Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries

We consider a semilinear Dirichlet problem with an unbounded and indefinite potential and a superlinear reaction which need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. Using a combination of variational methods (critical point theory) with truncation and comparison techniques, with Morse theory and with flow invariance arguments, we show that the problems has at least seven nontrivial smooth solutions and provide sign information for all of them.