Hardy Terms Research Articles

Analysis of degenerate [formula omitted]-Laplacian elliptic equations involving Hardy terms: Existence and numbers of solutions

This article investigates the existence of solutions to quasilinear degenerate elliptic equation with Hardy singular coefficient, in which the weighted function ω(x) is unbounded (singular), then we cannot use the classical space W01,p(Ω), so we have to find another space W01,p(ω,Ω) to deal with the difficulties caused by singularities or degeneracies. New criteria for the existence of at least one and at least two generalized solutions are established via variational methods and critical point theorems provided that the nonlinearity satisfies appropriate hypotheses.

Existence of positive solutions of elliptic equations with Hardy term

This paper is devoted to studying the existence of positive solutions of the problem: ( ∗ ) { − Δ u = u p | x | a + h ( x , u , ∇ u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is an open bounded smooth domain with boundary ∂ Ω , and 1 < p < N − a N − 2 , 0 < a < 2 . Under suitable conditions of h ( x , u , ∇ u ) , we get a priori estimates for the positive solutions of problem ( ∗ ) . By making use of these estimates and topological degree theory, we further obtain some existence results for the positive solutions of problem ( ∗ ) when 1 < p < N − a N − 2 .

The Existence of a Positive Solution for an Elliptical Equation with Hardy Terms and Sobolev Critical Exponents

The Existence of a Positive Solution for an Elliptical Equation with Hardy Terms and Sobolev Critical Exponents

Normalized ground state solutions for $ p $-Laplacian problems with a Hardy term

Normalized ground state solutions for $ p $-Laplacian problems with a Hardy term

Evaluation of collection samples of soft winter wheat of the Ivanivsk research and selection station as starting material for hybridization

Aim. Highlighting the results of the study of breeding value in terms of winter hardiness, plant height, weight of 1000 seeds and yield variety samples of soft winter wheat cultivars of domestic selection, selected for hybridization in the collection nursery of the Institute's Ivanivka research and breeding station of bioenergy crops and sugar beets of the National Agricultural Academy of Ukraine. Methods. The research was conducted in sowing in the 2020-2021 cropping year in the southern forest-steppe zone of Ukraine. The object of the study were variety samples of the domestic selection of the lutescens type (10) and erythrospermum (15), who grew and studied their biological and economic characteristics according to generally accepted classical methods. Results. The average value of winter hardiness of the studied variety samples of both types was above 8 points. All studied variety samples were medium-sized with a low level of plant height variability. The weight of 1,000 seeds of the studied variety samples was about 44 g on average, the level of variability for this characteristic was low in the lutescens type, and medium in the erythrospermum type. The average yield of variety samples of both types was higher than 6 t/ha. Conclusions. The studied variety samples were characterized by an optimal and high level of manifestation of economic and valuable traits, therefore they were recommended for practical use in breeding as a starting material for hybridization. The studied variety samples are suitable for use in production crops in the Southern Forest-Steppe zone of Ukraine.

Characteristics of winter spelt wheat (Triticum spelta L.) breeding lines in the eastern forest-steppe of Ukraine

Purpose and Objectives. The aim of our work was to characterize winter spelt lines selected from spelt/bread wheat hybrid combinations by major economically valuable parameters.
 Materials and Methods. Winter spelt (T. spelta) lines derived from crossing NSS 1/02 T. spelta var. duhamelianum line (Serbia) with Oktava T. aestivum variety (Ukraine) at the National Center for Plant Genetic Resources of Ukraine of the Plant Production Institute named after V.Y. Yuriev (PPI) of NAAS of Ukraine were studied. Plants were grown on the experimental farm Elite of the PPI NAAS of Ukraine (Kharkivska Oblast) in 2019-2022. The accessions were sown in plots of 10 m2 of preliminary and competitive trials in four replications within the optimal timeframe. Plants were analyzed for morphological parameters, performance constituents and grain quality. For structural analysis, plants were taken from an area of 0.25 m2 and 20 spikes were analyzed in detail. Studying the spelt lines, we took the following accessions as the check varieties: Yevropa winter spelt variety; Podolianka and Bunchuk winter bread wheat varieties (national standards), Harmonika and Doskonala bred at the PPI NAAS.
 Results and Discussion. The spelt lines were characterized by high winter hardiness, high resistance to powdery mildew and tan rot pathogens (8-9 points), and medium resistance to Septoria (5-6.4 points). They started spring growth more slowly than winter bread wheat. They came into ear simultaneously with Yevropa spelt and Podolianka bread wheat. Plants of the spelt lines were as tall as or slightly shorter than Yevropa plants (check variety). Our observations showed that spelt stems were quite strong and resistant to lodging, despite their length. The tillering coefficient of the spelt lines ranged 1.69 to 1.80, with the mean exceeding Yevropa’s value (1.31), but was lower in comparison with the winter bread wheat varieties. Spikes of the spelt lines were white, loose and awnless. In terms of the spike length and the number of spikelets in the main spike, the spelt lines were inferior to both Yevropa and bread wheat varieties. Spelt kernels were elongated, plump and resembled Doskonala and Podolianka kernels. As to the kernel weight from the main spike, the spelt lines were slightly inferior to Yevropa, Harmonika and Doskonala, but not inferior to Bunchuk and Podolianka (national standards). The thousand kernel weight was high, ranging 47.31 g to 50.44 g. In terms of yield, the spelt lines outperformed the check bread wheat varieties on average by 0.50 t/ha. 1140-16 T. spelta line was most productive, yielding by 0.82 t/ha more than Yevropa and by 0.40 t/ha more than the check winter bread wheat varieties, which was possible due to easy threshing of the accessions. As to quality indicators (protein content in grain, gluten content in flour), the spelt lines were similar to Yevropa and bested the bread wheat varieties, but their sedimentation index was worse than that in the bread wheat varieties.
 Conclusions. The “NSS 1/02 T. spelta var. duhamelianum line (Serbia) / Oktava T. aestivum variety (Ukraine)” turned out to be a promising combination for breeding improvement of winter spelt in terms of yield, winter hardiness and grain quality in the Eastern Forest-Steppe of Ukraine.

Improved Caffarelli–Kohn–Nirenberg inequalities in unit ball and sharp constants in dimension three

We show that the Caffarelli–Kohn–Nirenberg inequalities in unit ball Bn can be improved by subtraction of Hardy term. In three dimension and 0≤a<12, we show that the sharp constant coincides with that in R3. This is an analogous result to that of the sharp constant in the n−12-th order Hardy-Sobolev-Maz’ya inequality in the unit ball of dimension n when n is odd. As an application, we obtain a sharp Sobolev inequality on hyperbolic Caffarelli–Kohn–Nirenberg space introduced by L. Dupaigne, I. Gentil and S. Zugmeyer.

Radial solutions to critical elliptic systems involving p-Laplacian and different Hardy-type terms

We study a system of elliptic equations, which involves p-Laplacian operator, strongly coupled critical nonlinearities and different Hardy-type terms. By ODE analysis methods, the asymptotic properties at the origin and infinity of radially symmetric and strictly decreasing solutions to the system are proved completely. It is found that the properties of solutions depend not only on the coefficients of Hardy terms but also on the exponents of coupled critical terms. The conclusions are new even in the case of p=2.

Elliptic inequalities with nonlinear convolution and Hardy terms in cone-like domains

We study the inequality −Δu−μ|x|2u≥(|x|−α⁎up)uq in an unbounded cone CΩρ⊂RN (N≥2) generated by a subdomain Ω of the unit sphere SN−1⊂RN, p,q,ρ>0, μ∈R and 0≤α<N. In the above, |x|−α⁎up denotes the standard convolution operator in the cone CΩρ. We discuss the existence and nonexistence of positive solutions in terms of N,p,q,α,μ and Ω. Extensions to systems of inequalities are also investigated.

Source material for selection of winter soft wheat for winter hardiness and yield in the conditions of the forest-steppe of the Middle Volga region

Source material is the starting point of breeding any culture. Its careful study in specific environmental conditions largely determines breeding effectiveness. This article provides the results of a study of 102 soft winter wheat varieties of various ecological and geographical origin in terms of yield and winter hardiness. Sources of high winter hardiness, yield and their combinations have been determined and recommended as a source material for breeding use in the conditions of the forest-steppe of the Middle Volga region. Reliable correlations between yield and winter hardiness of winter soft wheat have been established. This is of practical importance both for crop breeding and for its production use.

Normalized ground state for the Sobolev critical Schrödinger equation involving Hardy term with combined nonlinearities

AbstractIn this paper, we study the existence and properties of normalized solutions for the following Sobolev critical Schrödinger equation involving Hardy term: with prescribed mass where 2* is the Sobolev critical exponent. For a L2‐subcritical, L2‐critical, or L2‐supercritical perturbation , we prove several existence results of normalized ground state when and non‐existence results when . Furthermore, we also consider the asymptotic behavior of the normalized solutions u as or .

A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term

A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term

Search for SNP markers of stress tolerance in the &lt;i&gt;TaDREB1&lt;/i&gt; and &lt;i&gt;TaWRKY19&lt;/i&gt; genes of bread wheat in the Cis-Ural steppe zone

BACKGROUND: Under the conditions of the Cis-Ural steppe zone, abiotic stress factors such as winter conditions and drought have a negative impact on the yield of bread wheat (Triticum aestivum L.). However, the efficiency of breeding to these stress factors is limited by wide variations depending on the weather conditions in different years. To collect stress-resistant forms in the selected material, genomic selection can be used, however, SNP markers of drought tolerance and winter hardiness of bread wheat, unique to this region, are unknown.&#x0D; AIM: The purpose of this study is to identify SNP markers in the genes of the key transcription factors of bread wheat TaDREB1 and TaWRKY19, which are involved in the regulation of plant resistance to abiotic stresses.&#x0D; MATERIALS AND METHODS: 16 varieties and lines of bread wheat were analyzed, contrasting in terms of winter hardiness and drought tolerance. DNA from wheat leaves was isolated by the standard CTAB method. Primers were selected, fragments of open reading frames of the studied genes were amplified, and nucleotide sequences were determined by automatic capillary sequencing.&#x0D; RESULTS: Among the analyzed bread wheat varieties and lines, 12 SNPs in the TaDREB1 gene and 5 SNPs in TaWRKY19 were found.&#x0D; CONCLUSIONS: The results of statistical analysis showed that substitution 866 T/A in the TaDREB1 gene is associated with winter hardiness, while substitution 587 A/G in the TaWRKY19 gene is associated with winter hardiness and drought resistance.

Critical equations with Hardy terms in the Heisenberg group

In this paper, we are concerned with the study of a critical (p, q) equation with Hardy terms on the Heisenberg group. Existence of entire solutions is obtained via an application of some concentration–compactness type results and the mountain pass theorem. Our results are presented in the model case of the (p, q) horizontal Laplacian equations, but the method can be extended to deal with a more general class of problems with operators of (p, q) growth.

INTRODUCTION TO THE COMPOSITAE – 3

INTRODUCTION TO THE COMPOSITAE – 3

Solutions to coupled critical elliptic systems involving attractive Hardy–type terms

In this paper, a system of elliptic equations is studied, which involves coupled critical nonlinearities and attractive Hardy terms. By the local compactness analysis of (PS)c sequence and estimates on extremal functions corresponding to the best Sobolev constant, the existence of Mountain–Pass solutions is proved. Secondly, the asymptotic properties at the origin of solutions to the system are proved by the analytic technics and variational arguments.

Existence results for perturbed fourth-order Kirchhoff type elliptic problems with singular term

Under appropriate growth conditions on the nonlinearity, the existence of multiple solutions for a perturbednonlocal fourth-order Kirchhoff-type problem involving the Hardy term:$$\Delta_p ^2 u-\big[M(\int_{\Omega}|\nabla u|^{p}dx)\big]^{p-1}\Delta_{p}u-\mu\frac{|u|^{p-2}u}{|x|^{2p}}= \lambda f(x,u),$$is established. Our main tools are based on variational methods and some critical pointstheorems. We give some examples to illustrate the obtained results.

Solutions for nonhomogeneous singular fractional p-Laplacian equations via fixed point theorem

In this paper, we study the following fractional p-Laplacian equations with exponential growth and Hardy term: where the nonlinear term satisfies exponential growth, , , λ is a positive parameter, is the fractional p-Laplacian, is a scalar potential, the weight function either falls into some Lebesgue space or is a convolution function, and g is a perturbation term. With the help of a fixed point result and singular Trudinger–Moser inequality in , we obtain the existence of a nontrivial weak solution for the above equations.

On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces

&lt;abstract&gt;&lt;p&gt;In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.&lt;/p&gt;&lt;/abstract&gt;

Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces

Using the Helgason-Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we prove in a unified way that the sharp constant in the n−12-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best n−12-th order Sobolev constant when n is odd and n≥5. We will also establish a lower bound of the coefficient of the Hardy term for the k-th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension n and k-th order derivatives. As a consequence, we thus show that the sharp constant in the k-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n is strictly less than the best k-th order Sobolev constant for all n≥2k+2. Precise expressions and optimal bounds for Green's functions of the operator −ΔH−(n−1)24 on the hyperbolic space Bn and operators of the product form are given, where (n−1)24 is the spectral gap for the Laplacian −ΔH on Bn. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space in terms of hypergeometric functions. In fact, we will establish the precise formulas of the Green's functions of the operator∏j=0k−1((ν+j)2−(n−1)2/4−ΔH),ν≥0, in terms of hypergeometric function F(a,b;c,z).Our approach to establish the main theorems is using the Helgason-Fourier analysis on hyperbolic spaces and are substantially different from that in dealing with the first order Hardy-Sobolev-Maz'ya inequalities on upper half spaces.