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Solvability of linear and semilinear eigenvalue problems with \(L^ 1\) data. (English) Zbl 0822.35106

Summary: We study the equation \[ A(u)= \lambda u+f \quad \text{in } \Omega, \qquad u=0 \quad \text{on } \partial \Omega, \] where \(A\) is a linear elliptic operator in divergence form, \(\lambda\) is a real number and \(f\) is a function belonging to \(L^ 1 (\Omega)\). We find existence results similar to those obtained in the case \(f\in L^ 2 (\Omega)\). Furthermore, we study the Landesmann-Lazer, Dolph and Ambrosetti-Prodi problems for the operator \(A\), always with \(L^ 1 (\Omega)\) data.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
47F05 General theory of partial differential operators
35J15 Second-order elliptic equations
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References:

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