme dear to Giuseppe Da Prato [23], is reviewed focusing on 3D Navier-Stokes equations and dyadic models of turbulence.

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes into account area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean and Gauss curvature, the classical counterexample by

Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong & Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the intrinsic

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. There is nowadays a large literature on the evolution of superoscillations under Schrödinger equation with different type of potentials. In this paper, we study the evolution of superoscillations under the Klein-Gordon equation and we describe in precise mathematical terms in what sense

In this work, we study the existence of a weak solution to the following quasi linear elliptic problem involving the fractional p-Laplacian operator, a Hardy potential, and multiple critical Sobolev nonlinearities with singularities,$$(-{\Delta}_{p})^{s}u - {\mu}\frac{|u|^{p-2}u}{|x|^{ps}}=\frac{|u|^{p^{*}_{s}(\beta)-2_{u}}}{|x|^{\beta}}+\frac{|u|^{p^{*}_{s}{(\alpha)-2_u}}}{|x|^{\alpha}},$$where \(x

The author regrets to point out some errors in [1], even thought these do not influence the validity of the results obtained. The detailed corrections are in the following.

In this paper we investigate the existence of a positive vector solution for a class of non-linear strongly coupled Schrödinger system in \(\mathbb{R}^N\), which is non-autonomous and asymptotically linear at infinity. Using topological arguments, combined with sharp exponential decay estimates, we obtain a bound state solution when the ground state is not attained.

This paper concerns in building extremal curves with respect to the parameters \(\lambda,\mu \geq 0\) for existence and multiplicity of \(D^{1,2}(\mathbb{R}^N)\)-solutions for the multi-parameter elliptic system$$ \left\{\begin{array}{lll} -{\Delta}{u} = {\lambda}{w}(x)f_{1}(u)g_{1}(v)\quad {\rm in}\quad \mathbb{R}^N,\\ -{\Delta}{v} = {\mu}w(x)f_{2}(v)g_{2}(u)\quad {\rm in} \quad \mathbb{R}^N,\\ u

In this paper we address the problem of continuous dependence on initial and boundary data for a one-dimensional dynamic debonding model describing a thin film peeled away from a substrate. The system underlying the process couples the (weakly damped) wave equation with a Griffith’s criterion which rules the evolution of the debonded region. We show that under general convergence assumptions on the

We study the triviality of the solutions of weighted superlinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We prove a Liouville-type theorem for solutions bounded from below with nonnegative initial data, under an integral growth condition on the weight.

In this paper we study the Ostrovsky–Hunter equation for the case where the flux function f(x, u) may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (namely fx(x, u) is uniformly Lipschitz locally in u and fu(x, u) is uniformly bounded), then there exists a unique entropy solution. To show the existence, after proving some a priori

Through the use of linearized bundles, we prove the stability of tautological bundles over the symmetric product of a curve and of the kernel of the evaluation map on their global sections.

In this manuscript it is proved existence results for some singular problems involving an anisotropic operator. In the approach we combine sub-supersolutions, truncation arguments and the Schaefer’s Fixed Point Theorem [23]. In this work it is not used approximation arguments as in [33, 37]

We address the problem of understanding the dynamics around typical singular points of 3D piecewise smooth vector fields. A model Z0 in 3D presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: (i) Z0 has an invariant plane \(\pi_0\) filled up with periodic orbits (this means that the restriction \(Z_{0|\pi_0}\) is a center around the singularity);

In this paper, we show the existence and multiplicity of positive solutions of the fractional Kirchhoff system$$\begin{aligned} {\left\{ \begin{array}{ll}\mathcal {L}_{M}(u) = {\lambda }f(x)|u|^{q-2}{u} + \frac{2{\alpha }}{{\alpha }+{\beta }} |u|^{\alpha -2} \,u|v|^{\beta} &{}\quad \mathrm{in}\, \Omega ,\\ \mathcal {L}_M(v) = {\mu }g(x)|v|^{q-2}v + \frac{2{\beta }}{{\alpha }+{\beta }}|u|^{\alpha} \

Fluid flows around an obstacle generate vortices which are difficult to locate and to describe. A variational formulation for a class of mixed and nonstandard boundary conditions on a smooth obstacle is discussed for the Stokes equations. Possible boundary data are then derived through separation of variables of biharmonic equations in a planar region having an internal concave corner. Explicit singular

The graph Merriman–Bence–Osher scheme produces, starting from an initial node subset, a sequence of node sets obtained by iteratively applying graph diffusion and thresholding to the characteristic (or indicator) function of the node subsets. One result in [14] gives sufficient conditions on the diffusion time to ensure that the set membership of a given node changes in one iteration of the scheme

We are interested in finding nontrivial solutions for a Hamiltonian elliptic system in dimension two involving a potential function which can be coercive and nonlinearities that have maximal growth with respect to the Trudinger–Moser inequality. To establish the existence of solutions, we use variational methods combined with Trudinger–Moser type inequalities in Lorentz–Sobolev spaces and a finite-dimensional

We prove that the law of the minimum \({m := {\rm min}_{t\in[0,1]}\xi(t)}\) of the solution \({\xi}\) to a one-dimensional stochastic differential equation with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets \({\{x \in C([0,1]) : {\rm inf} x \geq r\}}\) have finite perimeter with respect to the law \({\nu}\) of the

Let C be a closed, convex and nonempty subset of a Banach space X. Let \({T : C \rightarrow X}\) be a nonexpansive inward mapping. We consider the boundary point map \({h_{C,T } : C \rightarrow \mathbb{R}}\) depending on C and T defined by \({h_{C,T} = {\rm max}\{\lambda \in [0,1] : [(1-\lambda)x + \lambda Tx] \in C\}}\), for all \({x \in C}\). Then for a suitable step-by-step construction of the control

We discuss the difference between orthogonal polynomials on finite and infinite dimensional vectors spaces. In particular we prove an infinite dimensional extension of Favard lemma.

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration \({\pi : \mathbb{CP}^3 \rightarrow S^4}\). We prove three results about the topology of the twistor discriminant locus of an algebraic surface in \({\mathbb{CP}^3}\). First of all we prove that, with the exception of two special cases, the real dimension of the twistor discriminant

In this paper we prove existence and uniqueness results for a lower order perturbation of elliptic Dirichlet problems with a singular convection term in divergence form and L1 data.

This paper concerns with the existence and regularity of solutions for the following Choquard type equation,$$-\Delta_u = \big(I_{\mu} * F(u)\big) f(u) {\rm in} \mathbb{R}^3, \quad \quad (P)$$where \({I_\mu = \frac{1}{|x|^\mu}, 0 < \mu < 3}\), is the Riesz potential, \({F(s)}\) is the primitive of the continuous function f(s), and \({I_{\mu} * F(u)}\) denotes the convolution of \({I_{\mu}}\) and F(u)

We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional:

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the \({H^\infty}\) functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials

The paper is devoted to the study of some properties of the first eigenvalue of the anisotropic p-Laplace operator with Robin boundary condition involving a function \({\beta}\) which in general is not constant. In particular we obtain sharp lower bounds in terms of the measure of the domain and we prove a monotonicity property of the eigenvalue with respect the set inclusion.

We deal with a severe ill posed problem, namely the reconstruction process of an image during tomography acquisition with (very) few views. We present different methods that we have been investigated during the past decade. They are based on variational analysis. This is a survey paper and we refer to the quoted papers for more details.

Let \({P_{\rm MAX}(d, s)}\) denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree d in \({\mathbb{P}^3}\) that is not contained in a surface of degree < s. A bound P(d, s) for \({P_{\rm MAX}(d, s)}\) has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family \({\mathcal{C}}\)

We prove that nontrivial weak solutions of a class of fractional magnetic Schrödinger equations in \({\mathbb{R}^N}\) are bounded and vanish at infinity.

We consider a mathematical model which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. We model the material’s behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition. We derive variational formulation for the model which is in the form of a system

We give a self-contained and simple approach to prove the existence and uniqueness of a weak solution to a linear elliptic boundary value problem with drift in divergence form. Taking advantage of the method of continuity, we also deal with the relative dual problem.

This study deals with the convergence properties of Struve matrix functions within complex analysis. Certain new classes of matrix differential recurrence relations, matrix differential equations, the various families of integral representations and integrals obtained here are believed to be new in the theory of Struve matrix functions, and the several properties of the modified Struve matrix functions

We consider a generalized Ostrovsky–Hunter equation and the corresponding generalized Ostrovsky one with nonlinear dispersive effects. For the first equation, we study the well-posedness of entropy solutions for the Cauchy problem. For the second equation, we prove that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the

In this work we consider existence and uniqueness of solutions for a quasilinear elliptic problem, which may be singular at the origin. Furthermore, we consider a comparison principle for subsolutions and supersolutions just in \({W^{1, \Phi}_{loc} (\Omega)}\) to the problem$$\left\{\begin{array}{ll}-\Delta_{\Phi}u=f(x,u)\, {\rm in}\, \Omega,\\u > 0\, {\rm in} \, \Omega, u = 0 \,{\rm on}\, \partial\Omega

In this work we state and prove versions of some classical results, in the framework of functionals defined in the space of functions of bounded variation in \({\mathbb{R}^N}\). More precisely, we present versions of the Radial Lemma of Strauss, the compactness of the embeddings of the space of radially symmetric functions of BV (\({\mathbb{R}^N}\)) in some Lebesgue spaces and also a version of the

We deal with the equation$$-\left( 1+\int\nolimits_{\mathbb{R}^3}|\nabla u|^2 dx\right)\Delta u + V(x)u=a(x)|u|^{p-1}u,\quad x\in {\mathbb{R}}^3,$$with p ∈ (3, 5). Under some conditions on the sign-changing potentials V and a we obtain a nonnegative ground state solution. In the radial case we also obtain a nodal solution.

We analyze global bifurcations along the family of radially symmetric vortices in the Gross–Pitaevskii equation with a symmetric harmonic potential and a chemical potential µ under the steady rotation with frequency \({\Omega}\). The families are constructed in the small-amplitude limit when the chemical potential µ is close to an eigenvalue of the Schrö dinger operator for a quantum harmonic oscillator

In this work, we use variational methods to prove the existence of a positive solution for the following class of elliptic problems,$$\left\{ \begin{array}{ll}-\Delta u + u = u^q + \epsilon u^{2^*-1}, \, \,{\rm in}\, \Omega,\\ u > 0, \,\,{\rm in}\,\, \Omega,\\ u \in H^1_0(\Omega),\end{array}\right. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (P_\epsilon) $$where \({\Omega \subset \mathbb{R}^N}\) is an

We give an overview of some results and techniques related to the Mumford–Tate conjecture for motives over finitely generated fields of characteristic 0. In particular, we explain how working in families can lead to non-trivial results.

In this note we present a new proof for the following Phragmen–Lindelöf type result: If \({{0 \leq u \in S_{\lambda,\Lambda}}}\) in the half space \({H^{+}_{n}}\) and u vanishes on the boundary \({\partial H^{+}_{n}}\), then necessarily \(u(x) = u(e_{n}) \cdot x_{n}\).

The aim of this article is to study the uniqueness of a complete spacelike hypersurface \({\sum^{n}}\) immersed with constant mean curvature H in a spatially closed generalized Robertson–Walker spacetime \({\overline{M}^{n+1} = -I {\times_{f}} {M^{n}}}\), whose Riemannian fiber \({M^n}\) has positive curvature. Supposing that the warping function f is such that −log f is convex and \({H{f^{\prime}}

An abstract version of the fourth-order equation$$\partial_{tttt}u+\alpha\partial_{ttt}u+\beta\partial_{tt}u-\gamma\Delta\partial_{tt}u-\delta\Delta\partial_{t}u-\varrho\Delta u=0$$subject to the homogeneous Dirichlet boundary condition is analyzed. Such a model encompasses the Moore–Gibson–Thompson equation with memory in presence of an exponential kernel. The stability properties of the related solution

We study nonhomogeneous elliptic problems considering a general linear elliptic operator with singular critical potentials and nonlinearities depending on multiplier operators that can be derivatives (even fractional) and singular integral operators. The general elliptic operator can contain derivatives of high-order and fractional type like polyharmonic operators and fractional Laplacian. We obtain

We consider the hyperbolic system ü \({ - {\rm div} (\mathbb{A} \nabla u) = f}\) in the time varying cracked domain \({\Omega \backslash \Gamma_t}\), where the set \({\Omega \subset \mathbb{R}^d}\) is open, bounded, and with Lipschitz boundary, the cracks \({\Gamma_t, t \in [0, T]}\), are closed subsets of \({\bar{\Omega}}\), increasing with respect to inclusion, and \({u(t) : \Omega \backslash \Gamma_t

Muramatu’s integral formula is a very useful tool for the study of Sobolev spaces, although this does not seem to be widely recognized. Most theorems in Sobolev spaces can be proved by this formula combined with basic inequalities in analysis, and it is possible to directly treat not only the whole space but also a special Lipschitz domain. In this paper, we present an introduction to L p -based Sobolev

In this paper, we establish existence and asymptotic behaviour of nontrivial weak solution of a class of quasilinear stationary Kirchhoff type equations involving the variable exponent spaces with critical growth, namely$$\begin{aligned}{\left\{ \begin{array}{ll} -M (\mathcal{A}(u)) {\rm div} (a(|\nabla u|^{p(x)}) | \nabla u|^{p(x) - 2} \nabla u) = \lambda f (x, u) + |u|^{s(x)-2} u \quad {\rm in} \quad

We report our recent results from [1, 2] on the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and provide an explicit representation formula. In the case of continuous Cartesian curves, i.e., of graphs \({c_{u}}\) of continuous functions u on an interval, the relaxed energy is finite if and only if the curve \({c_{u}}\)

This is a review of the work of John Forbes Nash, jr., and of his contributions to economics and mathematics. It is a written version of a lecture delivered in Varese, on September 14, 2015, on the occasion of the fourth RISM School (www.rism.it).