scholarly journals Strong Comparison Principle for Radial Solutions of Quasi-Linear Equations

2001 ◽  
Vol 258 (1) ◽  
pp. 366-370 ◽  
Author(s):  
S Prashanth
Keyword(s):  
Comparison Principle ◽  
Linear Equations ◽  
Radial Solutions ◽  
Strong Comparison Principle

scholarly journals Strong Comparison Principle for the Fractional p-Laplacian and Applications to Starshaped Rings

2018 ◽  
Vol 18 (4) ◽  
pp. 691-704 ◽  
Author(s):  
Sven Jarohs
Keyword(s):  
Half Space ◽  
Comparison Principle ◽  
Nonnegative Function ◽  
Open Set ◽  
Nonnegative Solutions ◽  
Strong Comparison Principle

AbstractIn the following, we show the strong comparison principle for the fractional p-Laplacian, i.e. we analyze\quad\left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p}v+q(x)\lvert v\rvert% ^{p-2}v&\displaystyle\geq 0&&\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle(-\Delta)^{s}_{p}w+q(x)\lvert w\rvert^{p-2}w&\displaystyle\leq 0&% &\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle v&\displaystyle\geq w&&\displaystyle\phantom{}\text{in ${\mathbb% {R}^{N}}$},\end{aligned}\right.where {s\in(0,1)}, {p>1}, {D\subset\mathbb{R}^{N}} is an open set, and {q\in L^{\infty}(\mathbb{R}^{N})} is a nonnegative function. Under suitable conditions on s, p and some regularity assumptions on v, w, we show that either {v\equiv w} in {\mathbb{R}^{N}} or {v>w} in D. Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.

The strong comparison principle in parabolic problems with thep-Laplacian in a domain

2019 ◽  
Vol 98 ◽  
pp. 365-373
Author(s):  
Jiří Benedikt ◽  
Petr Girg ◽  
Lukáš Kotrla ◽  
Peter Takáč
Keyword(s):  
Comparison Principle ◽  
Parabolic Problems ◽  
Strong Comparison Principle

scholarly journals Positive Solutions for Neutral Difference Equations with Continuous Arguments

2007 ◽  
Vol 14 (4) ◽  
pp. 699-710
Author(s):  
Xianyi Li
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Comparison Principle ◽  
Difference Equations ◽  
Linear Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Difference Equations ◽  
Necessary And Sufficient

Abstract Some “sharp” conditions are established for a kind of linear neutral difference equations with continuous arguments not to possess eventually positive solutions. The existence and asymptotic behavior are obtained for positive solutions of the kind of equations. The results for linear cases are further extended to nonlinear ones. A comparison principle, which is a necessary and sufficient condition, for linear equations not to possess eventually positive solutions is also presented.

scholarly journals Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jacques Giacomoni ◽  
Deepak Kumar ◽  
Konijeti Sreenadh
Keyword(s):  
Maximum Principle ◽  
Comparison Principle ◽  
Weak Solutions ◽  
Global Regularity ◽  
Boundary Regularity ◽  
Holder Continuity ◽  
Regularity Of Weak Solutions ◽  
Regularity Results ◽  
Boundary Estimates ◽  
Strong Comparison Principle

Abstract In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional ( p , q ) {(p,q)} -Laplacian, denoted by ( - Δ ) p s 1 + ( - Δ ) q s 2 {(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}} for s 2 , s 1 ∈ ( 0 , 1 ) {s_{2},s_{1}\in(0,1)} and 1 < p , q < ∞ {1<p,q<\infty} . We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional ( p , q ) {(p,q)} -problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish a new Hopf-type maximum principle and a strong comparison principle in both situations.

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