## On a Schrödinger equation with periodic potential and spectrum point zero.(English)Zbl 1030.35068

The authors prove the existence of solutions $$u$$ of the nonlinear Schrödinger equation $-\Delta u(x) + V(x)u(x) = g(x,u(x)) \quad \text{ for } x\in \mathbb R^N$ with $$u(x) \to 0$$ as $$|x|\to\infty$$. Here, $$x\mapsto V(x)$$ is 1-periodic and continuous. Furthermore, $$(x,t)\mapsto g(x,t)$$ is continuous, 1-periodic in the first argument and grows superlinearly but subcritically in the second argument. Besides some conditions on the relation between $$g$$ and its primitive function $$G$$ it is assumed that 0 is either a left end point or right end point of a spectral gap.
The hypotheses are weaker than the Ambrosetti-Rabinowitz condition $$0 < \gamma G(x,u) \leq g(x,u)u$$ for some $$\gamma >2$$, $$u\neq 0$$ which is known to imply a similar result [see T. Bartsch and Y. Ding, Math. Ann. 313, 15-37 (1999; Zbl 0927.35103)].
The proof is based on an abstract linking theorem in reflexive Banach spaces which is an extension of the results obtained by Y. Ding and M. Willem [Z. Angew. Math. Phys. 50, 759-778 (1999; Zbl 0997.37041)] and A. Szulkin and W. Zou [J. Funct. Anal. 187, 25-41 (2001; Zbl 0984.37072)].

### MSC:

 35J60 Nonlinear elliptic equations 35J10 Schrödinger operator, Schrödinger equation 35Q55 NLS equations (nonlinear Schrödinger equations) 47J30 Variational methods involving nonlinear operators

### Citations:

Zbl 0927.35103; Zbl 0997.37041; Zbl 0984.37072
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