A generalized min–max theorem for functionals of strongly indefinite sign

Abstract

We consider non-linear Schrodinger equations of the following type: $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u(x) + V(x)u(x)-q(x)|u(x)|^\sigma u(x) = \lambda u(x), \quad x\in \mathbb{R }^N \\ u\in H^1(\mathbb{R }^N)\setminus \{0\}, \end{array} \right. \end{aligned}$$ where \(N\ge 1\) and \(\sigma >0\). We will concentrate on the case where both \(V\) and \(q\) are periodic, and we will analyse what happens for different values of \(\lambda \) inside a spectral gap \(]\lambda ^-,\lambda ^+[\). We derive both the existence of multiple orbits of solutions and the bifurcation of solutions when \(\lambda \nearrow \lambda ^+\). Thereby we use the corresponding energy function \({I_\lambda }\) and we derive a new variational characterization of multiple critical levels for such functionals: in this way we get multiple orbits of solutions. One main advantage of our new view on some specific critical values \(c_0(\lambda )\le c_1(\lambda )\le \cdots \le c_n(\lambda )\le \cdots \) is a multiplicity result telling us something about the number of critical points with energies below \(c_n(\lambda )\), even if for example two of these values \(c_i(\lambda )\) and \(c_j(\lambda )\) (\(0\le i<j\le n\)) coincide. Let us close this summary by mentioning another main advantage of our variational characterization of critical levels: we present our result in an abstract setting that is suitable for other problems and we give some hints about such problems (like the case corresponding to a Coulomb potential \(V\)) at the end of the present paper.