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May/June 2011 Short proofs of results by Landesman, Lazer, and Leach on problems related to resonance
S.P. Hastings, J.B. Mcleod
Differential Integral Equations 24(5/6): 435-441 (May/June 2011).


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.


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S.P. Hastings. J.B. Mcleod. "Short proofs of results by Landesman, Lazer, and Leach on problems related to resonance." Differential Integral Equations 24 (5/6) 435 - 441, May/June 2011.


Published: May/June 2011
First available in Project Euclid: 20 December 2012

zbMATH: 1249.35080
MathSciNet: MR2809615

Primary: 34B15, 35J57

Rights: Copyright © 2011 Khayyam Publishing, Inc.


Vol.24 • No. 5/6 • May/June 2011
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