1. On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applicationsNabil Chems Eddine, Maria Alessandra Ragusa, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: 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.
Ključne besede: Sobolev embeddings, concentration-compactness principle, anisotropic variable exponent Sobolev spaces, p(x)-Laplacian, fractional Brezis-Nirenberg problem
Objavljeno v DiRROS: 20.03.2024; Ogledov: 138; Prenosov: 134
 Celotno besedilo (550,73 KB)
Gradivo ima več datotek! Več... |
2. On the Schrödinger-Poisson system with $(p,q)$-LaplacianYueqiang Song, Yuanyuan Huo, Dušan Repovš, 2023, izvirni znanstveni članek Povzetek: 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.
Ključne besede: double phase operator, Schrödinger-Poisson systems, (p, q)–Laplacian, fixed point theory
Objavljeno v DiRROS: 14.03.2024; Ogledov: 94; Prenosov: 46
Celotno besedilo (686,98 KB)
Gradivo ima več datotek! Več... |
3. Fractional Sobolev spaces with kernel function on compact Riemannian manifoldsAhmed Aberqi, Abdesslam Ouaziz, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: 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.
Ključne besede: nonlinear elliptic problem, fractional Sobolev spaces, kernel function, Lévy-integrability condition, compact Riemannian manifolds, existence of solutions, topological degree theory
Objavljeno v DiRROS: 19.02.2024; Ogledov: 146; Prenosov: 52
Celotno besedilo (507,44 KB)
Gradivo ima več datotek! Več... |
4. On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearityMin Zhao, Yueqiang Song, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: 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)(-\Delta)_{p, A}^{s} u+V(x)|u|^{p-2} u=\lambda\left(\int_\limits{\mathbb{R}^{N}} \frac{|u|^{p_{\mu, s}^{*}}}{|x-y|^{\mu}} \mathrm{d}y\right)|u|^{p_{\mu, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$ where $0 < s < 1 < p$, $ps < N$, $p < q < 2p^{*}_{s,\mu}$, $0 < \mu < N$, $\lambda$ and $k$ are some positive parameters, $p^{*}_{s,\mu}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ 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 using the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.
Ključne besede: Hardy-Littlewood-Sobolev nonlinearity, Schrödinger-Kirchhoff equations, variational methods, electromagnetic fields
Objavljeno v DiRROS: 16.02.2024; Ogledov: 175; Prenosov: 56
Celotno besedilo (2,62 MB)
Gradivo ima več datotek! Več... |
5. Nodal solutions for Neumann systems with gradient dependenceKamel Saoudi, Eadah Alzahrani, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: We consider the following convective Neumann systems: $\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-\Delta_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+\delta_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) \text{in}\;\Omega,\\ -\Delta _{p_2}u_2+\frac{|\nabla u_2|^{p_2}}{u_2+\delta_2}=f_2(x,u_1,u_2,\nabla u_1,\nabla u_2) \text{in}\;\Omega, \\ |\nabla u_1|^{p_1-2}\frac{\partial u_1}{\partial \eta }=0=|\nabla u_2|^{p_2-2}\frac{\partial u_2}{\partial \eta} \text{on}\;\partial\,\Omega,\end{array}\right.\end{equation*}$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$) with a smooth boundary $\partial\,\Omega, \delta_1, \delta_2 > 0$ are small parameters, $\eta$ is the outward unit vector normal to $\partial\,\Omega, f_1, f_2: \Omega \times \mathbb{R}^2 \times \mathbb{R}^{2N} \rightarrow \mathbb{R}$ are Carathéodory functions that satisfy certain growth conditions, and $\Delta _{p_i}$ ($1< p_i < N,$ for $i=1,2$) are the $p$-Laplace operators $\Delta _{p_i}u_i=\mathrm{div}(|\nabla u_i|^{p_i-2}\nabla u_i)$, for $u_i \in W^{1,p_i}(\Omega).$ 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.
Ključne besede: Neumann elliptic systems, gradient dependence, subsolution method, supersolution method, nodal solutions
Objavljeno v DiRROS: 16.02.2024; Ogledov: 152; Prenosov: 46
Celotno besedilo (1,48 MB)
Gradivo ima več datotek! Več... |
