Mathematische Nachrichten最新文献

We introduce the notion of localized operators. We extend the boundedness of the localized operators from the weighted Lebesgue spaces to the weighted Herz spaces. The localized operators include the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function. Therefore, this paper extends the mapping properties of the the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function to the weighted Herz spaces.

We study a superlinear elliptic boundary value problem involving the $�p$-Laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem.

Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to slightly subcritical nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces.

In this paper, we extend the well-known concentration–compactness principle of P.L. Lions to Orlicz spaces. As an application, we show an existence result to some critical elliptic problem with nonstandard growth.

We propose a notion of stability for constant $�k$-mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem.

We consider the fully nonlocal diffusion equations with nonnegative $�L^1$-data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time-space fractional Laplacian equations.

We prove new multiplicity results for some critical growth $�p$-biharmonic problems in bounded domains. More specifically, we show that each of the problems considered here has arbitrarily many solutions for all sufficiently large values of a certain parameter $��\lambda �>�0�$. In particular, the number of solutions goes to infinity as $��\lambda �\to �\infty �$. We also give an explicit lower bound on $�\lambda$ in order to have a given number of solutions. This lower bound will be in terms of an unbounded sequence of eigenvalues of a related eigenvalue problem. Our multiplicity results are new even in the semilinear case $��p�=�2�$. The proofs are based on an abstract critical point theorem.

The Gross–Pitaevskii (GP) equation is a model for the description of the dynamics of Bose–Einstein condensates. Here, we consider the GP equation in a two-dimensional setting with an external periodic potential in the $�x$-direction and a harmonic oscillator potential in the $�y$-direction in the so-called tight-binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one-dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non-trivial task due to the oscillations in the external periodic potential which become singular in the tight-binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non-resonance conditions have to be satisfied in the two-dimensional case.

First, we introduce a new notion of pseudo-anti commuting for real hypersurfaces in the complex quadric $��Q^m�=�S�O_{�m�+�2�}�/�S�O_m�S�O_2�$ and give a complete classification of Hopf pseudo-Ricci–Yamabe soliton real hypersurfaces in the complex quadric $�Q^m$. Next as an application we obtain a classification of gradient pseudo-Ricci–Yamabe solitons on Hopf real hypersurfaces in $�Q^m$.