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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 $-{\Delta }u=\lambda u±f\left(x,u\right)+h\left(x\right)$ 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 $f\left(x,u\right)$ behaves at infinity like ${a\left(x\right)|u|}^{q-2}u$, with $M>a\left(x\right)>\delta >0$, and $1.
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35P05 General topics in linear spectral theory of PDE 58J50 Spectral problems; spectral geometry; scattering theory