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.

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.

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.

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 On the extremal solutions of superlinear Helmholtz problems

2022 ◽  
Vol 40 ◽  
pp. 1-8
Author(s):  
Makkia Dammak ◽  
Majdi El Ghord ◽  
Saber Ali Kharrati
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Weak Sense ◽  
Critical Value ◽  
Extremal Solutions ◽  
Regular Solutions ◽  
Smooth Bounded Domain ◽  
Bifurcation Phenomena ◽  
Stable Solutions

Abstract: In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

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 Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

2008 ◽  
Vol 2 (2) ◽  
pp. 158-174 ◽  
Author(s):  
Qianqiao Guo ◽  
Pengcheng Niu ◽  
Jingbo Dou
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

2014 ◽  
Vol 16 (04) ◽  
pp. 1350048
Author(s):  
Shuangjie Peng ◽  
Yanfang Peng
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Variational Method ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Singular Elliptic Equation ◽  
Sign Changing Solutions ◽  
Least Energy

We study the following singular elliptic equation [Formula: see text] with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.

Sign in / Sign up

Export Citation Format

Share Document