Bifurcation and regularity of entire solutions for the planar nonlinear Schrödinger-Poisson system (Q6583573)

This paper is concerned with the study of the nonlinear Schrödinger-Poisson system in the plane \N\[\N\begin{cases} \N-\Delta u+\gamma\phi u =\lambda u +f(x,u), \quad \gamma=2\pi,\\\N\Delta\phi=u^2. \N\end{cases} \N\]\NA first difficulty of the problem is that the nonlinearity \(f\) has a critical growth in the sense of Trudinger-Moser. A second one is due to the fact that, different for example from the same problem in higher dimensions, the logarithmic kernel changes sign.\N\NThe authors are interested in bifurcations results as well regularity properties of the solutions.\N\NBy means of variational methods and the Lusternick-Schnirelmann Theory the authors first show a result on eigenvalues and eigenfunctions of the problem \N\[\N-\Delta u + [\ln(1+|\cdot|)*u^2]u =\lambda u \quad \text{in} \quad \mathbb R^2.\N\]\NThen by means of the degree theory the bifurcation result is obtained. Finally, under stronger assumption on the nonlinearity \(f\) a regularity result is shown.