Uniqueness of positive bound states with multiple bumps for Schrödinger-Poisson system. (English) Zbl 1480.35165
Summary: In this paper, we are concerned with the uniqueness of multi-bump solutions for the following Schrödinger-Poisson system
\[
\begin{cases}
-\varepsilon^2 \Delta u + Q(x)u + \Psi(x)u = |u|^{p-1} u, & \text{in } {\mathbb{R}}^N,\\
-\Delta\Psi = (N-2)\omega_{N-1} u^2, & \text{in } {\mathbb{R}}^N,
\end{cases}\tag{0.1}
\]
where \(\varepsilon\) is a parameter, \(Q(x)\) is a potential function in \(\mathbb{R}^N\), \(N \in [3, 6]\), \(p \in (1, \frac{N+2}{N-2})\) and \(\omega_{N-1}\) is the surface area of a unit ball in \(\mathbb{R}^N\). We prove the uniqueness of positive multi-bump solutions for Schrödinger-Poisson system concentrating at the critical points of \(Q(x)\), whenever \(Q(x)\) is degenerate or non-degenerate, or even not \(C^2\).
MSC:
| 35J47 | Second-order elliptic systems |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35J50 | Variational methods for elliptic systems |
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