scholarly journals FOUNTAIN-LIKE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS WITH CRITICAL GROWTH AND HARDY POTENTIAL

2005 ◽  
Vol 07 (06) ◽  
pp. 867-904 ◽  
Author(s):  
VERONICA FELLI ◽  
SUSANNA TERRACINI
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Critical Growth ◽  
Hardy Potential ◽  
Nonlinear Elliptic ◽  
Hardy Type ◽  
Type Potential

We prove the existence of fountain-like solutions, obtained by superposition of bubbles of different blow-up orders, for a nonlinear elliptic equation with critical growth and Hardy-type potential.

Nonlinear Elliptic Equations in Strip-Like Domains

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Jaeyoung Byeon ◽  
Kazunaga Tanaka
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

AbstractWe study the existence of a positive solution of a nonlinear elliptic equationwhere k ≥ 2 and D is a bounded domain domain in R

Nonlinear elliptic equations with critical potential and critical parameter

2006 ◽  
Vol 136 (5) ◽  
pp. 1041-1051 ◽  
Author(s):  
Yao Tian Shen ◽  
Yang Xin Yao
Keyword(s):  
Elliptic Equation ◽  
Hardy Space ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Critical Parameter ◽  
Nonlinear Elliptic Equations ◽  
Point Theory ◽  
Positive Answer ◽  
Critical Potential ◽  
Nonlinear Elliptic

We give a positive answer to an open problem about Hardy's inequality raised by Brézis and Vázquez, and another result obtained improves that of Vázquez and Zuazua. Furthermore, by this improved inequality and the critical-point theory, in a k-order Sobolev–Hardy space, we obtain the existence of multi-solution to a nonlinear elliptic equation with critical potential and critical parameter.

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

Multiple boundary blow-up solutions for nonlinear elliptic equations

2003 ◽  
Vol 133 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Amandine Aftalion ◽  
Manuel del Pino ◽  
René Letelier
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Positive Parameter ◽  
Number Of Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in RN, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ0 > 0 such that, for λ < λ0, there is no positive solution that crosses 1 and, for λ > λ0, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.

scholarly journals On a class of nonlinear elliptic equations with fast increasing weight and critical growth

2010 ◽  
Vol 249 (5) ◽  
pp. 1035-1055 ◽  
Author(s):  
Marcelo F. Furtado ◽  
Olímpio H. Myiagaki ◽  
João Pablo P. da Silva
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Critical Growth ◽  
Nonlinear Elliptic

On Existence of L∞-Ground States for Singular Elliptic Equations in the Presence of a Strongly Nonlinear Term

2007 ◽  
Vol 7 (3) ◽  
Author(s):  
J.V. Goncalves ◽  
A.L. Melo ◽  
C.A. Santos
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Term ◽  
Ground States ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
Singular Elliptic Equations

AbstractWe establish new results concerning existence and the behavior at infinity of solutions for the singular nonlinear elliptic equation −Δu = ρa(x)u

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