Multiple solitary wave solutions of nonlinear Schrödinger systems

Rushun Tian, Zhi-Qiang Wang

Abstract


Consider the $N$-coupled nonlinear elliptic system
$$
\cases
\displaystyle
-\Delta U_j+ U_j=\mu U_j^3+\beta U_j\sum_{k\neq j} U_k^2
\quad \text{in } \Omega,\\
U_j> 0 \quad\text{in } \Omega,\quad
U_j=0 \quad \text{on } \partial\Omega,\ j=1, \ldots, N.
\endcases
\tag P
$$
where $\Omega$ is a smooth and bounded (or unbounded if $\Omega$ is
radially symmetric) domain in $\mathbb R^n$, $n\leq3$. By using
a $Z_N$ index theory, we prove the existence of multiple solutions
of (P) and show the dependence
of multiplicity results on the coupling constant $\beta$.

Keywords


Nonlinear Schrödinger system; Nehari manifold; a $Z_N$-index theory

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