Min-Max Principles with Nonlinear Generalized Rayleigh Quotients for Nonlinear Equations

Critical points with prescribed energy for a class of functionals depending on a parameter: existence, multiplicity and bifurcation results

“…The key point in this approach is to note that for any c the functional u → µ(c, u) satisfies the following relation, which can be easily checked (see also [25]):…”

“…Here ∂µ ∂u (c, u) denotes the Fréchet derivative of the functional u → µ(c, u). The above relation provides us with the following equivalence: Let us focus now on finding points in K(c) by following the nonlinear generalized Rayleigh quotient method developed in [23,25]. We consider the fibering map associated to the functional u → µ(c, u), namely, the real-valued function ψ c,u given by It should be noted that both possibilities in (b) can occur, as we shall see in some applications.…”

“…To this end, we shall follow the nonlinear generalized Rayleigh quotient method, introduced by Y Il'yasov [23] and used successfully also in other papers, as e.g. [12,24,[26][27][28].…”

“…Let us note that some preliminary results on (1.3) can be found in [25], as well as in [36], where the case c = 0 has been treated.…”

Prescribed energy saddle-point solutions of nonlinear indefinite problems

“…The prescribed energy solutions of nonlinear problems was studied recently in [7,18,19] by using the nonlinear Rayleigh quotients [17]. The nonlinear Rayleigh quotients have the remarkable property that the critical points of these functionals correspond to the solutions of the equations while having a simpler structure than the corresponding energy functionals (see, e.g., [17]).…”