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 Radial symmetry of large solutions of semilinear elliptic equations with convection

2014 ◽  
Vol 144 (1) ◽  
pp. 139-147
Author(s):  
Ehsan Kamalinejad ◽  
Amir Moradifam
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Radial Solution ◽  
Radial Symmetry ◽  
Semilinear Elliptic Equations ◽  
Rate Of Growth ◽  
Large Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We study the radial symmetry of large solutions of the semilinear elliptic problem Δu + ∇h · ∇u = f (∣x∣, u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.

scholarly journals Existence of solutions for some degenerate semilinear elliptic equations with measure data

Mathematica ◽  
2021 ◽  
Vol 63 (86) (1) ◽  
pp. 23-31
Author(s):  
Badajena Arun Kumar ◽  
Pradhan Shesadev
Keyword(s):  
Weak Solution ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Semilinear Elliptic Equations ◽  
Measure Data ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We study the existence of a weak solution for a certain degenerate semilinear elliptic problem.

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 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>

NONNEGATIVE SOLUTIONS FOR INDEFINITE SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 14 (03) ◽  
pp. 1250021 ◽  
Author(s):  
FRANCISCO ODAIR DE PAIVA
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Solution Method ◽  
Mountain Pass Theorem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Nonnegative Solutions ◽  
Carathéodory Function ◽  
Multiplicity Of Positive Solutions

This paper is devoted to the study of existence, nonexistence and multiplicity of positive solutions for the semilinear elliptic problem [Formula: see text] where Ω is a bounded domain of ℝN, λ ∈ ℝ and g(x, u) is a Carathéodory function. The obtained results apply to the following classes of nonlinearities: a(x)uq + b(x)up and c(x)(1 + u)p (0 ≤ q < 1 < p). The proofs rely on the sub-super solution method and the mountain pass theorem.

scholarly journals Asymptotic Estimates and Qualitative Properties of an Elliptic Problem in Dimension Two

2004 ◽  
Vol 4 (1) ◽  
Author(s):  
Khalil El Mehdi ◽  
Massimo Grossi
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Asymptotic Estimates ◽  
Qualitative Properties ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

AbstractIn this paper we study a semilinear elliptic problem on a bounded domain in ℝ

scholarly journals Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

2008 ◽  
Vol 2 (2) ◽  
pp. 158-174 ◽  
Author(s):  
Qianqiao Guo ◽  
Pengcheng Niu ◽  
Jingbo Dou
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

scholarly journals Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

2018 ◽  
Vol 9 (1) ◽  
pp. 108-123 ◽  
Author(s):  
Claudianor O. Alves ◽  
Grey Ercole ◽  
M. Daniel Huamán Bolaños
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Ground State Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

Abstract We prove the existence of at least one ground state solution for the semilinear elliptic problem \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=u^{p(x)-1},\quad u% >0,\quad\text{in}\ G\subseteq\mathbb{R}^{N},\ N\geq 3,\\ \displaystyle u&\displaystyle\in D_{0}^{1,2}(G),\end{aligned}\right. where G is either {\mathbb{R}^{N}} or a bounded domain, and {p\colon G\to\mathbb{R}} is a continuous function assuming critical and subcritical values.

scholarly journals Planar vortices in a bounded domain with a hole

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Shusen Yan ◽  
Weilin Yu
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Positive Constant ◽  
Classical Solutions ◽  
Single Vortex ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Planar Flows ◽  
Simply Connected ◽  
Simply Connected Domains

<p style='text-indent:20px;'>In this paper, we consider the inviscid, incompressible planar flows in a bounded domain with a hole and construct stationary classical solutions with single vortex core, which is closed to the hole. This is carried out by constructing solutions to the following semilinear elliptic problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1111"> \begin{document}$ \begin{equation} \begin{cases} -\Delta \psi = \lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p,\quad &amp;\text{in}\; \Omega,\\ \psi = \rho_\lambda,\quad &amp;\text{on}\; \partial O_0,\\ \psi = 0,\quad &amp;\text{on}\; \partial\Omega_0, \end{cases} \;\;\;\;\;\;\;\;(1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> is a positive constant, <inline-formula><tex-math id="M3">\begin{document}$ \rho_\lambda $\end{document}</tex-math></inline-formula> is a constant, depending on <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Omega = \Omega_0\setminus \bar{O}_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ O_0 $\end{document}</tex-math></inline-formula> are two planar bounded simply-connected domains. We show that under the assumption <inline-formula><tex-math id="M8">\begin{document}$ (\ln\lambda)^\sigma\leq\rho_\lambda\leq (\ln\lambda)^{1-\sigma} $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ \sigma&gt;0 $\end{document}</tex-math></inline-formula> small, (1) has a solution <inline-formula><tex-math id="M10">\begin{document}$ \psi_\lambda $\end{document}</tex-math></inline-formula>, whose vorticity set <inline-formula><tex-math id="M11">\begin{document}$ \{y\in \Omega:\, \psi(y)-\kappa+\rho_\lambda\eta(y)&gt;0\} $\end{document}</tex-math></inline-formula> shrinks to the boundary of the hole as <inline-formula><tex-math id="M12">\begin{document}$ \lambda\to +\infty $\end{document}</tex-math></inline-formula>.</p>

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