Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum

We study the following nonlinear Schrödinger equation $-Δu + V(x) u = g(x,u),$ where V and g are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $-Δ+V$. The superlinear and subcritical term g satisfies a Nehari type monotonicity condition. We employ a Nehari manifold type technique in a strongly indefitnite setting and obtain the existence of a ground state solution. Moreover we get infinitely many geometrically distinct solutions provided that g is odd.

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