scholarly journals On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions

2021 ◽  
Author(s):  
K. Kefi ◽  
N. Irzi ◽  
M. M. Al-Shomrani ◽  
D. D. Repovš
Keyword(s):  
Variational Methods ◽  
Fourth Order ◽  
Weak Solutions ◽  
Growth Conditions ◽  
Indefinite Weight ◽  
Order Problem ◽  
Nonstandard Growth ◽  
Fourth Order Problem ◽  
Nonstandard Growth Conditions

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].

\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 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.

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.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

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