Multiple Normalized Solutions for Choquard equations involving Kirchhoff type perturbation
Keywords
Choquard equation, Kirchhoff equation, mountain pass geometry, normalized solutionsAbstract
In this paper we study the existence of critical points of the $C^1$ functional \begin{equation*} E(u)=\frac{a}{2}\int_{\bbrn}|\nabla u|^2dx+\frac{b}{4}\bigg(\int_{\bbrn}|\nabla u|^2dx\bigg)^2-\frac{1}{2p}\int_{\bbrn}(I_\alpha*|u|^p)|u|^pdx \end{equation*} under the constraint \begin{equation*} S_c=\bigg\{u\in H^1(\bbrn)\ \bigg\vert\ \int_{\bbrn}|u|^2dx=c^2\bigg\}, \end{equation*} where $a> 0$, $b> 0$, $N\geq3$, $\alpha\in(0,N)$, $ (N+\alpha)/{N}< p< (N+\alpha)/$ $(N-2)$ and $I_{\alpha}$ is the Riesz Potential. When $p$ belongs to different ranges, we obtain the threshold values separating the existence and nonexistence of critical points of $E$ on $S_c$. We also study the behaviors of the Lagrange multipliers and the energies corresponding to the constrained critical points when $c\to 0$ and $c\to +\infty$, respectively.References
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