scholarly journals Regularity results of solutions to elliptic equations involving mixed local and nonlocal operators

2021 ◽  
Vol 7 (3) ◽  
pp. 4199-4210
Author(s):  
CaiDan LaMao ◽  
◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Canyun Huang ◽  
...  
Keyword(s):  
Bounded Domain ◽  
Laplace Operator ◽  
Measurable Function ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Smooth Bounded Domain ◽  
Nonlocal Operators ◽  
Semilinear Elliptic ◽  
Regularity Results ◽  
Fractional Laplace Operator

<abstract><p>In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{matrix} - \Delta u(x)+{{(-\Delta )}^{s}}u(x)=f(x), &amp; x\in \Omega , \\ u(x)\ge 0,~~~~~ &amp; x\in \Omega , \\ u(x)=0,~~~~~ &amp; x\in {{\mathbb{R}}^{N}}\setminus \Omega , \\ \end{matrix} \right. $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 $, $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain, $ (-\Delta)^s $ is the fractional Laplace operator, $ f $ is a measurable function.</p></abstract>

Pointwise Lower Bounds for Solutions of Semilinear Elliptic Equations and Applications

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Asadollah Aghajani ◽  
Alireza Mosleh Tehrani ◽  
Nassif Ghoussoub
Keyword(s):  
Bounded Domain ◽  
Lower Bounds ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Smooth Bounded Domain ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

AbstractWe consider the semilinear elliptic problem −Δu = f (x, u), posed in a smooth bounded domain Ω of ℝ

scholarly journals On the number of critical points of solutions of semilinear elliptic equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Massimo Grossi
Keyword(s):  
Bounded Domain ◽  
Critical Points ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Nodal Solution ◽  
Smooth Bounded Domain ◽  
Semilinear Elliptic

<p style='text-indent:20px;'>In this survey we discuss old and new results on the number of critical points of solutions of the problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE0.1"> \begin{document}$ \begin{equation} \begin{cases} -\Delta u = f(u)&amp;in\ \Omega\\ u = 0&amp;on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ N\ge2 $\end{document}</tex-math></inline-formula> is a smooth bounded domain. Both cases where <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> is a positive or nodal solution will be considered.</p>

scholarly journals Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain

2018 ◽  
Vol 173 ◽  
pp. 123-145 ◽  
Author(s):  
Woocheol Choi ◽  
Younghun Hong ◽  
Jinmyoung Seok
Keyword(s):  
Bounded Domain ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic

Positive solutions for non-homogeneous semilinear elliptic equations with data that changes sign

2003 ◽  
Vol 133 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Qiuyi Dai ◽  
Yonggeng Gu
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Nonlinear Problem ◽  
Elliptic Equations ◽  
Linear Problem ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Image Position

Let Ω ⊂ RN be a bounded domain. We consider the nonlinear problem and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative solutions of the following linear problem: .In particular, if p > (N + 2)/(N − 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in § 1 of this paper).

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

Hopf boundary maximum principle violation for semilinear elliptic equations

2010 ◽  
Vol 72 (7-8) ◽  
pp. 3346-3355 ◽  
Author(s):  
Yavdat Il’yasov ◽  
Youri Egorov
Keyword(s):  
Maximum Principle ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic

Solutions of semilinear elliptic equations with asymptotic linear nonlinearity

2002 ◽  
Vol 50 (2) ◽  
pp. 275-283 ◽  
Author(s):  
Cheng-Hsiung Hsu ◽  
Yi-Wen Shih
Keyword(s):  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic
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