Normalized solutions to the Chern-Simons-Schrodinger system under the nonlinear combined effect

We investigate normalized solutions to a class of Chern-Simons-Schrodinger systems with combined nonlinearities f(u) = |u|(p-2)u + mu|u|(q-2)u in R-2, where mu is an element of{+/- 1} and 2 < p, q < infinity. The solutions correspond to critical points of the underlying energy functional subject to the L-2-norm constraint, namely, integral(R2) |u|(2)dx = c for c > 0 given. Of particular interest is the competing and double L-2-supercritical case, i.e., mu = -1 and min{p, q} > 4. We prove several existence and multiplicity results depending on the size of the exponents p and q. It is worth emphasizing that some of them are also new even in the study of the Schrodinger equations. In addition, the asymptotic behaviors of the solutions and the associated Lagrange multipliers lambda as c -> 0 are described.