Electronic Research Archive (Feb 2024)
Normalized ground states to the nonlinear Choquard equations with local perturbations
Abstract
In this paper, we considered the existence of ground state solutions to the following Choquard equation $ \begin{eqnarray*} \left\{ \begin{aligned} &-\Delta u = \lambda u + (I_{\alpha}\ast F(u))f(u) + \mu|u|^{q-2}u \hskip0.5cm \mbox{in} \hskip0.2cm\mathbb{R}^{N}, \\ & \int\limits_{\mathbb{R}^{N}}|u|^{2}dx = a >0, \end{aligned} \right. \end{eqnarray*} $ where $ N \geq 3 $, $ I_{\alpha} $ is the Riesz potential of order $ \alpha \in (0, N) $, $ 2 < q \leq 2+ \frac{4}{N} $, $ \mu > 0 $ and $ \lambda \in \mathbb{R} $ is a Lagrange multiplier. Under general assumptions on $ F\in \mathcal{C}^{1}(\mathbb{R}, \mathbb{R}) $, for a $ L^{2} $-subcritical and $ L^{2} $-critical of perturbation $ \mu|u|^{q-2}u $, we established several existence or nonexistence results about the normalized ground state solutions.
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