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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
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