Ground states for irregular and indefinite superlinear Schrödinger equations

We consider the existence of a ground state for the subcritical stationary semilinear Schrödinger equation −Δu+u=a(x)|u|p−2u−Δu+u=a(x)|u|p−2u in H1H1, where a∈L∞(RN)a∈L∞(RN) may change sign. Our focus is on the case where loss of compactness occurs at the ground state energy. By providing a new variant of the Splitting Lemma we do not need to assume the existence of a limit problem at infinity, be it in the form of a pointwise limit for a   as |x|→∞|x|→∞ or of asymptotic periodicity. That is, our problem may be irregular at infinity. In addition, we allow a to change sign near infinity, a case that has never been treated before.