Critical Points of Solutions to Elliptic Equations in Planar Domains with Corners

2015 ◽  
pp. 105-112
Author(s):  
Jaime Arango ◽  
Jairo Delgado
Keyword(s):  
Critical Points ◽  
Elliptic Equations ◽  
Planar Domains ◽  
Domains With Corners

scholarly journals Critical Points for Elliptic Equations with Prescribed Boundary Conditions

2017 ◽  
Vol 226 (1) ◽  
pp. 117-141 ◽  
Author(s):  
Giovanni S. Alberti ◽  
Guillaume Bal ◽  
Michele Di Cristo
Keyword(s):  
Boundary Conditions ◽  
Critical Points ◽  
Elliptic Equations

scholarly journals MULTIPLE CONDENSATIONS FOR A NONLINEAR ELLIPTIC EQUATION WITH SUB-CRITICAL GROWTH AND CRITICAL BEHAVIOUR

2001 ◽  
Vol 44 (3) ◽  
pp. 631-660 ◽  
Author(s):  
Juncheng Wei
Keyword(s):  
Elliptic Equation ◽  
Critical Points ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic Equations ◽  
Subject Classification ◽  
Critical Behaviour ◽  
Unknown Constant ◽  
Secondary 35J40 ◽  
Nonlinear Elliptic

AbstractWe consider the following nonlinear elliptic equations\begin{gather*} \begin{cases} \Delta u+u_{+}^{N/(N-2)}=0\amp\quad\text{in }\sOm, \\ u=\mu\amp\quad\text{on }\partial\sOm\quad(\mu\text{ is an unknown constant}), \\ \dsty\int_{\partial\sOm}\biggl(-\dsty\frac{\partial u}{\partial n}\biggr)=M, \end{cases} \end{gather*}where $u_{+}=\max(u,0)$, $M$ is a prescribed constant, and $\sOm$ is a bounded and smooth domain in $R^N$, $N\geq3$. It is known that for $M=M_{*}^{(N)}$, $\sOm=B_R(0)$, the above problem has a continuum of solutions. The case when $M>M_{*}^{(N)}$ is referred to as supercritical in the literature. We show that for $M$ near $KM_{*}^{(N)}$, $K>1$, there exist solutions with multiple condensations in $\sOm$. These concentration points are non-degenerate critical points of a function related to the Green's function.AMS 2000 Mathematics subject classification: Primary 35B40; 35B45. Secondary 35J40

Solutions with multiple peaks for nonlinear elliptic equations

1999 ◽  
Vol 129 (2) ◽  
pp. 235-264 ◽  
Author(s):  
Daomin Cao ◽  
Ezzat S. Noussair ◽  
Shusen Yan
Keyword(s):  
Positive Solutions ◽  
Critical Points ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Multiple Peaks ◽  
Nonlinear Elliptic ◽  
Image Position

Solutions with peaks near the critical points of Q(x) are constructed for the problemWe establish the existence of 2k −1 positive solutions when Q(x) has k non-degenerate critical points in ℝN

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