Normalized solutions for Choquard equations with general nonlinearities

Abstract

<p style="text-indent:20px;">In this paper, we prove the existence of positive solutions with prescribed <inline-formula><tex-math id="M1">$ L^{2} $</tex-math></inline-formula>-norm to the following Choquard equation:</p><p style="text-indent:20px;"><disp-formula> <label></label> <tex-math id="FE1"> $ \begin{equation*} -\Delta u-\lambda u = (I_{\alpha}*F(u))f(u), \ \ \ \ x\in \mathbb{R}^3, \end{equation*} $ </tex-math></disp-formula></p><p style="text-indent:20px;">where <inline-formula><tex-math id="M2">$ \lambda\in \mathbb{R}, \alpha\in (0,3) $</tex-math></inline-formula> and <inline-formula><tex-math id="M3">$ I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R} $</tex-math></inline-formula> is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any <inline-formula><tex-math id="M4">$ c>0 $</tex-math></inline-formula>, the above equation possesses at least a couple of weak solution <inline-formula><tex-math id="M5">$ (\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^- $</tex-math></inline-formula> such that <inline-formula><tex-math id="M6">$ \|\bar{u}_c\|_{2}^{2} = c $</tex-math></inline-formula>.</p>