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Povzetek: 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.
Ključne besede: multiphase spectral problem, stratified Lie group, Heisenberg group, left invariant vector field, (2, q)-eigenvalue problem
Objavljeno v DiRROS: 15.11.2024; Ogledov: 105; Prenosov: 41
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Povzetek: 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.
Ključne besede: p-biharmonic equation, variational methods, existence of solutions, Hardy potential, Nehari manifold, fibering map
Objavljeno v DiRROS: 08.07.2024; Ogledov: 379; Prenosov: 153
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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: 673; Prenosov: 579
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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: 569; Prenosov: 229
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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: 632; Prenosov: 224
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