Ground state for a stationary Schrödinger equation with several limit problems

We consider the stationary Schrödinger equation given by
\begin{equation} \label{es}
-\Delta u + u = a(x) |u|^{p-2} u, \quad u \in H^1(\mathbb{R}^N),
\end{equation}
where \(2 <p <2^*\) (\(2^*=2N/(N-2)\) for \(N \geq 3\) ,
\(2^*=\infty\) for \(N=2\)) and \(a \in L^\infty(\mathbb{R}^N)\). We
search for ground states for this equation using the Concentration
Compactness Principle. We are interested in cases where \(a\) is
sign-changing and \(\lim_{\|x\| \to \infty} a(x)\) doesn't exist,
so we will have to consider various limit problems for the
Concentration Compactness Principle. We will present some particular
cases and from them extract a set of more general hypothesis that
assure the existence of a ground state.