A semilinear elliptic eigenvalue problem, II. The plasma problem

1980 ◽  
Vol 87 (1-2) ◽  
pp. 83-109 ◽  
Author(s):  
Grant Keady ◽  
John Norbury
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Image Position ◽  
Plasma Problem ◽  
Simply Connected ◽  
Special Case

SynopsisThis paper continues the study of the boundary value problem, for (λ, ψ)Here δ denotes the Laplacian,kis a given positive constant, and λ1will denote the first eigenvalue for the Dirichlet problem for −δ on Ω. For λ ≦ λ1, the only solutions are those with ψ = 0. Throughout we will be interested in solutions (λ, ψ) with λ > λ1and with ψ > 0 in Ω.In the special case Ω =B(0,R) there is a branch ℱe, of explicit exact solutions which bifurcate from infinity at λ = λ1and for which the following conclusions are valid, (a) The setAψ,is simply-connected, (b) Along ℱe, ψm→k, ‖ψ‖1→ 0 and the diameter ofAψtends to zero as λ → ∞, whereHere it is shown that the above conclusions hold for other choices of Ω, and in particular, for Ω = (−a, a)×(−b, b). (Existence is settled in Part I, and elsewhere.)The results of numerical and asymptotic calculations when Ω = (−a, a)×(−b, b) are given to illustrate both the above, and some limitations in the conclusions of our analysis.

scholarly journals Necessary and Sufficient Condition for the Existence of Solutions to a Discrete Second-Order Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Chenghua Gao
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Sufficient Condition ◽  
First Eigenvalue ◽  
Necessary And Sufficient Condition ◽  
The First Eigenvalue ◽  
Necessary And Sufficient

This paper is concerned with the existence of solutions for the discrete second-order boundary value problemΔ2u(t-1)+λ1u(t)+g(Δu(t))=f(t),t∈{1,2,…,T},u(0)=u(T+1)=0, whereT>1is an integer,f:{1,…,T}→R,g:R→Ris bounded and continuous, andλ1is the first eigenvalue of the eigenvalue problemΔ2u(t-1)+λu(t)=0,t∈T,u(0)=u(T+1)=0.

scholarly journals Nonlinear elastic beam problems with the parameter near resonance

2018 ◽  
Vol 16 (1) ◽  
pp. 1176-1186
Author(s):  
Man Xu ◽  
Ruyun Ma
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Global Bifurcation ◽  
Elastic Beam ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Linear Eigenvalue Problem ◽  
Near Resonance ◽  
Nonlinear Elastic

AbstractIn this paper, we consider the nonlinear fourth order boundary value problem of the form$$ \begin{array}\text \left\{ \begin{aligned} &u^{(4)}(x)-\lambda u(x)=f(x, u(x))-h(x), \ \ x\in (0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0,\\ \end{aligned}\right. \end{array} $$which models a statically elastic beam with both end-points cantilevered or fixed. We show the existence of at least one or two solutions depending on the sign of λ−λ1, where λ1 is the first eigenvalue of the corresponding linear eigenvalue problem and λ is a parameter. The proof of the main result is based upon the method of lower and upper solutions and global bifurcation techniques.

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

2016 ◽  
Vol 16 (1) ◽  
pp. 51-65 ◽  
Author(s):  
Salvatore A. Marano ◽  
Sunra J. N. Mosconi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Morse Theory ◽  
Multiple Solutions ◽  
First Eigenvalue ◽  
Reaction Term ◽  
The First Eigenvalue

AbstractThe existence of multiple solutions to a Dirichlet problem involving the ${(p,q)}$-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of ${-\Delta_{p}}$ in ${W^{1,p}_{0}(\Omega)}$ and exhibits a concave behavior near zero.

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.

A Quasi-Linear Elliptic Boundary Value Problem

1966 ◽  
Vol 18 ◽  
pp. 1105-1112 ◽  
Author(s):  
R. A. Adams
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Classical Solutions ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Image Position ◽  
Linear Elliptic Boundary ◽  
Typical Point

Let Ω be a bounded open set in Euclidean n-space, En. Let α = (α1, … , an) be an n-tuple of non-negative integers;and denote by Qm the set ﹛α| 0 ⩽ |α| ⩽ m}. Denote by x = (x1, … , xn) a typical point in En and putIn this paper we establish, under certain circumstances, the existence of weak and classical solutions of the quasi-linear Dirichlet problem1

scholarly journals Eigenvalues for a combination between local and nonlocal p-Laplacians

2019 ◽  
Vol 22 (5) ◽  
pp. 1414-1436 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Raúl Ferreira ◽  
Julio D. Rossi
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Limit Problem ◽  
First Eigenvalue ◽  
Order Zero ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Homogeneous Operator ◽  
Dirichlet Eigenvalue Problem

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

scholarly journals Existence of positive solutions of some semilinear elliptic equations with singular coefficients

2001 ◽  
Vol 131 (6) ◽  
pp. 1275-1295 ◽  
Author(s):  
Nirmalendu Chaudhuri ◽  
Mythily Ramaswamy
Keyword(s):  
Linear Problem ◽  
First Eigenvalue ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Singular Functions ◽  
Palais Smale Condition ◽  
Existence Of Positive Solutions ◽  
Singular Coefficients ◽  
Image Position ◽  
The Hardy Inequality

In this paper we consider the semilinear elliptic problem in a bounded domain Ω ⊆ Rn, where μ ≥ 0, 0 ≤ α ≤ 2, 2α* := 2(n − α)/(n − 2), f : Ω → R+ is measurable, f > 0 a.e, having a lower-order singularity than |x|-2 at the origin, and g : R → R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions Ip for which the embedding is compact. When p = 2, α = 2, f ∈ I2 and 0 ≤ μ < (½(n − 2))2, we prove that the linear problem has -discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of I2, the first eigenvalue goes to a positive number as μ approaches (½(n − 2))2. Furthermore, when g is superlinear, we show that for the same subclass of I2, the functional corresponding to the differential equation satisfies the Palais-Smale condition if α = 2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0 ≤ α < 2.

Sign in / Sign up

Export Citation Format

Share Document