scholarly journals Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sami Aouaoui ◽  
Rahma Jlel
Keyword(s):  
Euclidean Space ◽  
Nonlinear Equations ◽  
Exponential Type ◽  
Growth Conditions ◽  
Concentration Compactness ◽  
Weighted Inequalities ◽  
Radial Functions ◽  
Compactness Arguments ◽  
Logarithmic Type ◽  
Conditions At Infinity

<p style='text-indent:20px;'>This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N,\ N \geq 2. $\end{document}</tex-math></inline-formula> The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.</p>

scholarly journals Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness

2018 ◽  
Vol 9 (1) ◽  
pp. 39-64 ◽  
Author(s):  
Ahmed Mohammed ◽  
Vicenţiu D. Rădulescu ◽  
Antonio Vitolo
Keyword(s):  
Nonlinear Equations ◽  
Blow Up ◽  
Growth Conditions ◽  
Primary Objective ◽  
Asymptotic Estimates ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Large Solutions ◽  
Boundary Estimates ◽  
Conditions At Infinity

Abstract The primary objective of the paper is to study the existence, asymptotic boundary estimates and uniqueness of large solutions to fully nonlinear equations {H(x,u,Du,D^{2}u)=f(u)+h(x)} in bounded {C^{2}} domains {\Omega\subseteq\mathbb{R}^{n}} . Here H is a fully nonlinear uniformly elliptic differential operator, f is a non-decreasing function that satisfies appropriate growth conditions at infinity, and h is a continuous function on Ω that could be unbounded either from above or from below. The results contained herein provide substantial generalizations and improvements of results known in the literature.

Breather degeneration and lump superposition for the (3 + 1)-dimensional nonlinear evolution equation

2021 ◽  
pp. 2150250X
Author(s):  
Wei Tan ◽  
Miao Li
Keyword(s):  
Evolution Equation ◽  
Exponential Type ◽  
Water Waves ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Nonlinear Phenomena ◽  
Degradation Behavior ◽  
Shallow Water Waves ◽  
Perturbation Parameters ◽  
Logarithmic Type

This paper is devoted to the study of a (3 + 1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. The breather solutions with different structures are obtained based on the bilinear form with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions, and we also study general lump soliton, lumpoff solution and superposition phenomenon between lump soliton and breather solution. Besides, some theorems about the superposition between lump soliton and [Formula: see text]-soliton ([Formula: see text] is a nonnegative integer) are given. Some examples, including lump-[Formula: see text]-exponential type, lump-[Formula: see text]-logarithmic type, higher-order lump-type [Formula: see text]-soliton, are given to illustrate the correctness of the theorems and corollaries described. Finally, some novel nonlinear phenomena, such as emergence of lump soliton, degeneration of breathers, fission and fusion of lumpoff, superposition of lump-[Formula: see text]-solitons, etc., are analyzed and simulated.

Weighted inequalities for the Stieltjes transform and the maximal spherical partial sum operator on radial functions

1995 ◽  
Vol 125 (1) ◽  
pp. 195-204
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Fourier Transform ◽  
Lorentz Space ◽  
Sufficient Conditions ◽  
Weighted Inequalities ◽  
Stieltjes Transform ◽  
Radial Functions ◽  
Partial Sum ◽  
Power Weights ◽  
The Fourier Transform ◽  
Spherical Partial Sum

If TRf(x) is the spherical partial sum of the Fourier transform of f and T*f(x) = SUPR > 0 | TRf(x)|, sufficient conditions are given on the non-negative weight function ω(x) which ensure that T* restricted to radial functionsis bounded on the Lorentz space Lp,s(Rn,ω) into Lp,q(Rn, ω) For power weights, these conditions are also necessary. The weight pairs (u,v) for which the generalised Stieltjes transform Sλ is bounded from LP,S(R+, v)into Lp,q(R+, u)are also characterised. These are an essential ingredient for the study of T*.

scholarly journals On weighted inequalities for fractional integrals of radial functions

2011 ◽  
Vol 55 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Pablo L. De Nápoli ◽  
Irene Drelichman ◽  
Ricardo G. Durán
Keyword(s):  
Fractional Integrals ◽  
Weighted Inequalities ◽  
Radial Functions

scholarly journals Harnack inequality for non-divergence structure semi-linear elliptic equations

2018 ◽  
Vol 7 (3) ◽  
pp. 259-269 ◽  
Author(s):  
Ahmed Mohammed ◽  
Giovanni Porru
Keyword(s):  
Elliptic Operator ◽  
Harnack Inequality ◽  
Elliptic Equations ◽  
Growth Conditions ◽  
Divergence Structure ◽  
Conditions At Infinity

AbstractIn this paper we establish a Harnack inequality for non-negative solutions of {Lu=f(u)} where L is a non-divergence structure uniformly elliptic operator and f is a non-decreasing function that satisfies an appropriate growth conditions at infinity.

scholarly journals Existence of Solutions for thep(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Yu Mei ◽  
Fu Yongqiang ◽  
Li Wang
Keyword(s):  
Bounded Domain ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Concentration Compactness ◽  
Laplacian Equation

This paper deals with thep(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness inW01,p(x)(Ω)space is established, then by applying it we obtain the existence of solutions for the followingp(x)-Laplacian problem:-div (|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u),  x∈Ω,  u=0,  x∈∂Ω,whereΩ⊂ℝNis a bounded domain,0∈Ω,1<p-≤p(x)≤p+<N, andf(x,u)satisfiesp(x)-growth conditions.

scholarly journals On the Range of the Radon Transform onZnand the Related Volberg’s Uncertainty Principle

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Ahmed Abouelaz ◽  
Abdallah Ihsane ◽  
Takeshi Kawazoe
Keyword(s):  
Euclidean Space ◽  
Fourier Analysis ◽  
Uncertainty Principle ◽  
Exponential Type ◽  
Radon Transform ◽  
Integral Geometry ◽  
Discrete Radon Transform

We characterize the image of exponential type functions under the discrete Radon transformRon the latticeZnof the Euclidean spaceRn  n≥2. We also establish the generalization of Volberg's uncertainty principle onZn, which is proved by means of this characterization. The techniques of which we make use essentially in this paper are those of the Diophantine integral geometry as well as the Fourier analysis.

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