Geometrically distinct solutions of nonlinear elliptic systems with periodic potentials

Archiv der Mathematik, volume 115, 703–716

In this paper, we study the following nonlinear elliptic systems:

\begin{equation*} -\Delta u_1+V_1(x)u_1=\partial_{u_1}F(x,u)\quad x\in \mathbb{R}^N, \end{equation*}

\begin{equation*} -\Delta u_2+V_2(x)u_2=\partial_{u_2}F(x,u)\quad x\in \mathbb{R}^N, \end{equation*}

where $u=(u_1,u_2):\mathbb{R}^N\rightarrow \mathbb{R}^2$, $F$ and $V_i$ are periodic in $x_1,\cdots,x_N$ and $0\notin \sigma(-\Delta+V_i)$ for $i=1,2$, where $\sigma(-\Delta +V_i)$ stands for the spectrum of the Schr"odinger operator $-\Delta +V_i$. Under some suitable assumptions on $F$ and $V_i$, we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the resultin Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).