## On a nonlinear Schrödinger equation with periodic potential.(English)Zbl 0927.35103

Summary: We find entire solutions of the semilinear elliptic problem $-\Delta u+ V(x)u= g(x,u)\quad\text{for }x\in \mathbb{R}^N;$
$u(x)\to 0\quad\text{as }| x|\to\infty;$ where $$V$$ and $$g$$ are assumed to be periodic in $$x$$. The spectrum $$\sigma(S)$$ of $$S= -\Delta+ V$$ on $$L^2(\mathbb{R}^N)$$ is purely absolutely continuous. We consider the singular case that $$0\in\sigma(S)$$ is a boundary point of $$\sigma(S)$$. Under certain conditions on $$g$$ we obtain one solution, and if $$g$$ is odd infinitely many solutions. The solutions lie in $$H^2_{\text{loc}}(\mathbb{R}^N)$$ but not necessarily in $$H^1(\mathbb{R}^N)$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 35J60 Nonlinear elliptic equations
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