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Nonselfadjoint resonance problems with unbounded perturbations. (English) Zbl 0599.35069

Let L be an elliptic but not necessarily self-adjoint operator on \(\Omega \subset R^ n\) and g be locally Lipschitzian. This paper studies the solvability of the Dirichlet problem \[ Lu=\lambda_ 1u+g(u)-h;\quad u=0\quad on\quad \partial \Omega, \] under the growth assumption \(\limsup_{| \xi | \to \infty} g(\xi)/\xi <d_ 0-\lambda_ 1\) where \(d_ 0>\lambda_ 1\) reduces to \(\lambda_ 2\) when L is self- adjoint and \(\lambda_ 1\) is the principal eigenvalue of L, and the Landesman-Lazer condition \[ \limsup_{u\to -\infty} g(u)\int_{\Omega}\theta^*_ 1(x) dx<\int_{\Omega}\theta^*_ 1(x) h(x) dx< \limsup_{u\to +\infty} g(u)\int_{\Omega}\theta^*_ 1(x) dx, \] where \(\theta^*_ 1\) is strictly positive eigenfunction of \(L^*\) corresponding to \(\lambda_ 1\). The proof uses a Leray-Schauder type argument whose applicability depends upon an interesting auxiliary result on the solutions of an associated linear problem.
Reviewer: J.Mawhin

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B20 Perturbations in context of PDEs
35B50 Maximum principles in context of PDEs
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