scholarly journals A Hardy–Littlewood–Sobolev-Type Inequality for Variable Exponents and Applications to Quasilinear Choquard Equations Involving Variable Exponent

2019 ◽  
Vol 16 (2) ◽  
Author(s):  
Claudianor O. Alves ◽  
Leandro S. Tavares
Keyword(s):  
Variable Exponent ◽  
Type Inequality ◽  
Sobolev Type ◽  
Variable Exponents ◽  
Sobolev Type Inequality

HERZ–MORREY SPACES ON THE UNIT BALL WITH VARIABLE EXPONENT APPROACHING AND DOUBLE PHASE FUNCTIONALS

2019 ◽  
pp. 1-34 ◽  
Author(s):  
YOSHIHIRO MIZUTA ◽  
TAKAO OHNO ◽  
TETSU SHIMOMURA
Keyword(s):  
Unit Ball ◽  
Variable Exponent ◽  
Type Inequality ◽  
Morrey Spaces ◽  
Maximal Functions ◽  
Sobolev Type ◽  
Riesz Potentials ◽  
Hölder Continuous ◽  
Double Phase ◽  
Sobolev Type Inequality

Our aim in this paper is to deal with integrability of maximal functions for Herz–Morrey spaces on the unit ball with variable exponent$p_{1}(\cdot )$approaching$1$and for double phase functionals$\unicode[STIX]{x1D6F7}_{d}(x,t)=t^{p_{1}(x)}+a(x)t^{p_{2}}$, where$a(x)^{1/p_{2}}$is nonnegative, bounded and Hölder continuous of order$\unicode[STIX]{x1D703}\in (0,1]$and$1/p_{2}=1-\unicode[STIX]{x1D703}/N>0$. We also establish Sobolev type inequality for Riesz potentials on the unit ball.

scholarly journals A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

2017 ◽  
Vol 29 (3) ◽  
pp. 515-542 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
T. RICCIARDI ◽  
G. ZECCA
Keyword(s):  
Total Population ◽  
Critical Mass ◽  
Type System ◽  
Type Inequality ◽  
Chemical Substance ◽  
Sobolev Type ◽  
Lyapunov Functionals ◽  
Single Chemical ◽  
Chemotaxis System ◽  
Sobolev Type Inequality

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

Existence and isoperimetric characterization of steady spherical vortex rings in a uniform flow in RN

2004 ◽  
Vol 134 (3) ◽  
pp. 449-476 ◽  
Author(s):  
Geoffrey R. Burton ◽  
Luca Preciso
Keyword(s):  
Type Inequality ◽  
Fluid Flows ◽  
Semilinear Elliptic Equation ◽  
Vortex Rings ◽  
Sobolev Type ◽  
Semilinear Elliptic ◽  
Stream Functions ◽  
Spherical Vortex ◽  
Sobolev Type Inequality

We study a boundary-value problem for a particular semilinear elliptic equation on Rn (n ≥ 2), whose solutions represent generalized stream functions for steady axisymmetric ideal fluid flows. Solutions are shown to exist that generalize those already known in dimensions 2 and 3. An isoperimetric characterization is given for our solutions, which represent generalized spherical vortex-rings. As a corollary, a sharp Sobolev-type inequality is obtained.

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