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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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