Multiple positive bound state solutions of a critical Choquard equation
| Autor: | Alves, Claudianor O., Figueiredo, Giovany M., Molle, Riccardo |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | IIn this paper we consider the problem $$ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u=(I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right.\leqno{(P_{\lambda})} $$ where $V_{\lambda}=\lambda+V_{0}$ with $\lambda \geq 0$, $V_0\in L^{N/2}(\R^N)$, $I_{\mu}=\frac{1}{|x|^\mu}$ is the Riesz potential with $0<\mu<\min\{N,4\}$ and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ with $N\geq 3$. Under some smallness assumption on $V_0$ and $\lambda$ we prove the existence of two positive solutions of $(P_\lambda)$. In order to prove the main result, we used variational methods combined with degree theory. Comment: In this new version, we change the title and prove two new main results with the collaboration of Professor Dr Riccardo Molle |
| Databáze: | arXiv |
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