Differential and Integral Equations

Short proofs of results by Landesman, Lazer, and Leach on problems related to resonance

S.P. Hastings and J.B. Mcleod

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study two boundary-value problems originally considered by A.C. Lazer and coauthors. The first is an elliptic problem \[ Lu+\alpha u+g \left( u\right) =h\left( x\right) \] in a bounded domain $\Omega\subset R^{N},$ with $u=0$ on the boundary $\partial\Omega.$ It is assumed that $Lu+\alpha u=0$ has a one-dimensional set of solutions satisfying the same boundary condition$.$ The second is an ODE problem \[ u^{\prime\prime}+n^{2}u+g\left( u\right) =e\left( t \right) , \] where $e$ has period $2\pi$ and a $2\pi$-periodic solution is sought. Here, the corresponding linear homogeneous equation, $u^{\prime\prime}+n^{2}u=0,$ has a two-dimensional set of $2\pi$-periodic solutions. In each case, conditions are sought which guarantee the existence of at least one solution to the original problem. We give short proofs of theorems first proved by E.M. Landesman and Lazer, and by Lazer and D.E. Leach, on these two problems.

Article information

Source
Differential Integral Equations, Volume 24, Number 5/6 (2011), 435-441.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356018912

Mathematical Reviews number (MathSciNet)
MR2809615

Zentralblatt MATH identifier
1249.35080

Subjects
Primary: 34B15: Nonlinear boundary value problems 35J57: Boundary value problems for second-order elliptic systems

Citation

Hastings, S.P.; Mcleod, J.B. Short proofs of results by Landesman, Lazer, and Leach on problems related to resonance. Differential Integral Equations 24 (2011), no. 5/6, 435--441. https://projecteuclid.org/euclid.die/1356018912


Export citation