scholarly journals Parametric superlinear double phase problems with singular term and critical growth on the boundary

2021 ◽  
Author(s):  
Ángel Crespo‐Blanco ◽  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Singular Term ◽  
Critical Growth ◽  
Double Phase

scholarly journals Anisotropic singular double phase Dirichlet problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rǎdulescu ◽  
Youpei Zhang
Keyword(s):  
Positive Solutions ◽  
Singular Term ◽  
Independent Interest ◽  
Phase Problem ◽  
Type Result ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools ◽  
Superlinear Perturbation ◽  
Double Phase Problem

<p style='text-indent:20px;'>We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on <inline-formula><tex-math id="M1">\begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{document}</tex-math></inline-formula>. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.</p>

scholarly journals Positive Solutions for Resonant (p, q)-equations with convection

2020 ◽  
Vol 10 (1) ◽  
pp. 217-232 ◽  
Author(s):  
Zhenhai Liu ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Positive Solutions ◽  
Smooth Solution ◽  
Singular Term ◽  
Phase Problem ◽  
Drift Term ◽  
Double Phase ◽  
Double Phase Problem ◽  
Point Argument

Abstract We consider a nonlinear parametric Dirichlet problem driven by the (p, q)-Laplacian (double phase problem) with a reaction exhibiting the competing effects of three different terms. A parametric one consisting of the sum of a singular term and of a drift term (convection) and of a nonparametric perturbation which is resonant. Using the frozen variable method and eventually a fixed point argument based on an iterative asymptotic process, we show that the problem has a positive smooth solution.

Compactness Property of a Singular Quasilinear Elliptic Equation

2008 ◽  
Vol 8 (4) ◽  
Author(s):  
Jianqing Chen
Keyword(s):  
Elliptic Equation ◽  
Quasilinear Elliptic Equation ◽  
Singular Term ◽  
Quasilinear Equation ◽  
Critical Growth ◽  
Compactness Property ◽  
Quasilinear Elliptic

AbstractWe characterize a compactness property for a quasilinear equation with critical growth and singular term. Some applications of the compactness property are also pointed out.

Quantitative nano-characterization of heterogeneous catalysts

1995 ◽  
Vol 53 ◽  
pp. 398-399
Author(s):  
P.A. Crozier ◽  
M. Pan
Keyword(s):  
Heterogeneous Catalysts ◽  
Low Dose ◽  
Imaging Techniques ◽  
Structural Features ◽  
Catalyst Characterization ◽  
New Developments ◽  
Double Phase ◽  
Phase Systems ◽  
Dose Imaging

Heterogeneous catalysts can be of varying complexity ranging from single or double phase systems to complicated mixtures of metals and oxides with additives to help promote chemical reactions, extend the life of the catalysts, prevent poisoning etc. Although catalysis occurs on the surface of most systems, detailed descriptions of the microstructure and chemistry of catalysts can be helpful for developing an understanding of the mechanism by which a catalyst facilitates a reaction. Recent years have seen continued development and improvement of various TEM, STEM and AEM techniques for yielding information on the structure and chemistry of catalysts on the nanometer scale. Here we review some quantitative approaches to catalyst characterization that have resulted from new developments in instrumentation.HREM has been used to examine structural features of catalysts often by employing profile imaging techniques to study atomic details on the surface. Digital recording techniques employing slow-scan CCD cameras have facilitated the use of low-dose imaging in zeolite structure analysis and electron crystallography. Fig. la shows a low-dose image from SSZ-33 zeolite revealing the presence of a stacking fault.

Probing the Versatility of Shape-Persistent Tetraphenylmethane Dendrimers by Modification of the Skeleton

2019 ◽  
Author(s):  
Julio Ignacio Urzúa ◽  
Sandra Campana ◽  
Massimo Lazzari ◽  
Mercedes Torneiro
Keyword(s):  
Design Principle ◽  
Focal Point ◽  
Sonogashira Coupling ◽  
Terminal Alkyne ◽  
Double Phase ◽  
Rigid Rod ◽  
Convergent Approach ◽  
First And Second Generation ◽  
Oxidative Homocoupling ◽  
Hypercrosslinked Polymers

Tetraphenylmethane has emerged as a recurrent building block for advanced porous materials such as COFs, PAFs and hypercrosslinked polymers. Guided by a similar design principle, we have previously synthesized shape-persistent dendrimers with tetraphenylmethane nodes and ethynylene linkers. Here we report the generality of our approach by describing new dendritic architectures built from tetraphenylmethane. First, we prepared expanded dendrimers where the tetrahedral units are bonded through larger rigid rod spacers. Among the different synthetic strategies tested, the convergent route, with alternating steps of Pd-catalyzed Sonogashira coupling and alkyne activation by removal of TMS masking groups, efficiently afforded the first- and second-generation dendrimers. A second type of compounds having a linear diyne at the core is also described. The dendrimers of generations 1-2 were also synthesized by a convergent approach, with the diyne being assembled in the last step of the synthesis by a Glaser oxidative homocoupling of the corresponding dendrons bearing a terminal alkyne at the focal point. A third-generation dendrimer was also successfully prepared by a double-phase strategy.<br>

Differentiation of Tuberculous and Pyogenic Spondylitis Using Double Phase F-18 FDG PET

2008 ◽  
Vol 2 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Jung Sub Lee ◽  
Seong-Jang Kim ◽  
Kuen Tak Suh ◽  
In-Ju Kim ◽  
Yong-Ki Kim
Keyword(s):  
Fdg Pet ◽  
Pyogenic Spondylitis ◽  
Double Phase

scholarly journals Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 464
Author(s):  
Jichao Wang ◽  
Ting Yu
Keyword(s):  
Poisson Equation ◽  
Existence Result ◽  
Critical Growth ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Number Of Solutions ◽  
Concentration Behavior ◽  
The Relationship ◽  
Concentration Phenomenon

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.

Sign in / Sign up

Export Citation Format

Share Document