| Abstract: |
| Quantitative and qualitative informations on nonlinear Schr\odinger equations strongly coupled with Poisson`s equation can be derived from nonlocal Choquard type equations. Limiting cases appear when the underlying function space setting is not well defined for the equation, as a consequence of the limiting Sobolev emedding which provides logarithmic kernels competing with exponential nonlinearities. We present two possible approaches to overcome this difficulty. The first one by establishing a suitable weighted Trudinger-Moser type inequality which eventually yields a proper functional setting. Alternatively, one can exploit a uniform approximation of the $\log$-kernel and then pass to the limit in the approximating equations. Both methods reveal new aspects which throw some light on the problem. |
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