Short proofs of results by Landesman, Lazer, and Leach on problems related to resonance. (English) Zbl 1249.35080

Summary: We study two boundary-value problems originally considered by A. C. Lazer and coauthors. The first one is an elliptic problem \(Lu+\alpha u+g(u)=h(x)\) in a bounded domain \(\Omega \subset \mathbb {R}^N\), 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 one is an ODE problem \(u''+n^2u+g(u)=e(t)\), where \(e\) has period \(2\pi \) and a \(2\pi \)-periodic solution is sought. Here, the corresponding linear homogeneous equation, \(u''+n^2u=0\), 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 A. C. Lazer [J. Math. Mech. 19, 609–623 (1970; Zbl 0193.39203)] and by A. C. Lazer and D. E. Leach [Ann. Mat. Pura Appl., IV. Ser. 82, 49–68 (1969; Zbl 0194.12003)] on these two problems.


35J25 Boundary value problems for second-order elliptic equations
34C25 Periodic solutions to ordinary differential equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian