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We study the following nonlinear Schr\"odinger equation $$-\Delta u + V(x) u= g(x,u),$$ where V and g are periodic in x. We assume that 0 is a rightboundary point of the essential spectrum of $-\Delta+V$. The superlinear andsubcritical term g satisfies a Nehari type monotonicity condition. We employ aNehari manifold type technique in a strongly indefitnite setting and obtain theexistence of a ground state solution. Moreover we get infinitely manygeometrically distinct solutions provided that g is odd.

Solutions to a nonlinear Schr\"odinger equation with periodic potential and zero on the boundary of the spectrum