arXiv subjects
Publications and source records attributed to Dušan D. Repovš.
This study is devoted to proving the existence of weak solutions for a nonlinear elliptic problem with Neumann-type boundary data. The problem is driven by a discontinuous power nonlinearity and a nonsmooth prescribed data. Additionally, we aim to derive an estimate that proves the well-posedness of the problem. This estimate serves as an evidence for the uniqueness of the existing solution when the boundary term is ``smooth".
We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} Δ^{2}u+V(\varepsilon x)u=λu+μ|u|^{q-2}u+|u|^{2^{**}-2}u \mbox{ in }\ \mathbb{R}^{N}, \quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2}, \end{equation*} where $Δ^{2}$ is the biharmonic operator, $N\geq5$, $μ,c>0$, $\varepsilon>0,$ $λ\in\mathbb{R}$, $q\in(2,2+\frac{8}{N}),$ and $2^{**}=\frac{2N}{N-4}$ is the Sobolev critical exponent. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. Using minimization techniques and a truncation argument, we show that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\varepsilon$ is sufficiently small. To overcome the loss of compactness of the energy functional due to the critical growth, we apply the concentration-compactness principle. To the best of our knowledge, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$. To some extent, the main results included in this paper complement several recent contributions to the study of biharmonic equations with combined nonlinearities.
This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-Δ)_{p}^{s}u+\varepsilon^{qs}(-Δ)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{μ-N}[|x|^{-μ}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac{N}{s} 0$ small enough. In a certain sense, we generalize some previously known results.
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 normal degree one maps with a reference map to $B$. More precisely, we prove that for every $n \ge 6$ and every finite complex $B,$ there exists a map $Γ: H_n(B;\mathbb{L}) \longrightarrow \mathcal{N}^{H}_{n}(B,\partial).$
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 $Br(k_1, \ldots, k_n)$ that generalize examples constructed by Debrunner in 1964. We are interested in hyperbolic invariants of 3-manifolds $S^3 \setminus Br(k_1, \ldots, k_n)$ and we obtain upper bounds for their volumes. Our approach is based on Dehn fillings on cusped manifolds with volumes related to volumes of ideal right-angled hyperbolic antiprisms.
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 productively Hurewicz subspaces of the Cantor space which are not productively Menger.
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 regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.
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.
In this article, we deal with the following $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$ M\left([u]_{s,A}^{p}\right)(-Δ)_{p, A}^{s} u+V(x)|u|^{p-2} u=λ\left(\int_{\mathbb{R}^{N}} \frac{|u|^{p_{μ, s}^{*}}}{|x-y|^μ} \mathrm{d}y\right)|u|^{p_{μ, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$$ where $0<s<1<p$, $ps < N$, $p<q<2p^{*}_{s,μ}$, $0<μ<N$, $λ$ and $k$ are some positive parameters, $p^{*}_{s,μ}=\frac{pN-p\fracμ{2}}{N-ps}$ is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions $V$, $M$ satisfy the suitable conditions. By proving the compactness results with the help of the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.
We consider the following convective Neumann systems:\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-Δ_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+δ_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) & \text{in}\;Ω,\\ -Δ_{p_2}u_2+\frac{|\nabla u_2|^{p_2}}{u_2+δ_2}=f_2(x,u_1,u_2,\nabla u_1,\nabla u_2)&\text{in}\;Ω, \\ |\nabla u_1|^{p_1-2}\frac{\partial u_1}{\partial η}=0=|\nabla u_2|^{p_2-2}\frac{\partial u_2}{\partial η}&\text{on}\;\partialΩ,\end{array}\right.\end{equation*}where $Ω$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$) with a smooth boundary $\partialΩ$,$δ_1,\,δ_2 >0$ are small parameters, $η$ is the outward unit vector normal to $\partial Ω,$ $f_1,\,f_2:Ω\times\mathbb{R}^2\times\mathbb{R}^{2N}\rightarrow \mathbb{R}$ are Carathéodory functions that satisfy certain growth conditions, and $Δ_{p_i}$ ($1<p_i<N,$ for $i=1,2$) are the $p$-Laplace operators $Δ_{p_i}u_i=\mathrm{div}(|\nabla u_i|^{p_i-2}\nabla u_i)$,for every $\,u_i\in W^{1,p_i}(Ω).$ In order to prove the existence of solutions to such systems, we use a sub-supersolution method. We also obtain nodal solutions by constructing appropriate sub-solution and super-solution pairs. To the best of our knowledge, such systems have not been studied yet.
We study the following critical Choquard equation on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle {-Δ_H u }=μ |u|^{q-2}u+\int_Ω \frac{|u(η)|^{Q_λ^{\ast}}} {|η^{-1}ξ|^λ} dη|u|^{Q_λ^{\ast}-2}u &\mbox{in }\ Ω, u=0 &\mbox{on }\ \partialΩ, \end{cases} \end{equation*} where $Ω\subset \mathbb{H}^N$ is a smooth bounded domain, $Δ_H$ is the Kohn-Laplacian on the Heisenberg group $\mathbb{H}^N$, $1 0$, $0<λ<Q=2N+2$, and $Q_λ^{\ast}=\frac{2Q-λ}{Q-2}$ is the critical exponent. Using the concentration compactness principle and the critical point theory, we prove that the above problem has the least two positive solutions for $1<q<2$ in the case of low perturbations (small values of $μ$), and has a nontrivial solution for $2<q<Q_λ^\ast$ in the case of high perturbations (large values of $μ$). Moreover, for $1<q<2$, we also show that there is a positive ground state solution, and for $2<q<Q_λ^\ast$, there are at least $n$ pairs of nontrivial weak solutions.
In this article, we investigate the Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group of the following form: \begin{equation*} \left\{ \begin{array}{lll} {-(a+b\int_Ω|\nabla_{H} u|^{p}dξ)Δ_{H,p}u-μϕ|u|^{p-2}u}=λ|u|^{q-2}u+|u|^{Q^{\ast}-2}u &\mbox{in}\ Ω, \\ -Δ_{H}ϕ=|u|^{p} &\mbox{in}\ Ω, \\ u=ϕ=0 &\mbox{on}\ \partialΩ, \end{array} \right. \end{equation*} where $a,b$ are positive real numbers, $Ω\subset \mathbb{H}^N$ is a bounded region with smooth boundary, $1<p<Q$, $Q = 2N + 2$ is the homogeneous dimension of the Heisenberg group $\mathbb{H}^N$, $Q^{\ast}=\frac{pQ}{Q-p}$, $q\in(2p, Q^{\ast})$, and $Δ_{H,p}u=\mbox{div}(|\nabla_{H} u|^{p-2}\nabla_{H} u)$ is the $p$-horizontal Laplacian. Under some appropriate conditions for the parameters $μ$ and $λ$, we establish existence and multiplicity results for the system above. To some extent, we generalize the results of An and Liu (Israel J. Math., 2020) and Liu et al. (Adv. Nonlinear Anal., 2022).
This paper is concerned with 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 is obtained by a special technique, which borrows from the ideas of Soave (J. Funct. Anal. 279 (6) (2020) 1086102020). This paper represents an extension of previously known results - in the local and the nonlocal cases.
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 sequence of $C^1$ functionals and a {\it cutoff function}. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.
We investigate the degenerate fractional Schrödinger-Kirchhoff-Poisson equation in $\mathbb{R}^3$ with critical nonlinearity and electromagnetic fields $\varepsilon^{2s} M([u]_{s,A}^2)(-Δ)_{A}^su + V(x)u + ϕu = k(x)|u|^{r-2}u + \left(\mathcal{I}_μ*|u|^{2_s^\sharp}\right)|u|^{2_s^\sharp-2}u$ and $(-Δ)^tϕ= u^2,$ where $\varepsilon > 0$ is a parameter, $3/4<s<1$, $0 < t < 1$, $V$ is an electric potential satisfying some suitable assumptions, $0 < k_\ast \leq k(x) \leq k^\ast$, $\mathcal{I}_μ(x) = |x|^{3-μ}$ with $0<μ<3$, $2_s^\sharp =\frac{3+μ}{3-2s},$ and $2 < r < 2_s^\sharp$. With the help of the concentration compactness principle and variational methods, together with some fine analytical tools, we establish the existence and multiplicity of solutions for the above problem when $\varepsilon \rightarrow 0$ in the degenerate cases, i.e. when the Kirchhoff term $M$ vanishes at zero.
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 smooth solutions, two of which are of constant sign (positive and negative).
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 oscillating behaviour either at the origin or at infinity.
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.