Standing waves with a critical frequency for nonlinear Choquard equations. (English) Zbl 1370.35032
Summary: We study the nonlocal Choquard equation
\[
- \varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon = (I_\alpha \ast | u_\varepsilon |^p) | u_\varepsilon |^{p - 2} u_\varepsilon \quad \text{ in } \mathbb{R}^N
\]
where \(N \geq 1\), \(I_\alpha\) is the Riesz potential of order \(\alpha \in(0, N)\) and \(\varepsilon > 0\) is a parameter. When the nonnegative potential \(V \in C(\mathbb{R}^N)\) achieves \(0\) with a homogeneous behavior or on the closure of an open set but remains bounded away from \(0\) at infinity, we show the existence of groundstate solutions for small \(\varepsilon > 0\) and exhibit the concentration behavior as \(\varepsilon \rightarrow 0\).
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
| 35R09 | Integro-partial differential equations |
Keywords:
nonlinear Choquard equations; nonlocal semilinear elliptic equation; semi-classical limit; groundstate solutions; concentration behaviorReferences:
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