Stationary waves with prescribed $L^2$-norm for the planar Schr\"odinger-Poisson system

23 Aug 2019 Cingolani Silvia Jeanjean Louis

The paper deals with the existence of standing wave solutions for the Schr\"odinger-Poisson system with prescribed mass in dimension $N=2$. This leads to investigate the existence of normalized solutions for an integro-differential equation involving a logarithmic convolution potential, namely $$ \left \{ \begin{aligned} - \Delta u & + \lambda u + \gamma \Bigl(\log {| \cdot |} * |u|^2 \Bigr) u =a |u|^{p-2} u \qquad \text{in $\mathbb R^2$,} \\ &\int_{\mathbb R^2} |u|^2 dx = c \end{aligned} \right... (read more)

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