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On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition
Clustered layers for the Schrödinger-Maxwell system on a ball
| 1. | Department of Mathematics, Nanjing University, Nanjing 210093, China, China |
$-\varepsilon^2\Delta v+v+\omega\phi v-
\varepsilon^{\frac{p-1}{2}} v^p=0
\quad \text{ in}\ B_1,$
$-\Delta \phi=4\pi \omega v^2
\quad \text{in}\ B_1$,
$v,\ \phi>0 \ \text{in}\ B_1
\quad \text{and}\quad v=\phi=0
\quad \text{on}\ \partial B_1$,
where $B_1$ is the unit ball in $\mathbb{R}^3,\ \omega>0$ and $\ \frac{7}{3}$<$p\leq 5$ are constants, and $\varepsilon$>$0$ is a small parameter. Using the localized energy method, we prove that for every sufficiently large integer $N$, the system has a family of radial solutions $(v_\varepsilon, \phi_\varepsilon)$ such that $v_\varepsilon$ has $N$ sharp spheres concentrating on a sphere $\{|x|=r_N\}$ as $\varepsilon\to 0$.
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