Normalized Solutions for a Class of the Nonlinear Elliptic Systems with Mixed-Power Nonlinearities on Bounded Domains

In this paper, we discuss the existence of normalized solutions for a class of the nonlinear elliptic systems with mixed-power nonlinearities on bounded domains. When the lower-order nonlinearity is L2-subcritical and the leading term is L2-critical, the global solutions can be found by minimum principle. If the lower-order nonlinearity is L2-subcritical or L2-critical and the leading term is L2-supercritical and Sobolev-subcritical, we can find the local minimum solutions by providing some proper conditions about the parameters. Furthermore, if the lower-order nonlinearity is L2-critical and the leading term is Sobolev-critical, we restore the compactness of a minimizing sequence so that the local solution can be achieved.