\mathfrak{B}_{1} classes of De Giorgi, Ladyzhenskaya, and Ural'tseva and their application to elliptic and parabolic equations with nonstandard growth

2019 ◽  
Vol 16 (3) ◽  
pp. 403-447
Author(s):  
Igor Skrypnik ◽  
Mykhailo Voitovych
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Nonstandard Growth ◽  
Elliptic And Parabolic Equations ◽  
Functional Classes ◽  
Nonstandard Growth Conditions ◽  
Quasilinear Elliptic

The article provides an application of the generalized De Giorgi functional classes to the proof of the Hölder continuity of weak solutions to quasilinear elliptic and parabolic equations with nonstandard growth conditions.

scholarly journals Local subestimates of solutions to double-phase parabolic equations via nonlinear parabolic potentials

2019 ◽  
Vol 16 (1) ◽  
pp. 28-45
Author(s):  
Kateryna Buryachenko
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Growth Conditions ◽  
Local Boundedness ◽  
Nonstandard Growth ◽  
Right Hand ◽  
Double Phase ◽  
Nonlinear Parabolic ◽  
Nonstandard Growth Conditions ◽  
The Right

For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation.

scholarly journals Multiplicity results for nonlocal elliptic transmission problem with variable exponent

2014 ◽  
Vol 33 (2) ◽  
pp. 187-201
Author(s):  
Abdesslem Ayoujil ◽  
Mimoun Moussaoui
Keyword(s):  
Variational Principle ◽  
Nonlinear Equations ◽  
Weak Solutions ◽  
Variable Exponent ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Transmission Problem ◽  
Multiplicity Results ◽  
Nonstandard Growth ◽  
Nonstandard Growth Conditions

In this paper, a transmission problem given by a system of two nonlinear equations of p(x)-Kirchho type with nonstandard growth conditions are studied. Using the mountain pass theorem combined with the Ekeland's variational principle, we obtain at least two distinct, non-trivial weak solutions.

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