On the domain monotonicity of the first eigenvalue of a boundary value problem

1976 ◽  
Vol 27 (4) ◽  
pp. 487-491 ◽  
Author(s):  
Miriam Bareket
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
First Eigenvalue ◽  
The First Eigenvalue

THE REGION OF SOLVABILITY OF A PARAMETERIZED BOUNDARY VALUE PROBLEM CAN BE DISCONNECTED

2004 ◽  
Vol 06 (06) ◽  
pp. 901-912
Author(s):  
ANTONIO J. UREÑA
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Second Eigenvalue

A celebrated result by Amann, Ambrosetti and Mancini [1] implies the connectedness of the region of existence for some parameter-depending boundary value problems which are resonant at the first eigenvalue. The analogous thing does not hold for problems which are resonant at the second eigenvalue.

scholarly journals A note on resonance problems with nonlinearity bounded in one direction

1995 ◽  
Vol 52 (2) ◽  
pp. 183-188 ◽  
Author(s):  
To Fu Ma ◽  
Luís Sanchez
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
First Eigenvalue ◽  
Existence Of A Solution ◽  
The First Eigenvalue ◽  
At Resonance ◽  
Growth Restrictions ◽  
Problem At Resonance ◽  
Bounded Below

We prove the existence of a solution for a semilinear boundary value problem at resonance in the first eigenvalue. The nonlinearity is assumed to be bounded below or above; no further growth restrictions are assumed.

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 On the existence threshold for positive solutions of p-Laplacian equations with a concave–convex nonlinearity

2015 ◽  
Vol 17 (06) ◽  
pp. 1450044 ◽  
Author(s):  
Fernando Charro ◽  
Enea Parini
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Boundary Value ◽  
First Eigenvalue ◽  
Limit Case ◽  
The First Eigenvalue ◽  
Laplacian Equations

We study the following boundary value problem with a concave–convex nonlinearity: [Formula: see text] Here Ω ⊂ ℝnis a bounded domain and 1 < q < p < r < p*. It is well known that there exists a number Λq, r> 0 such that the problem admits at least two positive solutions for 0 < Λ < Λq, r, at least one positive solution for Λ = Λq, r, and no positive solution for Λ > Λq, r. We show that [Formula: see text] where λ1(p) is the first eigenvalue of the p-Laplacian. It is worth noticing that λ1(p) is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case q = p.

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