scholarly journals Bifurcation for some quasilinear operators

2001 ◽  
Vol 131 (4) ◽  
pp. 733-765 ◽  
Author(s):  
David Arcoya ◽  
José Carmona ◽  
Benedetta Pellacci
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Complete Analysis ◽  
Multiplicity Results ◽  
Elliptic Boundary Value ◽  
The Matrix ◽  
Image Position ◽  
Quasilinear Elliptic

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem , where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

Non-Local Elliptic Boundary-Value Problems

1968 ◽  
Vol 20 ◽  
pp. 1365-1382 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Differential Operators ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Operators ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Non Local ◽  
Image Position

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.

scholarly journals Positive solutions of an asymptotically planar system of elliptic boundary value problems

1998 ◽  
Vol 21 (3) ◽  
pp. 549-554 ◽  
Author(s):  
F. J. S. A. Corrêa
Keyword(s):  
Maximum Principle ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Bifurcation Theory ◽  
Existence Result ◽  
Elliptic Boundary ◽  
Elliptic Boundary Value Problems ◽  
Planar System ◽  
Elliptic Boundary Value

We will prove an existence result of positive solutions for an asymptotically planar system of two elliptic equations. It will be used as main tools for a Maximum Principle and a result on Bifurcation Theory.

scholarly journals The power concavity of solutions of some semilinear elliptic boundary-value problems

1985 ◽  
Vol 31 (2) ◽  
pp. 181-184 ◽  
Author(s):  
Grant Keady
Keyword(s):  
Boundary Value Problems ◽  
Convex Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Bounded Convex Domain ◽  
Semilinear Elliptic ◽  
Image Position ◽  
Bounded Convex

Let Ω be a bounded convex domain in R2 with a smooth boundary. Let 0 < γ < 1. Let be a solution, positive in Ω, ofThen the function uα is concave for α = (l–γ)/2.

scholarly journals UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

2013 ◽  
Vol 55 (2) ◽  
pp. 399-409 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problems ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problems \[ \{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega, \] where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, λ is a positive parameter and f:(0,∞) → (0,∞) is sublinear at ∞ and is allowed to be singular at 0.

scholarly journals Existence and Nonexistence of Positive Solutions for Quasilinear Elliptic Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
K. Saoudi
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Quasilinear Elliptic Problem ◽  
Elliptic Boundary Value ◽  
Existence And Nonexistence ◽  
Quasilinear Elliptic

Using variational arguments we prove some existence and nonexistence results for positive solutions of a class of elliptic boundary-value problems involving thep-Laplacian.

UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS II

2015 ◽  
Vol 58 (2) ◽  
pp. 461-469 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problem \begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb{R}$n with smooth boundary ∂ Ω, λ is a large positive parameter, f:(0,∞) → [0,∞) is nonincreasing for large t and is allowed to be singular at 0.

scholarly journals Multiplicity results for sublinear elliptic equations with sign-changing potential and general nonlinearity

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Wei He ◽  
Qingfang Wu
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Critical Point Theorem ◽  
Elliptic Boundary Value ◽  
General Nonlinearity

Abstract In this paper, we study the following elliptic boundary value problem: $$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x, u),\quad x\in \Omega , \\ u=0, \quad x \in \partial \Omega , \end{cases} $$ { − Δ u + V ( x ) u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $\Omega \subset {\mathbb {R}}^{N}$ Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, and f is allowed to be sign-changing and is of sublinear growth near infinity in u. For both cases that $V\in L^{N/2}(\Omega )$ V ∈ L N / 2 ( Ω ) with $N\geq 3$ N ≥ 3 and that $V\in C(\Omega , \mathbb {R})$ V ∈ C ( Ω , R ) with $\inf_{\Omega }V(x)>-\infty $ inf Ω V ( x ) > − ∞ , we establish a sequence of nontrivial solutions converging to zero for above equation via a new critical point theorem.

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