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Nonresonance conditions at the two first eigenvalues for semilinear equations. (English) Zbl 0816.35029

From the introduction: We establish an existence result for the problem \(LU= NU\), where \(L\) is a linear operator in \(L^ 2 (\Omega)\), \(\Omega\) being a bounded open set in \(\mathbb{R}^ n\) and \(N\) is the Nemitskii operator associated with some Carathéodory function \(f: \Omega\times \mathbb{R}\to \mathbb{R}\) with linear growth and that interacts, in some sense, with the two first eigenvalues of \(L\). Let us denote by \(\lambda_ 1\) the first positive eigenvalue of \(L\). About the function \(f\), and relatively to \(\lambda_ 1\), we consider a condition, in such a way that \(f\) can “touch” this eigenvalue. Relatively to the eigenvalue \(\lambda=0\) a different hypothesis is made, in such a way that \(f\) can cross it. Here we make use of the Leray-Schauder continuation principle, and so, a key ingredient in our proof is an apriori bound for the solutions of some class of equations.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
47J25 Iterative procedures involving nonlinear operators
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