Superlinear, Noncoercive Asymmetric Robin Problems with Indefinite, Unbounded Potential

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. The reaction term exhibits an asymmetric behavior, namely it is superlinear in the positive direction but without satisfying the Ambrosetti-Rabinowitz condition and it is sublinear but noncoercive in the negative direction. Using variational methods together with suitable truncation and perturbation techniques and Morse theory (critical groups), we prove a multiplicity theorem producing three nontrivial smooth solutions two of which have constant sign (one positive and the other negative).