Estimates for Blow-Up Solutions to Nonlinear Elliptic Equations with p-Growth in the Gradient

2010 ◽  
pp. 219-234 ◽  
Author(s):  
Vincenzo Ferone ◽  
Ester Giarrusso ◽  
Basilio Messano ◽  
Maria Rosaria Posteraro
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

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.

Liouville-type Theorems for Fully Nonlinear Elliptic Equations and Systems in Half Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Jiuyi Zhu
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Decay Estimates ◽  
Fully Nonlinear Elliptic Equations ◽  
Liouville Type ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems

AbstractThe main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [18], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors in an earlier work [15] when no boundedness assumption was given on the solutions.

scholarly journals Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in $R^{n}\backslash\{0\}$

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Imed Bachar ◽  
Entesar Aljarallah
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Continuous Solution ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Nonlinear Elliptic Equations ◽  
Weighted Function ◽  
Distributional Sense ◽  
Nonlinear Elliptic ◽  
Semilinear Problem

AbstractWe consider the following singular semilinear problem $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in  D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , where $\gamma <1$ γ < 1 , $D=\mathbb{R}^{n}\backslash \{0\}$ D = R n ∖ { 0 } ($n\geq 3$ n ≥ 3 ) and p is a positive continuous function in D, which may be singular at $x=0$ x = 0 . Under sufficient conditions for the weighted function $p(x)$ p ( x ) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

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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