Normalized ground states to the nonlinear Choquard equations with local perturbations

Abstract

<abstract><p>In this paper, we considered the existence of ground state solutions to the following Choquard equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>