Bartsch, Thomas; Ding, Yanheng On a nonlinear Schrödinger equation with periodic potential. (English) Zbl 0927.35103 Math. Ann. 313, No. 1, 15-37 (1999). 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 \(H^1(\mathbb{R}^N)\). Cited in 1 ReviewCited in 89 Documents 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 Keywords:nonlinear stationary Schrödinger equation; spectrum of the Schrödinger operator; entire solutions; infinitely many solutions PDFBibTeX XMLCite \textit{T. Bartsch} and \textit{Y. Ding}, Math. Ann. 313, No. 1, 15--37 (1999; Zbl 0927.35103) Full Text: DOI