scholarly journals Existence and multiplicity of solutions for nonlinear elliptic equations of p-Laplace type in RN

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 2029-2043 ◽  
Author(s):  
Ji Lee ◽  
Yun-Ho Kim
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Real Constant ◽  
Multiplicity Of Solutions ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity ◽  
Type V ◽  
Laplace Type

In this paper, we discuss the following elliptic equation: -div(?(x,?u)) = ?f (x,u) in RN, where the function ? : RN x RN ? RN is of type |v|p-2 v with a real constant p > 1 and f : RN x R ? R satisfies a Carath?odory condition.

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

ON A CLASS OF FOURTH ORDER NONLINEAR ELLIPTIC EQUATIONS UNDER NAVIER BOUNDARY CONDITIONS

2010 ◽  
Vol 08 (02) ◽  
pp. 185-197 ◽  
Author(s):  
F. J. S. A. CORRÊA ◽  
J. V. GONCALVES ◽  
ANGELO RONCALLI
Keyword(s):  
Boundary Conditions ◽  
Fixed Points ◽  
Fourth Order ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Compact Operators ◽  
Schmidt Method ◽  
Navier Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity

We employ arguments involving continua of fixed points of suitable nonlinear compact operators and the Lyapunov–Schmidt method to prove existence and multiplicity of solutions in a class of fourth order non-homogeneous resonant elliptic problems. Our main result extends even similar ones known for the Laplacian.

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

scholarly journals An oscillation theorem for the nonlinear degenerate elliptic equation in the Heisenberg group

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Duan Wu ◽  
Pengcheng Niu
Keyword(s):  
Elliptic Equation ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Degenerate Elliptic Equation ◽  
Oscillation Theorem ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Oscillation Of Solutions ◽  
Nonlinear Degenerate

AbstractThe aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$ H n . We first derive a critical inequality in $H^{n}$ H n . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in $R^{n}$ R n .

Multiple Positive Solutions For a Class of Nonlinear Elliptic Equations

2006 ◽  
Vol 6 (1) ◽  
Author(s):  
Shenghua Weng ◽  
Yongqing Li
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nonlinear Elliptic Equations ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Multiplicity Results ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity

AbstractThis paper deals with a class of nonlinear elliptic Dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results.

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.

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