Fundamental Solution for Weighted Baouendi-Grushin Type Operators and a Class of Weighted Hardy Inequality

In this paper we obtain the fundamental solution for a class of weighted BaouendiGrushin type operator L_(p,γ,α)u = ▽_γ·(|▽_γu|^(p-2)ρ~α▽_γu) on R^(m+n )with singularity at the origin,where ▽_γ is the gradient operator defined by ▽_γ =(▽_x,|x|~γ▽_y) and ρ is the distance function.As an application,we get some Hardy type inequalities associated with ▽_γ.

Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Biharmonic Equation and an Improved Hardy Inequality

An improved Hardy inequality will be proven in the present work. Using the improved Hardy inequality and variational techniques, we also discuss the existence of nontrivial solution for following the weighted eigenvalue problem:

An Interpolation of Hardy Inequality and Moser–Trudinger Inequality on Riemannian Manifolds with Negative Curvature

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser-Trudinger inequality on M. Furthermore, the constant we obtain is sharp.

AN IMPROVED HARDY'S INEQUALITY

In this paper we shall extend Hardy＇s inequality associated with Fourier trans- form to the strip n（2-p） ≤σ〈 n＋p（N＋ 1） where N = [n（1/p- 1）], the greatest integer not exceeding n（1/p - 1）.

A MATHEMATICAL PROOF OF A PROBABILISTIC MODEL OF HARDY'S INEQUALITY

In this paper using an argument from [1] , we prove one of the probabilistic version of Hardy＇s inequality.

SOLUTIONS TO ELLIPTIC SYSTEMS INVOLVING DOUBLY CRITICAL NONLINEARITIES AND HARDY-TYPE POTENTIALS

In this article, an elliptic system is investigated, which involves Hardy-type potentials, critical Sobolev-type nonlinearities, and critical Hardy-Sobolev-type nonlinearities. By a variational global-compactness argument, the Palais-Smale sequences of related approximation problems is analyzed and the existence of infinitely many solutions to the system is established.

On Hardy-type integral inequalities

The Hardy integral inequality is one of the most important inequalities in analysis. The present paper establishes some new Copson-Pachpatte （C-P） type inequalities, which are the generalizations of the Hardy integral inequalities on binary functions.

NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESàRO TYPE

This paper deals with a general class of weighted multilinear Hardy-Cesaro op- erators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro oper- ators （with symbols in central BMO spaces） are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

GRADIENT ESTIMATES FOR SOLUTIONS TO QUASILINEAR ELLIPTIC EQUATIONS WITH CRITICAL SOBOLEV GROWTH AND HARDY POTENTIAL

This note is a continuation of the work[17].We study the following quasilinear elliptic equations- △pu-μ/｜x｜p ｜u｜p-2 u=Q（x）｜u｜Np/N-p -2u,x∈R N,where 1 〈 p 〈 N,0 ≤ μ 〈（（N-p）/p）p and Q ∈ L∞（RN）.Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.

EIGENVALUE PROBLEM OF ELLIPTIC EQUATIONS WITH HARDY POTENTIAL

Consider the eigenvalue problem of elliptic equations with Hardy potential. Improve the results of references by introducing a new Hilbert space and using integral inequality.

Improved Hardy–Littlewood–Sobolev Inequality on S^(n)under Constraints

In this paper,we establish an improved Hardy–Littlewood–Sobolev inequality on Snunder higher-order moments constraint.Moreover,by constructing precise test functions,using improved Hardy–Littlewood–Sobolev inequality on S^(n),we show such inequality is almost optimal in critical case.As an application,we give a simpler proof of the existence of the maximizer for conformal Hardy–Littlewood–Sobolev inequality.

NON-NEWTON FILTRATION EQUATION WITH SPECIAL MEDIUM VOID

This paper considers the existence and asymptotic estimates of global solutions and finite time blowup of local solution of non-Newton filtration equation with special medium void of the following form:where , ft is a smooth bounded domain in RN(N≥3), 0∈Ω, The result of asymptotic estimate of global solution depends on the best constant in Hardy inequality.

ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p

This article deals with the problem-△pu=λ｜u｜p/-2｜x｜pIn^p R/｜x｜＋f（x,u）,x∈Ω;u=0,x∈δΩ,where n = p. The authors prove that a Hardy inequality and the constant （p/p-1）^p is optimal. They also prove the existence of a nontrivial solution of the above mentioned problem by using the Mountain Pass Lemma.

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient △γ=（△x,｜x｜γ [x△y） by a direct and simple approach. Moreover, similar method is applied to ob- tain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L^p. We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.

Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β（u）-div（α（x, Du）＋F（u）） ∈ f in fΩ, where f ∈ L1 （Ω）. A vector field a（.,.） is a Carath6odory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.

A Struwe Type Decomposition Result for a Singular Elliptic Equation on Compact Riemannian Manifolds

On a compact Riemannian manifold, we prove a decomposition theorem for arbitrarily bounded energy sequence of solutions of a singular elliptic equation.

Some integral inequalities on time scales

In this article, we study the reverse HSlder type inequality and HSlder inequality in two dimensional case on time scales. We also obtain many integral inequalities by using HSlder inequalities on time scales which give Hardy＇s inequalities as spacial cases.

A Hardy–Trudinger–Moser Inequality Involving LpNorm in the Hyperbolic Space

Let B be the unit disc in R2,H be the completion of C0∞(B)under the norm||u||=(∫B|▽U|^2dx-∫Bu^2/(1-|x|^2)^2dx)^1/2,U∈C^∞0(B).By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric dv=dx/(1-|x|^2)^2,we prove that,for any fixedα,0≤α<λp(B)-infu∈■,u≠0||U||^2■/||U||^2p,the supremum u∈■、||su||■≤1∫B^e^4π(1+α||su||^2p)U^2dx<+∞,■p>1.This is an analog of early results of Lu–Yang(Discrete Contin.Dyn.Syst.,2009)and Yang(Trans.Amer.Math.Soc.,2007),and extends those of Wang–Ye(Adv.Math.,2012)and Yang–Zhu(Ann.Global Anal.Geom.,2016).

DEGENERATE BOUNDARY LAYER SOLUTIONS TO THE GENERALIZED BENJAMIN-BONAMAHONY-BURGERS EQUATION

This paper is concerned with the convergence rates of the global solutions of the generalized Benjamin-Bona-Mahony-Burgers（BBM-Burgers） equation to the corresponding degenerate boundary layer solutions in the half-space.It is shown that the convergence rate is t-（α/4） as t →∞ provided that the initial perturbation lies in H α 1 for α 〈 α（q）：= 3 ＋（2/q）,where q is the degeneracy exponent of the flux function.Our analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant introduced in [1]