Searcharxiv⌕ Search

arXiv subjects

Dušan D. Repovš

Publications and source records attributed to Dušan D. Repovš.

At least 73 records · Page 4Linked to original sources
arXiv2020

Nonlinear, nonhomogeneous Robin problems with indefinite potential and general reaction

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator plus an indefinite potential. On the reaction term we impose conditions only near zero. Using variational methods, together with truncation and perturbation techniques and critical groups, we produce three nontrivial solutions with sign information. In the semilinear case we improve this result by obtaining a second nodal solution for a total of four nontrivial solutions. Finally, under a symmetry condition on the reaction term, we generate a whole sequence of distinct nodal solutions.

math.AP↗
arXiv2020

Relaxation methods for optimal control problems

We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map $A:\mathbb{R}^N\rightarrow2^{\mathbb{R}^N}$. We do not assume that $D(A)=\mathbb{R}^N$, incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.

math.AP↗
arXiv2020

Nonlinear singular problems with indefinite potential term

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term is parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $λ$ varies. This work continues our research published in arXiv:2004.12583, where $ξ\equiv 0 $ and in the reaction the parametric term is the singular one.

math.AP↗
arXiv2020

Combined effects for non-autonomous singular biharmonic problems

We study the existence of nontrivial weak solutions for a class of generalized $p(x)$-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.

math.AP↗
arXiv2020

Ground state and nodal solutions for a class of double phase problems

We consider a double phase problem driven by the sum of the $p$-Laplace operator and a weighted $q$-Laplacian ($q<p$), with a weight function which is not bounded away from zero. The reaction term is $(p-1)$-superlinear. Employing the Nehari method, we show that the equation has a ground state solution of constant sign and a nodal (sign-changing) solution.

math.AP↗
arXiv2020

Nonlinear nonhomogeneous singular problems

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $(p-1)$ near $+\infty$ and with a reaction which has the competing effects of a parametric singular term and a $(p-1)$-superlinear perturbation which does not satisfy the usual Ambrosetti-Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a "bifurcation-type" theorem that describes the set of positive solutions as the parameter $λ$ moves on the positive semiaxis. We also show that for every $λ>0$, the problem has a smallest positive solution $u^*_λ$ and we demonstrate the monotonicity and continuity properties of the map $λ\mapsto u^*_λ$.

math.AP↗
arXiv2020

Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities

In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-Δ)_{A}^su + V(x)u = (|x|^{-α}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-Δ)_{A}^s$ is the fractional magnetic operator with $0 0$ is a positive parameter. The electric potential $V\in C(\mathbb{R}^N, \mathbb{R}^+_0)$ satisfies $V(x) = 0$ in some region of $\mathbb{R}^N$, which means that this is the critical frequency case. We first prove the $(PS)_c$ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term $M$ can vanish at zero.

math.AP↗
arXiv2020

Combined effects in nonlinear problems arising in the study of anisotropic continuous media

The paper deals with the study of a Lane-Emden-Fowler equation with Dirichlet boundary condition and variable potential functions. The analysis developed in this paper combines monotonicity methods with variational arguments. Remark (April 21, 2020): Our results were later studied also by D.-P. Covei, Quasilinear problems with the competition between convex and concave nonlinearities and variable potentials, Internat. J. Math. 24:1 (2013), art. 1350005), arXiv:1104.4626v1.

math.AP↗
arXiv2020

Controlled homotopy equivalences and structure sets of manifolds

For a closed topological $n$--manifold $K$ and a map $p:K\to B$ inducing an isomorphism $π_1(K)\toπ_1(B)$, there is a canonicaly defined morphism $b:H_{n+1}(B,K,\mathbb{L})\to \mathbb{S} (K)$, where $\mathbb{L}$ is the periodic simply-connected surgery spectrum and $\mathbb{S} (K)$ is the topological structure set. We construct a refinement $a:H_{n+1}^{+}(B,K,\mathbb{L} )\to \mathbb{S}_{\varepsilon ,δ}(K)$ in the case when $p$ is $UV^1$, and we show that $a$ is bijective if $B$ is a finite-dimensional compact metric ANR. Here, $H_{n+1}^{+}(B,K,\mathbb{L} )\subset H_{n+1}(B,K,\mathbb{L} )$, and $\mathbb{S}_{\varepsilon ,δ}(K)$ is the controlled structure set. We show that the Pedersen-Quinn-Ranicki controlled surgery sequence is equivalent to the exact $\mathbb{L}$-homology sequence of the map $p:K \to B$, i.e. that $$H_{n+1}(B,\mathbb{L})\to H_{n+1}^{+}(B,K,\mathbb{L} )\to H_n(K,\mathbb{L}^{+})\to H_n(B,\mathbb{L} ), \ \mathbb{L}^{+}\to \mathbb{L},$$ is the connected covering spectrum of $\mathbb{L}$. By taking for $B$ various stages of the Postnikov tower of $K$, one obtains an interesting filtration of the controlled structure set.

math.GT↗
arXiv2020

Existence and localization of solutions for nonlocal fractional equations

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the fractional Laplacian, finding a nontrivial weak solution of the equation \begin{eqnarray*} \begin{cases} (-Δ)^s u=h(x)f(u) & {\mbox{ in }} Ω\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus Ω, \end{cases} \end{eqnarray*} where $h\in L^{\infty}_+(Ω)\setminus\{0\}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a suitable continuous function. These problems have a variational structure and we find a nontrivial weak solution for them by exploiting a recent local minimum result for smooth functionals defined on a reflexive Banach space. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.

math.AP↗
arXiv2020

Positive solutions for nonlinear Neumann problems with singular terms and convection

We consider a nonlinear Neumann problem driven by the $p$-Laplacian. In the reaction term we have the competing effects of a singular and a convection term. Using a topological approach based on the Leray-Schauder alternative principle together with suitable truncation and comparison techniques, we show that the problem has positive smooth solutions.

math.AP↗
arXiv2020

Dynamics and oscillations of models for differential equations with delays

By developing new efficient techniques and using an appropriate fixed point theorem, we derive several new sufficient conditions for the pseudo almost periodic solutions with double measure for some system of differential equations with delays. As an application, we consider certain models for neural networks with delays.

math.AP↗
arXiv2019

Parametric nonlinear resonant Robin problems

We consider a nonlinear Robin problem driven by the $p$-Laplacian. In the reaction we have the competing effects of two nonlinearities. One term is parametric, strictly $(p-1)$-sublinear and the other one is $(p-1)$-linear and resonant at any nonprincipal variational eigenvalue. Using variational tools from the critical theory (critical groups), we show that for all large enough values of parameter $λ$ the problem has at least five nontrivial smooth solutions.

math.AP↗
arXiv2019

Exponential growth of codimensions of identities of algebras with unity

The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly one when an outer unity is adjoined to the original algebra. We show that the PI-exponents of unital algebras can take any value greater than two, and the exponents of finite-dimensional unital algebras form a dense subset of the domain [2,$\infty$).

math.RA↗
arXiv2019

On nerves of fine coverings of acyclic spaces

The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $\mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $\mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded into $(2n+1)$-dimensional Euclidean space as a cellular subset; and (3) There exists a locally compact planar set which is acyclic with respect to Čech homology and whose fine coverings are all nonacyclic.

math.GT↗
arXiv2019

Gradient-type systems on unbounded domains of the Heisenberg group

The purpose of this paper is to study the existence of weak solutions for some classes of one-parameter subelliptic gradient-type systems involving a Sobolev-Hardy potential defined on an unbounded domain $Ω_ψ$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$ ($n\geq 1$) whose geometrical profile is determined by two real positive functions $ψ_1$ and $ψ_2$ that are bounded on bounded sets. The treated problems have a variational structure and thanks to this, we are able to prove the existence of an open interval $Λ\subset (0,\infty)$ such that, for every parameter $λ\in Λ$, the system has at least two nontrivial symmetric weak solutions that are uniformly bounded with respect to the Sobolev $HW^{1,2}_0$-norm. Moreover, the existence is stable under certain small subcritical perturbations of the nonlinear term. The main proof, crucially based on the Palais principle of symmetric criticality, is obtained by developing a group-theoretical procedure on the unitary group $\mathbb{U}(n)=U(n)\times\{1\}$ and by exploiting some compactness embedding results into Lebesgue spaces, recently proved for suitable $\mathbb{U}(n)$-invariant subspaces of the Folland-Stein space $HW^{1,2}_0(Ω_ψ)$. A key ingredient for our variational approach is a very general min-max argument valid for sufficiently smooth functionals defined on reflexive Banach spaces.

math.AP↗