Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. (English) Zbl 1415.35115
Summary: In this paper we study a class of weakly coupled Schrödinger system arising in several branches of sciences, such as nonlinear optics and Bose-Einstein condensates. Instead of the well known super-quadratic condition that \(\lim_{|z|\rightarrow\infty}\frac{F(x,z)}{|z|^2} = \infty\) uniformly in \(x\), we introduce a new local super-quadratic condition that allows the nonlinearity \(F\) to be super-quadratic at some \(x\in \mathbb{R}^N\) and asymptotically quadratic at other \(x\in \mathbb{R}^N\). Employing some analytical skills and using the variational method, we prove some results about the existence of ground states for the system with periodic or non-periodic potentials. In particular, any nontrivial solutions are continuous and decay to zero exponentially as \(|x| \rightarrow \infty\). Our main results extend and improve some recent ones in the literature.
MSC:
| 35J50 | Variational methods for elliptic systems |
| 35J47 | Second-order elliptic systems |
| 35J10 | Schrödinger operator, Schrödinger equation |
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