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Normalized solutions for Choquard equations with general nonlinearities. (English) Zbl 1447.49011

Summary: In this paper, we prove the existence of positive solutions with prescribed \( L^2 \)-norm to the following Choquard equation: \[ -\Delta u-\lambda u = (I_{\alpha}*F(u))f(u),\quad x\in \mathbb{R}^3, \] where \(\lambda\in \mathbb{R}\), \(\alpha\in (0,3)\) and \( I_\alpha :\mathbb{R}^3\rightarrow \mathbb{R}\) is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any \(c>0\), the above equation possesses at least a couple of weak solution \((\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_c\times \mathbb{R}^- \) such that \(\|\bar{u}_c\|_2^2 = c\).

MSC:

49J35 Existence of solutions for minimax problems
35B09 Positive solutions to PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
35D30 Weak solutions to PDEs
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