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Semilinear elliptic problems near resonance with a nonprincipal eigenvalue. (English) Zbl 1149.35044

Summary: We consider the Dirichlet problem for the equation \(-\Delta u=\lambda u\pm f(x,u)+h(x)\) in a bounded domain, where \(f\) has a sublinear growth and \(h\in L^2\). We find suitable conditions on \(f\) and \(h\) in order to have at least two solutions for \(\lambda\) near to an eigenvalue of \(-\Delta\). A typical example to which our results apply is when \(f(x,u)\) behaves at infinity like \(a(x)|u|^{q-2}u\), with \(M>a(x)>\delta>0\), and \(1<q<2\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35P05 General topics in linear spectral theory for PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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