Double resonance for Robin problems with indefinite and unbounded potential

Abstract

We study a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential term. The nonlinearity \begin{document}$f(x, s)$\end{document} is a Caratheodory function which is asymptotically linear as \begin{document}$ s\to ± ∞$\end{document} and resonant. In fact we assume double resonance with respect to any nonprincipal, nonnegative spectral interval \begin{document}$ \left[ \hat{λ}_k, \hat{λ}_{k+1}\right]$\end{document} . Applying variational tools along with suitable truncation and perturbation techniques as well as Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of constant sign.