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In the paper we consider the following quasilinear Schrödinger–Poisson system in the whole space R 3 - ε 2 Δ u + (V + ϕ) u = u u p - 1 - Δ ϕ - β Δ 4 ϕ = u 2 , where 1 < p < 5 , β > 0 , V : R 3 → ] 0 , ∞ [ , and look for solutions u , ϕ : R 3 → R in the semiclassical regime, namely when ε → 0. By means of the Lyapunov–Schmidt method we estimate the number of solutions by the cup-length of the critical manifold of the external potential V. [ABSTRACT FROM AUTHOR]

ABSTRACT In this study, we explore volume constrained nonlocal problem with integrable kernels. A central challenge addressed is the discontinuity of true solutions across the boundary, posing difficulties for designing efficient numerical methods. To address this issue, we introduce an auxiliary function which serves as the solution of an auxiliary problem. This approach effectively transforms the original problem to a new nonlocal problem with a true solution which is continuous across the boundary. We propose two numerical schemes to solve the new nonlocal problem: one is exclusively applicable to one‐dimensional case, while the other is suitable for general dimensional cases. Experimental results show that both schemes exhibit optimal convergence order under mild condition. [ABSTRACT FROM AUTHOR]

A class of non-local problems, Dρtu(t) + Au(t) = f(t) (1 < ρ < 2, 0 < t ≤ T), αu(0) + βu(ξ) = φ, u'(0) = ψ (α, β are constants and ξ ∈ (0, T) is a fixed number), in an arbitrary separable Hilbert space H is considered. Here A is a strongly positive selfadjoint operator with the domain of definition D(A) and Dρt is the Caputo derivative. The main goal of the work is to study the correctness of the problem depending on the pair of parameters (α, β) ∈ R 2 . If |α| > |β| and φ, ψ ∈ D(A), then the solution of the problem exists and is unique. Inequalities of coercivity type are obtained. [ABSTRACT FROM AUTHOR]

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we aim to tackle the questions of existence and multiplicity of solutions to a new class of κ (ξ) -Kirchhoff-type equation utilizing a variational approach. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of space ψ -fractional H κ (ξ) μ , ν ; ψ (Λ) . In this sense, we present a few special cases and remark on the outcomes explored. [ABSTRACT FROM AUTHOR]

In this paper, we use variational methods to study the existence of nontrivial solutions for a class of Kirchhoff-type elliptic problems driven by the p(x)-Laplacian with triple regime and sign-changing nonlinearity. The main novelty of this paper is our ability to establish an existence result for a class of Kirchhoff-type problems in which the reaction term is sign-changing and exhibits a triple regime (subcritical, critical, and supercritical).

In the present paper, we study the existence and asymptotic behaviour of solutions of a quasilinear Dirac-Poisson system$ \begin{cases} i\alpha\cdot\nabla u-a\beta u-\omega u = \phi u+K(x)|u|^{p-2}u ,\; \; \; \; \; \; x\in\mathbb{R}^3,\\ -\Delta\phi-\varepsilon^4\Delta_4\phi = u^2 ,\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in\mathbb{R}^3, \end{cases} $where $ \alpha\cdot\nabla = \sum^3_{k = 1}\alpha_k\partial_k $, $ \partial_k = \partial/\partial x_k $, $ k = 1,2,3 $, $ \alpha_1, \alpha_2, \alpha_3 $ and $ \beta $ are $ 4\times 4 $ Pauli-Dirac matrices, $ a>0 $ is a constant, the operator $ \Delta_4 $ is the 4-Laplacian, defined as $ \Delta_4 \phi: = div(|\nabla \phi|^2\nabla \phi) $. By applying the variational argument, we establish the existence result of least energy solutions and obtain the regularity and the exponential decay estimate. Furthermore, we also prove that the least energy solutions of the quasilinear Dirac-Poisson system converge to a solution of the limit Dirac-Maxwell system. [ABSTRACT FROM AUTHOR]

This research work deals with two spectral matrix collocation algorithms based on (novel) clique functions to solve two classes of nonlinear nonlocal elliptic two-points boundary value problems (BVPs) arising in diverse physical models. The nonlinearity together with nonlocality makes the models so difficult to solve numerically. In both matrix methods by expanding the unknown solutions in terms of clique polynomials the nonlocality in the models is handled. In the first and direct technique, the clique coefficients are determined in an accurate way through solving an algebraic system of nonlinear equations. To get rid of the nonlinearity of the models and in order to gain efficacy, the quasilinearization method (QLM) is utilized in the second approach. In the latter algorithm named QLM-clique procedure, the first directly clique collocation method is applied to a family of linearized equations in an iterative manner. In particular, we show that the series expansion of clique polynomials is convergent exponentially in a weighted L2 norm. Numerous numerical simulations validate the performance of two matrix collocation schemes and demonstrate the accuracy as well as the gain in computational efficiency in terms of elapsed CPU time. The proposed matrix algorithms for computation are easy to implement and straightforward, and provide more accuracy compared to other available computational values in the literature.

In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems.

In this work, we establish some abstract results on the perspective of the fractional Musielak–Sobolev spaces, such as: uniform convexity, Radon–Riesz property with respect to the modular function, (S +) -property, Brezis–Lieb type Lemma to the modular function and monotonicity results. Moreover, we apply the theory developed to study the existence of solutions to the following class of nonlocal problems (- Δ) Φ x , y s u = f (x , u) , in Ω , u = 0 , on R N \ Ω , where N ≥ 2 , Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω and f : Ω × R → R is a Carathéodory function not necessarily satisfying the Ambrosetti–Rabinowitz condition. Such class of problems enables the presence of many particular operators, for instance, the fractional operator with variable exponent, double-phase and double-phase with variable exponent operators, anisotropic fractional p-Laplacian, among others. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems. [ABSTRACT FROM AUTHOR]

The solvability of spatially nonlocal boundary value problems for one-dimensional parabolic equations, as well as for some equations of the Sobolev type, is studied. We prove theorems on the existence and uniqueness of regular solutions, namely, solutions having all Sobolev generalized derivatives involved in the corresponding equation. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with some qualitative properties of the new fractional Musielak–Sobolev spaces W s L Φ x , y such that the generalized Poincaré type inequality and some continuous and compact embedding results. Moreover, we prove that any function in W s L Φ x , y (Ω) may be extended to a function in W s L Φ x , y (R N) , with Ω ⊂ R N is a bounded domain of class C 0 , 1 . In addition, we establish a result that relates to the complemented subspace in W s L Φ x , y R N . As an application, using the mountain pass theorem and some variational methods, we investigate the existence of a nontrivial weak solution for a class of nonlocal fractional type problems with Dirichlet boundary data. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic system of differential equations with nonlocal discrete diffusion. For two types of noises, we study the existence of either positive or probability solutions. Also, we analyze the asymptotic behavior of solutions in the long term, showing that under suitable assumptions they tend to a neighborhood of the unique deterministic fixed point. Finally, we perform numerical simulations and discuss the application of the results to life tables for mortality in Spain. [ABSTRACT FROM AUTHOR]

In this paper, based on the monotone iterative technique and the operator semigroup theory, a class of nonlocal problem of structural damped elastic systems with delay is studied in the case of noncompact semigroups in ordered Banach space. First, on the premise of the existence of upper and lower mild solutions, the existence and uniqueness of mild solutions for the elastic system are obtained. Then, an existence theorem of positive mild solutions for elastic system is obtained without assuming lower and upper mild solutions. Finally, as the application of abstract results, the existence and uniqueness of mild solutions and positive mild solutions for two classes of nonlocal damped beam vibration equations with delay are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of some non-local analogs of the second initial-boundary value problem for multidimensional hyperbolic and parabolic equations of the second order. We prove the existence and uniqueness theorems of regular solutions (which have all Sobolev generalized derivatives that are summable with a square and are included in the equation). Some generalization and amplification of the obtained results are also given.

We establish a stochastic representation for a class of nonlocal parabolic terminal–boundary value problems, whose terminal and boundary conditions depend on the solution in the interior domain; in particular, the solution is represented as the expectation of functionals of a diffusion process with random jumps from boundaries. We discuss three applications of the representation, the first one on the pricing of dual-purpose funds, the second one on the connection to regenerative processes, and the third one on modeling the entropy on a one-dimensional nonrigid body. [ABSTRACT FROM AUTHOR]

We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the Lq-norm of the solution and the p-Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their Lq-norms.

In this paper, we first introduce the new fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces. Then, using the direct variational approach, we investigate the existence of a nontrivial weak solution for a class of fractional type problems with Dirichlet boundary data of the following form (P a) (− Δ) a (x ,.) s u + a ˆ x (| u |) u = f (x , u) in Ω , u = 0 in R N ∖ Ω , where s ∈ (0 , 1) , Ω is an open bounded subset in R N with Lipschitz boundary ∂ Ω , and f : Ω × R ⟶ R is a Carathéodory function with suitable growth conditions. [ABSTRACT FROM AUTHOR]

We study the solvability of initial-boundary value problems for second-order hyperbolic and parabolic equations with a boundary condition that integrally connects the values of the solution on the lateral boundary with the values of the solution inside the domain. To study such problems, it was previously established that their solvability is ensured by the bijectivity of a certain Fredholm operator constructed from an integral condition. In this paper, we show that the condition of predecessors is not required for the existence and uniqueness of regular solutions (solutions with all derivatives generalized in the sense of Sobolev that are contained in the equation) of integral analogues of the first initial-boundary value problem for second-order hyperbolic and parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the topological structure of the solution set to a fractional differential inclusion with delay defined on the half-line. We first prove that the solution set to the inclusion is an R δ -set on compact intervals. Then, by means of the inverse limit method, we generalize our results to noncompact intervals. Moreover, under convex and nonconvex conditions, an R δ -property solution set is obtained for some nonlocal problems, where the nonlocal function is set-valued. Further, we study the symmetry of the solution set under some conditions. [ABSTRACT FROM AUTHOR]

In the setting of metric measure spaces satisfying the doubling condition and the (1, p)-Poincaré inequality, we prove a metric analogue of the Bourgain–Brezis–Mironescu formula for functions in the Sobolev space W 1 , p (X , d , ν) , under the assumption that for ν -a.e. point the tangent space in the Gromov–Hausdorff sense is Euclidean with fixed dimension N. [ABSTRACT FROM AUTHOR]

For abstract singular equations, nonlocal problems belonging to the class of ill-posed problems are considered. A uniqueness criterion for solutions is established. [ABSTRACT FROM AUTHOR]

In this article we study an equation driven by the nonlocal integrodifferential operator -LK in presence of an asymmetric nonlinear term f. Among the main results of the paper we prove the existence of at least a weak solution for this problem, under suitable assumptions on the asymptotic behavior of the nonlinearity f at 1. Moreover, we show the uniqueness of this solution, under additional requirements on f. We also give a non-existence result for the problem under consideration. All these results were obtained using variational techniques and a monotonicity property of the eigenvalues of -LK with respect to suitable weights, that we prove along the present paper. This monotonicity property is of independent interest and represents the nonlocal counterpart of a famous result obtained by de Figueiredo and Gossez [14] in the setting of uniformly elliptic operators. [ABSTRACT FROM AUTHOR]

Wc formulate and study the nonlocal and local least gradient problem, which is the Dirichlet problem for the I-Laplace operator, in the non-Euclidcan setting of Carnot groups. Wc study the passage from the nonlocal problem to the local problem as the range of the interaction goes to zero. During this procedure, we prove a total variation estimate of independent interest and give an existence result for the local problem. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence of two non-negative solutions for a class of fractional p(x,.)-Laplacian problems with non-negative weight functions. The main tool is the Nehari manifold approach. Moreover, under some suitable assumptions, continuous and compact embeddings results are established. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of weak solutions for a Kirchhoff–Schrödinger type problem in R 4 involving a critical nonlinearity and a suitable small perturbation. When N = 4 , the Sobolev exponent is 2 ∗ = 4 and, as a consequence, there is a tie between the growth for the nonlocal term and critical nonlinearity. Such behaviour causes new difficulties to treat our study from an exclusively variational point of view, besides those already known for the local operators. Some tools we used in this paper are the mountain-pass and Ekeland's Theorems and the Lions' Concentration Compactness Principle. [ABSTRACT FROM AUTHOR]

In this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to investigate the existence of periodic solutions for a class of nonlocal p(t) $p(t)$-Laplacian systems. When the nonlinear term is p+ $p^{+}$-superlinear at infinity, some new solvability conditions of nontrivial periodic solutions are obtained by using a version of the local linking theorem. A major point is that we ensure compactness without the well-known Ambrosetti–Rabinowitz type superlinearity condition. In addition, by applying the saddle point theorem, we established the existence of at least one periodic solution for such problems with a p− $p^{-}$-subquadratic potential.

Abstract We give a relatively short proof of the fact that the solution set of a nonlocal semilinear differential inclusion is dense in the weak solution set of the corresponding convexified system. Moreover, we prove a similar result for the solutions with continuous pseudoderivatives when the right-hand side is continuous with nonempty convex closed and bounded values.

Abstract In this paper, we study the existence of multiple positive solutions to the following Kirchhoff equation with competing potential functions: {−(ε2a+εb∫R3|∇v|2)Δv+V(x)v=K(x)|v|p−1vin R3,v>0,v∈H1(R3), $$ \textstyle\begin{cases} -(\varepsilon ^{2}a+\varepsilon b{\int _{\mathbb{R}^{3}}}{ \vert \nabla v \vert } ^{2})\Delta v+V(x)v=K(x) \vert v \vert ^{p-1}v \quad \mbox{in }\mathbb{R}^{3}, \\ v>0, \quad v\in H^{1}(\mathbb{R}^{3}), \end{cases} $$ where ε>0 $\varepsilon >0$ is a small parameter, a,b>0 $a,b>0$ are constants, 3

In this article, we deal with the nonlinear Schrödinger equation with nonlocal regional diffusion 0.1(−Δ)ραu+V(x)u=f(x,u)inℝN,u∈Hα(ℝN),where 0 < α < 1, n ≥ 2, and f:ℝN×ℝ→ℝ is a continuous function. The operator (−Δ)ρα is a variational version of the nonlocal regional Laplacian defined as ∫ℝN(−Δ)ραu(x)φ(x)dx=∫ℝN∫B(0,ρ(x))[u(x+z)−u(x)][φ(x+z)−φ(x)]|z|N+2αdzdx,where ρ∈C(ℝN,ℝ+) be a positive function. Considering that ρ, V, and f(· , t) are periodic or asymptotically periodic at infinity, we prove the existence of ground state solution of (1) by using Nehari manifold and comparison method. Furthermore, in the periodic case, by combining deformation‐type arguments and Lusternik–Schnirelmann theory, we prove that problem (1) admits infinitely many geometrically distinct solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhoff-type equations with Robin boundary data as follows: − M ∫ Ω 1 p (x) ∇ u p (x) d x + ∫ ∂ Ω β (x) p (x) ∇ u p (x) d σ div( ∇ u p(x)-2 ∇ u)=f(x,u)in Ω , ∇ u p (x) − 2 ∂ u ∂ v + β (x) u p (x) − 2 u = 0 on ∂ Ω , Where β ∈ L ∞ (∂ Ω) and f : Ω × ℝ → ℝ satisfies Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the topological structure of the solution set to a fractional differential inclusion with delay defined on the half-line. We first prove that the solution set to the inclusion is an Rδ-set on compact intervals. Then, by means of the inverse limit method, we generalize our results to noncompact intervals. Moreover, under convex and nonconvex conditions, an Rδ-property solution set is obtained for some nonlocal problems, where the nonlocal function is set-valued. Further, we study the symmetry of the solution set under some conditions.

In this article, our aim is to obtain weak solutions for a class of Kirchhoff-type problems − M (∫ Ω A (x , ∇ u) d x) div a (x , ∇ u) = f (x , u) in Ω , by using the theory of Young measures. [ABSTRACT FROM AUTHOR]

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is u t = ∫ ℝ N J (x − y) (u (y , t) − u (x , t)) 𝒢 (u (y , t) − u (x , t)) d y in  Ω × (0 , T) , u (x , 0) = u 0 (x) in  Ω , where we take, as the most important instance, 𝒢 (s) ∼ 1 + μ 2 s 1 + μ 2 s 2 with μ ∈ ℝ as well as u 0 ∈ L 1 (Ω) , J is a smooth symmetric function with compact support and Ω is either a bounded smooth subset of ℝ N , with nonlocal Dirichlet boundary condition, or ℝ N itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation. [ABSTRACT FROM AUTHOR]

Preface to the BPMAS special volume dedicated to the workshop: Something about nonlinear problems, Bologna, June 13-14, 2019.

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0

In the rectangular domain, the initial-boundary value problem for the Helmholtz equation and its non-local modifications are studied and the inverse problems for finding its right-hand side are studied. The solutions of direct problems with nonlocal boundary conditions and inverse problems are constructed in explicit form as the sums of orthogonal series in the system of eigenfunctions of the one-dimensional Sturm-Liouville spectral problem. The corresponding uniqueness theorems for the solution of all set problems are proved. Sufficient conditions for boundary functions are established, which are guaranteed by the existence and stability theorems for the solution of the proposed new problem statements.

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular L1. The commonly used Lipschitz condition on the right-hand side is weakened to a one-sided Lipschitz one. No compactness assumptions are required. We consider the cases of an arbitrary one-sided Lipschitz condition and the case of a negative one-sided Lipschitz constant. Illustrative examples, which can be modifications of real models, are provided.

In this paper, we consider the regularization of a class of elliptic variational‐hemivariational inequalities driven by the fractional Laplace operator. First, we demonstrate characterizations of nonlocal elliptic variational‐hemivariational inequalities. Next, we provide coercivity conditions that guarantee the existence and uniqueness of solution. Finally, by virtue of the so‐called Browder–Tikhonov regularization method, we introduce a strongly convergent approximation procedure for the considered nonlocal problems considered without any coercivity condition. [ABSTRACT FROM AUTHOR]

Nonlocal problems have been used to model very different applied scientific phenomena which involve the fractional Laplacian when one looks at the Lévy processes and stochastic interfaces. This paper deals with the nonlocal problems on a bounded domain where the stiffness matrices of the resulting systems are Toeplitz plus tridiagonal or far from being diagonally dominant as occurs when dealing with linear finite element approximations. By exploiting a weakly diagonally dominant Toeplitz property of the stiffness matrices, the optimal convergence of the two-grid method is well established in [Fiorentino and Serra-Capizzano, SIAM J. Sei. Comput., 17 (1996), pp. 1068-1081; Chen and Deng, SIAM J. Matrix Ano,I. Appl., 38 (2017), pp. 869-890], and there are still questions about the best ways to define the coarsening and interpolation operators when the stiffness matrix is far from being weakly diagonally dominant [Stuben, J. Comput. Appl. Math., 128 (2001), pp. 281-309]. In this work, using spectral indications from our analysis of the involved matrices, the simple (traditional) restriction operator and prolongation operator are employed in order to handle general algebraic systems which are neither Toeplitz nor weo,kly d,io,gono,lly domino,nt corresponding to the fractional Laplacian kernel and the constant kernel, respectively. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such situations. Moreover, the convergence of the full multigrid is also discussed with the constant kernel. The numerical experiments are performed to verify the convergence with only O(NlogN) complexity by the fast Fourier transform, where N is the number of the grid points. [ABSTRACT FROM AUTHOR]

In this paper, starting from a non-convex and nonlocal 3D variational mathematical model for the electric polarization in a ferroelectric material, and using an asymptotic process based on dimensional reduction, we analyze junction phenomena for two orthogonal joined ferroelectric wires. Depending on the initial boundary conditions, we get several different limit problems, sometimes uncoupled. We point out that all the limit problems remain non-convex, but the nonlocality disappears. [ABSTRACT FROM AUTHOR]

In this paper we consider the following quasilinear Schrödinger–Poisson system - Δ u + u + ϕ u = λ f (x , u) + | u | 4 u in R 3 - Δ ϕ - ε 4 Δ 4 ϕ = u 2 in R 3 , depending on the two parameters λ , ε > 0 . We first prove that, for λ larger than a certain λ ∗ > 0 , there exists a solution for every ε > 0 . Later, we study the asymptotic behaviour of these solutions whenever ε tends to zero, and we prove that they converge to the solution of the Schrödinger–Poisson system associated. [ABSTRACT FROM AUTHOR]