Dušan D. Repovš

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• Ph.D.
• Professor at University of Ljubljana

This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-\Delta)_{p}^{s}u+\varepsilon^{qs}(-\Delta)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{\mu-N}[|x|^{-\mu}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac...

This paper deals with the following fractional [Formula: see text]-Choquard equation with exponential growth of the form: [Formula: see text] [Formula: see text] where [Formula: see text] [Formula: see text] is a parameter, [Formula: see text] and [Formula: see text] The nonlinear function [Formula: see text] has an exponential growth at infinity a...

In this chapter we give a geometric representation of $H_{n}(B;\mathbb{L})$ classes, where $\mathbb{L}$ is the $4$-periodic surgery spectrum, by establishing a relationship between the normal cobordism classes ${\mathcal{N}}^{H}_{n}(B,\partial)$ and the $n$-th $\mathbb{L}$-homology of $B$, representing the elements of $H_{n}(B;\mathbb{L})$ by norma...

An $n$-component link $L$ is said to be \emph{Brunnian} if it is non-trivial but every proper sublink of $L$ is trivial. The simplest and best known example of a hyperbolic Brunnian link is the 3-component link known as "Borromean rings". For $n\geq 2,$ we introduce an infinite family of $n$-component Brunnian links with positive integer parameters...

This article is devoted to the interplay between productively Menger and productively Hurewicz subspaces of the Cantor space. In particular, we show that in the Laver model for the consistency of the Borel's conjecture these two notions coincide and characterize Hurewicz spaces. On the other hand, it is consistent with CH that there are productivel...

This paper is concerned with the existence of normalized ground state solutions for the mass supercritical fractional nonlinear Schrödinger equation involving a critical growth in the fractional Sobolev sense. The compactness of Palais–Smale sequences will be obtained by a special technique, which borrows from the ideas of Soave (J. Funct. Anal. 27...

We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into...

We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into...

The article is about an elliptic problem defined on a {\it stratified Lie group}. Both sub- and superlinear cases are considered whose solutions are guaranteed to exist in light of the interplay between the nonlinearities and the weak $L^1$ datum. The existence of infinitely many solutions is proved for suitable values of $\lambda,p,q$ by using the...

The article is about an elliptic problem defined on a stratified Lie group. Both sub and superlinear cases are considered whose solutions are guaranteed to exist in light of the interplay between the nonlinearities and the weak L 1 datum. The existence of infinitely many solutions is proved for suitable values of λ , p , q by using the Symmetric Mo...

We study identities of Lie superalgebras over a field of characteristic zero. We construct a series of examples of finite-dimensional solvable Lie superalgebras with non-nilpotent commutator subalgebra for which the PI-exponent of codimension growth exists and is an integer number. We study identities of Lie superalgebras over a field F of characte...

We study identities of Lie superalgebras over a field of characteristic zero. We construct a series of examples of finite-dimensional solvable Lie superalgebras with a non-nilpotent commutator subalgebra for which PI-exponent of codimension growth exists and is an integer number.

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results.

В статье изучаются тождества супералгебр Ли над полем нулевой характеристики. Построена серия примеров конечномерных разрешимых супералгебр с ненильпотентным коммутантом, для которых PI-экспонента роста коразмерностей сцществует и является целым числом.

The aim of this paper is to study existence results for a singular problem involving the $p$-biharmonic operator and the Hardy potential. More precisely, by combining monotonicity arguments with the variational method, the existence of solutions is established. By using the Nehari manifold method, the multiplicity of solutions is proved. An example...

The aim of this paper is to study existence results for a singular problem involving the p-biharmonic operator and the Hardy potential. More precisely, by combining monotonicity arguments with the variational method, the existence of solutions is established. By using the Nehari manifold method, the multiplicity of solutions is proved. An example i...

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results.

We consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of (2,q)-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.

We consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of (2, q )-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.

We study the following critical Choquard equation on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle {-\Delta_H u }={\mu}|u|^{q-2}u+\int_{\Omega}\frac{|u(\eta)|^{Q_{\lambda}^{\ast}}} {|\eta^{-1}\xi|^{\lambda}}d\eta|u|^{Q_{\lambda}^{\ast}-2}u &\mbox{in }\ \Omega, \\ u=0 &\mbox{on }\ \partial\Omega, \end{cases} \end{equation*} whe...

We consider a class of noncooperative Schrödinger-Kirchhoff type system which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration-compac...

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parame...

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parame...

We consider a class of noncooperative Schrödinger–Kirchhof–type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration–compac...

In this paper, we study existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional p-Laplacian on the Heisenberg group: M(u p µ)(µ(−∆) s p u +V (ξ)|u| p−2 u) = f (ξ , u) + H N |u(η)| Q * λ |η −1 ξ | λ dη|u| Q * λ −2 u in H N , where (−∆) s p is the fractional p-Laplacian on the Heisenberg gro...

We show that every Hausdorff Baire topology $\tau$ on $\mathcal{C}=\langle a,b\mid a^2b=a, ab^2=b\rangle$ such that $(\mathcal{C},\tau)$ is a semitopological semigroup is discrete and we construct a nondiscrete Hausdorff semigroup topology on $\mathcal{C}$. We also discuss the closure of a semigroup $\mathcal{C}$ in a semitopological semigroup and...

In this article, we deal with the following p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M [u] p s,A (−∆) s p,A u + V (x)|u| p−2 u = λ R N |u| p * µ,s |x − y| µ dy |u| p * µ,s −2 u + k|u| q−2 u, x ∈ R N , where 0 < s < 1 < p, ps < N , p < q < 2p * s,µ , 0 < µ < N , λ and k a...

In this article, we deal with the following p p -fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M ( [ u ] s , A p ) ( − Δ ) p , A s u + V ( x ) ∣ u ∣ p − 2 u = λ ∫ R N ∣ u ∣ p μ , s * ∣ x − y ∣ μ d y ∣ u ∣ p μ , s * − 2 u + k ∣ u ∣ q − 2 u , x ∈ R N , M({\left[u]}_{s,A}^{p}){\le...

We consider the following convective Neumann systems: ∂η = 0 = |∇u 2 | p 2-2 ∂u 2 ∂η on ∂ , where is a bounded domain in R N (N ≥ 2) with a smooth boundary ∂ , δ 1 , δ 2 > 0 are small parameters, η is the outward unit vector normal to ∂ , f 1 , f 2 : × R 2 × R 2N → R are Carathéodory functions that satisfy certain growth conditions, and p i (1 < p...

In this paper, we study the existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional p-Laplacian on the Heisenberg group: M(u p µ)(µ(−∆) s p u +V (ξ)|u| p−2 u) = f (ξ , u) + H N |u(η)| Q * λ |η −1 ξ | λ dη|u| Q * λ −2 u in H N , where (−∆) s p is the fractional p-Laplacian on the Heisenberg...

In this paper we construct consistent examples of subgroups of $2^\omega$ with Menger remainders which fail to have other stronger combinatorial covering properties. This answers several open questions asked by Bella, Tokgoz and Zdomskyy (Arch. Math. Logic 55 (2016), 767-784).

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability type condition on compact Riemann-ian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic...

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic...

This paper is concerned with existence results for the singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. More precisely , by using variational methods combined with the Mountain pass theorem and the Ekeland variational principle, we establish the existence and multiplicity of solutions. To illustra...

This article concerns the existence and multiplicity of solutions for the singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. To this end we use variational methods combined with the Mountain pass theorem and the Ekeland variational principle. We illustrate the usefulness of our results with and exam...

In this article, we investigate the Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group of the following form: \begin{document}$ \begin{equation*} \left\{ \begin{array}{lll} {-(a+b\int_{\Omega}|\nabla_{H} u|^{p}d\xi)\Delta_{H, p}u-\mu\phi |u|^{p-2}u} = \lambda |u|^{q-2}u+|u|^{Q^{\ast}-2}u &\mbox{in}\ \Omega, \\ -\Delta_{H}\phi = |u|^...

In this article, we investigate the Kirchhoff-Schr\"{o}dinger-Poisson type systems on the Heisenberg group of the following form: \begin{equation*} \left\{ \begin{array}{lll} {-(a+b\int_{\Omega}|\nabla_{H} u|^{p}d\xi)\Delta_{H,p}u-\mu\phi |u|^{p-2}u}=\lambda |u|^{q-2}u+|u|^{Q^{\ast}-2}u &\mbox{in}\ \Omega, \\ -\Delta_{H}\phi=|u|^{p} &\mbox{in}\ \Om...

We study a class of Schrödinger-Poisson systems with (p,q)-Laplacian. Using fixed point theory, we obtain a new existence result for nontrivial solutions. The main novelty of the paper is the combination of a double phase operator and the nonlocal term. Our results generalize some known results.

We establish the existence of at least two solutions of the Prandtl-Batchelor like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondiffer-entiable and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequenc...

We establish the existence of at least two solutions of the {\it Prandtl-Batchelor} like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a se...

We establish the existence of at least two solutions of the Prandtl-Batchelor like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable, and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequenc...

This paper intends to study the following degenerate fractional Schrödinger-Kirchhoff-Poisson equations with critical nonlinearity and electromagnetic fields in R 3      ε 2s M ([u] 2 s,A)(−∆) s A u + V (x)u + φu = k(x)|u| r−2 u + I µ * |u| 2 ♯ s |u| 2 ♯ s −2 u, x ∈ R 3 , (−∆) t φ = u 2 , x ∈ R 3 , where ε > 0 is a positive parameter, 3/4 < s...

In this paper, we study certain critical Schrödinger-Kirchhoff type systems involving the fractional p-Laplace operator on a bounded domain. More precisely, using the properties of the associated functional energy on the Nehari manifold sets and exploiting the analysis of the fibering map, we establish the multiplicity of solutions for such systems...

We investigate the degenerate fractional Schr\"{o}dinger-Kirchhoff-Poisson equation in $\mathbb{R}^3$ with critical nonlinearity and electromagnetic fields $\varepsilon^{2s} M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u + \phi u = k(x)|u|^{r-2}u + \left(\mathcal{I}_\mu*|u|^{2_s^\sharp}\right)|u|^{2_s^\sharp-2}u$ and $(-\Delta)^t\phi = u^2,$ where $\vareps...

In this article, we study certain critical Schrödinger-Kirchhoff-type systems involving the fractional p-Laplace operator on a bounded domain. More precisely, using the properties of the associated functional energy on the Nehari manifold sets and exploiting the analysis of the fibering map, we establish the multiplicity of solutions for such syste...

In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term f has a suitable oscilla...

We consider a parametric Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and with a reaction which exhibits the combined effects of a superlinear (convex) term and of a negative sublinear term. Using variational tools and critical groups we show that for all small values of the parameter, the problem has at least three nontrivial smoo...

We consider a parametric Dirichlet problem driven by the anisotropic (p, q)-Laplacian and with a reaction which exhibits the combined effects of a superlinear (convex) term and of a negative sublinear term. Using variational tools and critical groups we show that for all small values of the parameter, the problem has at least three nontrivial smoot...

In this paper we prove the existence of solutions to a double phase problem with a prescribed nonlinear boundary condition which is nonlinear in nature. The driving nonlinear perturbations obey a suitable condition at the origin and on the boundary. To the best of our knowledge, a double phase measure data problem with nonlinear boundary condition...

In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401-410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term $f$ has a suitable oscil...

Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain $\Omega$ and is H\"{o}lder continuous.

We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is ``superlinear". We prove a global (with respect to the parameter) multiplicity result for positive solutions. Also, we show the existence of a minimal positive solution and finally, we produce a nodal solution.

Let G := (V, E) be a weighted locally finite graph whose finite measure µ has a positive lower bound. Motivated by wide interest in the current literature, in this paper we study the existence of classical solutions for a class of elliptic equations involving the µ-Laplacian operator on the graph G , whose analytic expression is given by ∆µu(x) :=...

Let $\mathscr G:= (V,E)$ be a weighted locally finite graph whose finite measure $\mu$ has a positive lower bound. Motivated by wide interest in the current literature, in this paper we study the existence of classical solutions for a class of elliptic equations involving the $\mu$-Laplacian operator on the graph $\mathscr G$, whose analytic expres...

We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is ``superlinear". We prove a global (with respect to the parameter) multiplicity result for positive solutions. Also, we show the existence of a minimal positive solution and finally, we produce a nodal solution.

Let $\mathscr G:= (V,E)$ be a weighted locally finite graph whose finite measure $\mu$ has a positive lower bound. Motivated by a wide interest in the current literature, we study the existence of classical solutions for a class of elliptic equations involving the $\mu$-Laplacian operator on the graph $\mathscr G$, whose analytic expression is give...

We consider an anisotropic $(p,2)$-equation, with a parametric and superlinear reaction term. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the fifth nodal (sign-changing). The proofs use tools from critical point theory, truncation and comparison techniques...

In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the re...

We study the Neumann problem with Leray-Lions type operator. Using the classical variational theory, we prove the existence, uniqueness and multiplicity of solutions. As far as we know, this is the first attempt to investigate such a fourth-order problem involving Leray-Lions type operators.

We study a class of Schr\"{o}dinger-Poisson systems with $(p,q)$-Laplacian. Using fixed point theory, we obtain a new existence result for nontrivial solutions. The main novelty of the paper is the combination of a double phase operator and the nonlocal term. Our results generalize some known results.

We apply the Gromov-Hausdorff metric $d_G$ for characterization of certain generalized manifolds. Previously, we have proved that with respect to the metric $d_G,$ generalized $n$-manifolds are limits of spaces which are obtained by gluing two topological $n$-manifolds by a controlled homotopy equivalence (the so-called $2$-patch spaces). In the pr...

This paper is concerned with existence of normalized ground state solutions for the mass supercritical fractional nonlinear Schr\"{o}dinger equation involving a critical growth in the fractional Sobolev sense. The compactness of Palais-Smale sequences is obtained by a special technique, which borrows from the ideas of Soave (J. Funct. Anal. 279 (6)...

We apply the Gromov–Hausdorff metric \(d_G\) for characterization of certain generalized manifolds. Previously, we have proven that with respect to the metric \(d_G,\) generalized n-manifolds are limits of spaces which are obtained by gluing two topological n-manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the pres...

We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\R^{36}$ (resp. $\R^{83}$).

We study the following singular problem involving the p(x)-Laplace operator Δp(x)u=div(|∇u|p(x)−2∇u), where p(x) is a nonconstant continuous function, (Pλ){−Δp(x)u=a(x)|u|q(x)−2u(x)+λb(x)uδ(x)inΩ,u>0inΩ,u=0on∂Ω. Here, Ω is a bounded domain in RN≥2 with C2-boundary, λ is a positive parameter, a(x),b(x)∈C(Ω¯) are positive weight functions with compac...

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{\sigma-1}(-\Delta)^s_p u = \frac{\lambda}{u^{\gamma}}+u^{ p_s^{*}-1 }\quad \text{in }\Omega,\\ u>0,\;\;\;\;\quad \text{in }\Omega,\\ u=0,\;\;\;\;\quad \text{in }\mathbb{R}^{N}\setmin...

We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-\Delta)^{s(\cdot)}u+b(-\Delta)u&=\lambda |u|^{-\gamma(x)-1}u+\left(\int_{\Omega}\frac{F(y,u(y))}{|x-y|^{\mu(x,y)}}dy\right)f(x,u) & +\eta H(u-\alpha)|u|^{r(x...

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities: ( [ u ] s , p p ) σ − 1 ( − Δ ) p s u = λ u γ + u p s ∗ − 1 in Ω , u > 0 , in Ω , u = 0 , in R N ∖ Ω , where Ω is a bounded domain in R N with the smooth boundary ∂ Ω, 0 < s < 1 < p < ∞, N > s p, 1 < σ < p s ∗ / p, with p s ∗ = N p N − p...

Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain Ω and is Hölder continuous.

We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent a(-Δ)s(·)u+b(-Δ)u=λ|u|-γ(x)-1u+∫ΩF(y,u(y))|x-y|μ(x,y)dyf(x,u)+ηH(u-α)|u|r(x)-2u,inΩ,u=0,inRN\Ω.where a(-Δ)s(·)+b(-Δ) is a mixed operator with variable order s(·):R2N→(0,1), a,b≥0 with a+b>0...

We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of $*$-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional $*$-PI-exponent. We also construct a fa...

We consider a class of nonautonomous cellular neural networks (CNNs) with mixed delays, to study the solutions of these systems which are type pseudo almost periodicity. Using general measure theory and the Mittag-Leffler function, we obtain the existence of unique solutions for cellular neural equations and investigate the Mittag-Leffler stability...

We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of ⁎-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional ⁎-PI-exponent. We also construct a family...

In this paper we establish existence of a solution to a semilinear equation with free boundary conditions on stratified Lie groups. In the process, a monotonicity condition is proved, which is quintessential in establishing the regularity of the solution.

In this paper, existence of solutions is established for critical exponential Kirchhoff systems on the Heisenberg group by using the variational method. The novelty of our paper is that not only the nonlinear term has critical exponential growth, but also that Kirchhoff function covers the degenerate case. Moreover, our result is new even for the E...

In this paper, existence of solutions is established for critical exponential Kirchhoff systems on the Heisenberg group by using the variational method. The novelty of our paper is that not only the nonlinear term has critical exponential growth, but also that Kirchhoff function covers the degenerate case. Moreover, our result is new even for the E...

In this paper we establish existence of a solution to a semilinear equation with free boundary conditions on stratified Lie groups. In the process, a monotonicity condition is proved, which is quintessential in establishing the regularity of the solution.

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray-Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano (Appl. Anal. 89 (2010), 1-10).

This paper is concerned with existence of normalized ground state solutions for the mass supercritical fractional nonlinear Schr\"{o}dinger equation involving a critical growth in the fractional Sobolev sense. The compactness of Palais-Smale sequences will be obtained by a special technique, which borrows from the ideas of Soave (J. Funct. Anal. 27...

This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: $$ \left\{ \begin{array}{lll} -\mathfrak{M}(\|u\|_{s,A}^2)[(-\Delta)^s_Au+u] = G_u(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_\mu*|u|^{p^*}\right)|u|^{p^*-2}u \ &\mbox{in}\,\,\mathbb{R}^N,\\ \mathfrak{M}(\|v\|_{s,A})[(-\Delta)^s_Av+v]...

This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ −M(u 2 s,A)[(−Δ) s A u + u] = Gu(|x|, |u| 2 , |v| 2) + Iμ * |u| p * |u| p * −2 u in R N , M(v s,A)[(−Δ) s A v + v] = Gv(|x|, |u| 2 , |v| 2) + Iμ * |v| p * |v| p * −2 v in R N , where u s,A = R 2N |u(x) − e...

We study a class of $p(x)$-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions....

We study a class of p(x)-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We...

In this article, a space-dependent epidemic model equipped with a constant latency period is examined. We construct a delay partial integro-differential equation and show that its solution possesses some biologically reasonable features. We propose some numerical schemes and show that, by choosing the time step to be sufficiently small, the schemes...

We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and compa...

In this paper we construct consistent examples of subgroups of $2^\omega$ with Menger remainders which fail to have other stronger combinatorial covering properties. This answers several open questions asked by Bella, Tokgoz and Zdomskyy (Arch. Math. Logic 55 (2016), 767-784).

We establish a continuous embedding $W^{s(\cdot),2}(\Omega)\hookrightarrow L^{\alpha(\cdot)}(\Omega)$, where the variable exponent $\alpha(x)$ can be close to the critical exponent $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\bar{\Omega}$. Subsequently, this continuous embedding is used to prove the multiplicity of...

We introduce a class of proper posets which is preserved under countable support iterations, includes $\omega^\omega$-bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be...

We establish a continuous embedding Ws(⋅),2(Ω)↪Lα(⋅)(Ω), where the variable exponent α(x) can be close to the critical exponent 2s⁎(x)=2NN−2s¯(x), with s¯(x)=s(x,x) for all x∈Ω¯. Subsequently, this continuous embedding is used to prove the multiplicity of solutions for critical nonlocal degenerate Kirchhoff problems with a variable singular exponen...

We investigate the boundary value problem for biharmonic operators on the Heisenberg group. The inherent features of Hn make it an appropriate environment for studying symmetry rules and the interaction of analysis and geometry with manifolds. The goal of this paper is to prove that a weak solution for a biharmonic operator on the Heisenberg group...

For every finitely generated free group $F$, we construct an irreducible open $3$-manifold $M_F$ whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of $M_F$ isomorphic to $F$. The end homogeneity group is the group of all self-homeomorphisms of the end set that extend to homeomorphisms of the entire $3$-manifold. This...

We introduce a class of proper posets which is preserved under countable support iterations, includes ωω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ^\omega $...

This paper intend to study the following degenerate fractional Schr\"{o}dinger--Kirchhoff-Poisson equations with critical nonlinearity and electromagnetic fields in $\mathbb{R}^3$: \begin{equation*} \left\{\begin{array}{lll} \varepsilon^{2s}M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u + \phi u = k(x)|u|^{r-2}u + \left(\mathcal{I}_\mu*|u|^{2_s^\sharp}\rig...

We present the theory of a new fractional Sobolev space in complete manifolds with variable exponent. As a result, we investigate some of our new space’s qualitative properties, such as completeness, reflexivity, separability, and density. We also show that continuous and compact embedding results are valid. We apply the conclusions of this study t...

For every finitely generated free group F , we construct an irreducible open 3-manifold M_F whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of M_F isomorphic to F . The end homogeneity group is the group of all self-homeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends a...

We study the behavior of solutions for the parametric equation \begin{document}$ -\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0, $\end{document} under Dirichlet condition, where $ \Omega \subseteq \mathbb{R}^N $ is a bounded domain with a $ C^2 $-boundary $ \partial \Omega $,...

We show that every Hausdorff Baire topology τ on C=⟨ a,b| a 2 b=a, ab 2 =b⟩ such that (C,τ) is a semitopological semigroup is discrete and we construct a nondiscrete Hausdorff semigroup topology on C . We also discuss the closure of a semigroup C in a semitopological semigroup and prove that C does not embed into a topological semigroup with the co...

In this article a space-dependent epidemic model equipped with a constant latency period is examined. We construct a delay partial integro-differential equation and show that its solution possesses some biologically reasonable features. We propose some numerical schemes and show that by choosing the time step to be sufficiently small the schemes pr...

We investigate existence and multiplicity of weak solutions for fourth-order problems involving the Leray-Lions type operators in variable exponent spaces and improve a result of Bonanno and Chinn\`{i} (2011). We use variational methods and apply a multiplicity theorem of Bonanno and Marano (2010).