Bifurcation and regularity of entire solutions for the planar nonlinear Schrodinger-Poisson system

This paper deals with bifurcation and regularity properties of the entire solutions of the planar nonlinear Schrodinger-Poisson system in R-2 { - Delta u + gamma phi u =gamma u + f ( x, u), gamma = 2 pi, Delta phi = u(2). equivalent to the planar nonlinear logarithmic Schrodinger-Poisson equation - Delta u + (log | center dot | * |u|(2))u =lambda u + f ( x, u) in R-2, where the nonlinearity f has undefined sign and exhibits critical exponential growth in the sense of Trudinger-Moser at infinity. To this aim we have to overcome several difficulties in the proof of the bifurcation theorem and to establish for the first time existence and properties of the first eigenvalue of a related Laplacian equation with logarithmic kernel. The novelties of the paper lie in the appearance of the critical exponential growth and possibly the Hardy nonlinearity. Finally, a regularity result is also presented, where f has polynomial growth, and it guarantees the L-infinity-bound of any (weak) solution of the planar nonlinear Schrodinger-Poisson system (S) via the solution of the planar nonlinear logarithmic Schrodinger-Poisson equation (epsilon).