arXiv subjects
Publications and source records attributed to D. D. Repovš.
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 $λ,p,q$ by using the Symmetric Mountain Pass Theorem.
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 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.
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 is also given, to illustrate the importance of these results.
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-compactness principle for weighted-variable exponents Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz.
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 parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis-Nirenberg problem.
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: \begin{equation*} \begin{array}{lll} M(\|u\|_μ^{p})(μ(-Δ)^{s}_{p}u+V(ξ)|u|^{p-2}u)= f(ξ,u)+\int_{\mathbb{H}^N}\frac{|u(η)|^{Q_λ^{\ast}}}{|η^{-1}ξ|^λ}dη|u|^{Q_λ^{\ast}-2}u &\mbox{in}\ \mathbb{H}^N, \\ \end{array} \end{equation*} where $(-Δ)^{s}_{p}$ is the fractional $p$-Laplacian on the Heisenberg group $\mathbb{H}^N$, $M$ is the Kirchhoff function, $V(ξ)$ is the potential function, $0 0$, $f(ξ,u)$ is the nonlinear function, $0<λ<Q$, $Q=2N+2$, and $Q_λ^{\ast}=\frac{2Q-λ}{Q-2}$ is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if $μ$ is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has $m$ pairs of solutions if $μ$ is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case.
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 problem involving the fractional $p(\cdot, \cdot)$-Laplacian operator. As one of the main tools, topological degree theory is applied.
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 illustrate the usefulness of our results, an illustrative example is also presented.
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.
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)^{σ-1}(-Δ)^s_p u = \fracλ{u^γ}+u^{ p_s^{*}-1 }\quad \text{in }Ω,\\ u>0,\;\;\;\;\quad \text{in }Ω,\\ u=0,\;\;\;\;\quad \text{in }\mathbb{R}^{N}\setminus Ω,\end{array} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with the smooth boundary $\partial Ω$, $0 < s< 1 sp$, $1<σ<p^*_s/p,$ with $p_s^{*}=\frac{Np}{N-ps},$ $ (- Δ)_p^s$ is the nonlocal $p$-Laplace operator and $[u]_{s,p}$ is the Gagliardo $p$-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.
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).
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 also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results.
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 Chinnì (2011). We use variational methods and apply a multiplicity theorem of Bonanno and Marano (2010).
We prove the existence of multiple solutions for the following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_Ω\frac{1}{p(x)}|\nabla Δu|^{p(x)}dx)Δ^3_{p(x)} u = λf(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) \ \ \mbox{on} \ Ω$ and $ \ u=Δu=Δ^2 u=0 \ \ \mbox{on} \ \partialΩ,$ where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N > 3$, $Δ_{p(x)}^3u = \operatorname{div}\Big(Δ(|\nabla Δu|^{p(x)-2}\nabla Δu)\Big)$ is the $p(x)$-triharmonic operator, $p,q,r \in C(\overlineΩ)$, $1< p(x) < \frac N3$ for all $x\in \overlineΩ$, $M(s) = a - bs^γ$, $a,b,γ>0$, $λ>0$, $g: Ω\times \mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function while $f,h : Ω\times \mathbb{R} \to \mathbb{R}$ are sign-changing continuous functions in $Ω$. To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order $p(x)$-Kirchhoff type problems with sign changing Kirchhoff functions.
We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{Ω\times Ω} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-Δ)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in Ω, \\ \\ u= 0, \quad x\in \partialΩ, \end{array}\right.$$ where $Ω\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overlineΩ\times \overlineΩ \rightarrow (1, \infty)$, $(-Δ)^{s}_{p(x,\cdot)}$ denotes the $p(x,\cdot)$-fractional Laplace operator, $M: [0,\infty) \to [0, \infty),$ and $f: Ω\times \mathbb{R} \to \mathbb{R}$ are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo-Benci-Fortunato (Nonlinear Anal. 7:9 (1983), 981-1012), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti-Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.
We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator, with reaction which exhibits the competition of two Carathéodory terms. One is parametric, $(p-1)$-sublinear with a partially concave nonlinearity near zero. The other is $(p-1)$-superlinear and has almost critical growth. Exploiting the special geometry of the problem, we prove a bifurcation-type result, describing the changes in the set of positive solutions as the parameter $λ>0$ varies.
In this paper, we study the following $p(x)$-curl systems: \begin{eqnarray*} \begin{cases} \nabla\times(|\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u})+a(x)|\mathbf{u}|^{p(x)-2}\mathbf{u}=λf(x,\mathbf{u})+μg(x,\mathbf{u}),\quad\nabla\cdot \mathbf{u}=0,\; \mbox{ in } Ω, \\ |\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u}\times \mathbf{n}=0,\quad \mathbf{u}\cdot \mathbf{n}=0, \mbox{ on } \partialΩ, \end{cases} \end{eqnarray*} where $Ω\subset \mathbb{R}^{3}$ is a bounded simply connected domain with a $C^{1,1}$-boundary, denoted by $\partial Ω$, $p:\overlineΩ\to (1, +\infty)$ is a continuous function, $a \in L^\infty(Ω)$, $f,g : Ω\times \mathbb{R}^{3}\to \mathbb{R}^{3}$ are Carathéodory functions, and $λ,μ$ are two parameters. Using variational arguments based on Fountain theorem and Dual Fountain theorem, we establish some existence and non-existence results for solutions of this problem. Our main results generalize the results of Xiang et al. (J. Math. Anal. Appl., 2017), Bahrouni and Repovš (Complex Var. Elliptic Equ., 2018), and Ge and Lu (Mediterr. J. Math., 2019).