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.
Citation: Xudong Shang. Normalized ground states to the nonlinear Choquard equations with local perturbations[J]. Electronic Research Archive, 2024, 32(3): 1551-1573. doi: 10.3934/era.2024071
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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Xudong Shang. Normalized ground states to the nonlinear Choquard equations with local perturbations[J]. Electronic Research Archive, 2024, 32(3): 1551-1573. doi: 10.3934/era.2024071
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