BOUNDEDNESS OF THE EXTREMAL SOLUTION FOR SEMILINEAR ELLIPTIC PROBLEMS

2002 ◽  
Vol 04 (03) ◽  
pp. 547-558 ◽  
Author(s):  
DONG YE ◽  
FENG ZHOU
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Semilinear Elliptic Equation ◽  
Extremal Solution ◽  
Extremal Solutions ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

We investigate here the boundedness of extremal solutions for some semilinear elliptic equation -Δu=λf(u) posed on a bounded smooth domain of ℝN with Dirichlet boundary condition. Some sufficient conditions for f are established to ensure the regularity of extremal solutions when N ≤ 9, which cover all well-known cases.

scholarly journals On the extremal solutions of semilinear elliptic problems

2005 ◽  
Vol 2005 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Lamia Ben Chaabane
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Extremal Solutions ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

We investigate here the properties of extremal solutions for semilinear elliptic equation−Δu=λf(u)posed on a bounded smooth domain ofℝnwith Dirichlet boundary condition and withfexploding at a finite positive valuea.

A Note on Symmetry of Solutions for a Class of Singular Semilinear Elliptic Problems

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
Alessandro Trombetta
Keyword(s):  
Elliptic Equation ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Dirichlet Boundary Conditions ◽  
Moving Plane Method ◽  
Plane Method ◽  
Semilinear Elliptic ◽  
Moving Plane ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

AbstractWe prove symmetry and monotonicity properties for positive solutions of the singular semilinear elliptic equationin bounded smooth domains with zero Dirichlet boundary conditions. The well-known moving plane method is applied.

scholarly journals On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

2005 ◽  
Vol 2005 (2) ◽  
pp. 95-104
Author(s):  
M. Ouanan ◽  
A. Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Bounded Smooth

We study the existence of nontrivial solutions for the problemΔu=u, in a bounded smooth domainΩ⊂ℝℕ, with a semilinear boundary condition given by∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, whereWis a potential changing sign,ghas a superlinear growth condition, and the parameterλ∈]0,λ1];λ1is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

scholarly journals Global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains

1994 ◽  
Vol 124 (2) ◽  
pp. 371-388 ◽  
Author(s):  
Timothy J. Healey ◽  
Hansjörg Kielhöfer ◽  
Charles A. Stuart
Keyword(s):  
Boundary Condition ◽  
Weak Solutions ◽  
Dirichlet Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Order Elliptic Equation ◽  
Semilinear Elliptic ◽  
Nonsmooth Domains ◽  
Second Order Elliptic Equation ◽  
Semilinear Elliptic Problems ◽  
Parameter Dependent

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.

scholarly journals Blow up of solutions of semilinear heat equations in general domains

2015 ◽  
Vol 17 (02) ◽  
pp. 1350042 ◽  
Author(s):  
Valeria Marino ◽  
Filomena Pacella ◽  
Berardino Sciunzi
Keyword(s):  
Boundary Condition ◽  
Initial Data ◽  
Heat Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Initial Value ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

Consider the nonlinear heat equation vt - Δv = |v|p-1v in a bounded smooth domain Ω ⊂ ℝn with n > 2 and Dirichlet boundary condition. Given up a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑup blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent [Formula: see text]. Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.

MULTIPLICITY OF SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS WITH COMBINED NONLINEARITIES

2011 ◽  
Vol 13 (03) ◽  
pp. 389-405 ◽  
Author(s):  
XIAO JUN CHANG
Keyword(s):  
Upper And Lower Solutions ◽  
Degree Theory ◽  
Type Condition ◽  
Multiplicity Results ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth ◽  
Truncation Techniques

In this paper, we are concerned with the semilinear elliptic problem [Formula: see text] where Ω is a bounded smooth domain in Rm(m ≥ 2), h ∈ L∞(Ω), h(x) ≢ 0, 1 < q < 2, [Formula: see text]. When the nonlinearity f satisfies the Ambrosetti–Prodi type condition at infinity, two multiplicity results are established by using a combination of variational methods, upper and lower solutions, Leray–Schauder degree theory and suitable truncation techniques.

On the system of p-Laplacian equations with critical growth

2018 ◽  
Vol 29 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Xiangqing Liu ◽  
Junfang Zhao ◽  
Jiaquan Liu
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary ◽  
Critical Growth ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Laplacian Equations

In this paper, we consider the system of [Formula: see text]-Laplacian equations with critical growth [Formula: see text] where [Formula: see text] is a bounded smooth domain in [Formula: see text] the first eigenvalue of the [Formula: see text]-Laplacian operator [Formula: see text] with the Dirichlet boundary condition, [Formula: see text] for [Formula: see text]. The existence of infinitely many sign-changing solutions is proved by the truncation method and by the concentration analysis on the approximating solutions, provided [Formula: see text].

Uniqueness and local properties of positive solutions to a semilinear elliptic equation with singular potentials

2019 ◽  
Vol 21 (05) ◽  
pp. 1850032
Author(s):  
Lei Wei ◽  
Zengji Du
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Semilinear Elliptic Equation ◽  
Local Behavior ◽  
Singular Potentials ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic ◽  
Local Properties ◽  
Nonnegative Continuous Function ◽  
Bounded Smooth

In this paper, we study the uniqueness and local behavior at the origin of positive solutions to [Formula: see text] where [Formula: see text], [Formula: see text]) is a bounded smooth domain and [Formula: see text], and [Formula: see text] is a nonnegative continuous function over [Formula: see text]. In the two cases: (i) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text]; (ii) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text], we establish the uniqueness of positive solutions and the exact blow-up rate of positive solutions at the origin. This paper seems to be the first one to deal with the latter case.

EXTREMAL FUNCTIONS FOR A SHARP MOSER–TRUDINGER INEQUALITY

2006 ◽  
Vol 17 (03) ◽  
pp. 331-338 ◽  
Author(s):  
YUNYAN YANG
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Domain ◽  
Extremal Functions ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth ◽  
Trudinger Inequality

Let Ω be a bounded smooth domain in ℝ2, and λ1(Ω) the first eigenvalue of the Laplacian with Dirichlet boundary condition in Ω. Then Adimurthi and Druet show that for any 0 ≤ α < λ1(Ω)[Formula: see text] We prove in this paper that there exist extremal functions for the above inequality. In other words, we show that [Formula: see text] is attained for any 0 ≤ α < λ1(Ω).

scholarly journals A Lotka-Volterra Competition Model with Cross-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wenyan Chen ◽  
Ya Chen
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Cross Diffusion ◽  
Existence Of Positive Solutions

A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

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