PRESCRIBED ENERGY SADDLE-POINT SOLUTIONS OF NONLINEAR INDEFINITE PROBLEMS

. A minimax variational method for ﬁnding mountain pass-type solutions with prescribed energy levels is introduced. The method is based on application of the Linking Theorem to the energy-level nonlinear Rayleigh quotients which critical points correspond to the solutions of the equation with prescribed energy. An application of the method to nonlinear indeﬁnite elliptic problems with nonlinearities that does not satisfy the Ambrosetti-Rabinowitz growth conditions is also presented.

In the literature, solutions to the Schrödinger equations having a prescribed frequency λ and unknowns energy E and mass α = Q(u) are commonly studied (see, e.g, [10,26]).An alternative formulation which has also been actively investigated over the last decades consists of finding the solution u to (1.2) having prescribed mass α, while λ and E are unknown (see, e.g., [4,11,22,27]).Mathematically, all three approaches, namely, prescribed frequency, prescribed energy, and prescribed mass, are equally valid.Moreover, all of these approaches evidently are relevant from the physical point of view.In particular, the approach with prescribed energy arises in the study of inverse problems and the spectral and scattering control problems (see, e.g., [2,12,20,23,24,29]).
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]).They were particularly useful (see, e.g., [17,19]) for finding nonnegative solutions to zero-mass problems [6] and detecting S-shaped bifurcations of nonlinear partial differential equations [7].The nonlinear Rayleigh quotients method and solutions with prescribed energies were used to introduce a generalization of the Poincaré and Courant-Fischer-Weil minimization principles to nonlinear problems [18], as well as to study the orbital stability for ground states of the NLS equations [7].
There are at least two motivations to study prescribed energy solutions of (1.1), apart from the fact that it appears in some physical models.First, we develop the nonlinear Rayleigh quotient method for new classes of problems, in particular for equations with inhomogeneous and general forms of nonlinearities.Second, we develop the Mountain Pass methods in order to capture qualitative properties of the solutions that it generates.
The Mountain Pass Theorem introduced by Ambrosetti and Rabinowitz [1] and its generalization as the Benci-Rabinowitz Linking Theorem [5] is a powerful tool to establish the existence of solutions for nonlinear problems of the variational form.The solutions obtained by this method usually correspond to saddle critical points of the energy functional and are often referred to as mountain pass-type solutions or saddle-point solutions.In essence, this method is topological, which makes it possible to use it for solving problems of very general forms.On the other hand, this generality often makes it difficult to find out detailed information about the obtained solutions.The aim of this work is to show that the nonlinear Rayleigh quotient method can be applied to generate saddle-point solutions with prescribed energy within the framework of the Linking Theorem.
Let us state our main result.We seek for prescribed energy solutions using the energy level nonlinear Rayleigh quotient [7,17,19]: We assume that (A1) there exist The operator (−∆) with Dirichlet boundary conditions defines a self-adjoint operator in L 2 (Ω) (see, e.g., [14]) and its spectrum consists of an infinite sequence ordered 0 < λ 1 < λ 2 ≤ . . . of eigenvalues repeated according to their finite multiplicity.Now, with the convention that λ 0 = −∞, our main result is as follows.
To find solutions of (1.1), we apply the Benci-Rabinowitz Linking Theorem [5] to the energy level nonlinear Rayleigh quotient , while for the application of the linking theorem, in general, it is required that the functional belongs to C 1 ( W 1 2 (Ω), R).Below we overcome this difficulty by using an appropriate truncation function for R E λ which can be properly introduced in the case E > 0. In the zero-energy case E = 0, the solution is obtained by passing to the limit E → 0.
Remark 1.3.The zero-energy case E = 0 is particularly interesting, since the value μλ (0) in this case resembles linear eigenvalue.Indeed, for linear problems such as Lu = λu, where L is a self-adjoint linear operator on a Hilbert space H, any isolated eigenvalue λ n , n = 1, 2, . .., corresponds to an eigenfunction φ n of the zero-energy level, i. e., E = 1 2 φ n , Lφ n − λ n 1 2 φ n , φ n = 0. We shall use the following notation: This work is organized as follows.In Section 2, we introduce the nonlinear Rayleigh quotient together with its appropriate truncation function.In Section 3, we derive some properties of the nonlinear Rayleigh quotient R E λ and prove that the Cerami condition for R E λ is satisfied.In Section 4, we prove our main result.Conclusions are drawn in Section 5.In the Appendix, we state the Benci-Rabinowitz Linking Theorem and corresponding definitions.Then one can introduce the following equivalent norm where With this notation, we have To avoid the singularity at origin of R E , we define φ ρ ∈ C ∞ (R), for ρ > 0 such that and introduce Thus, R E ρ (u) ∈ C 1 (W ) for any ρ > 0. We define B r := {u ∈ W : u 1 ≤ r}, r > 0. We need the following result.
The functional R E satisfies the Cerami condition at the level c ∈ R, in short (Ce) condition, whenever any (Ce) sequence possesses a convergent subsequence.The definitions of the (Ce) sequence and the (Ce) condition for R E ρ are similar.
. Therefore (u n ) is also (Ce) sequence for R E and consequently, (u n ) possesses a convergent subsequence in W .The proof of opposite statement is similar.
Proof.By Corollary 2.3, it is sufficient to prove that the functional R E satisfies the (Ce) condition at any µ > 0. Assume that (u m ) is a (Ce) sequence for R E , i.e., µ m := R E (u m ) → µ > 0 and DR E (u m ) * (1 + u m 1 ) → 0 as m → +∞.Then , where 0 < c 1 < +∞ does not depend on m = 1, 2, . .., and therefore for some subsequence (m j ) such that m j → +∞ as j → +∞, then by (3.2) ≤ 0, and consequently, we obtain a contradiction: Thus, (u m ) is bounded and we may assume that u m u weakly in W and u m → u strongly in L r (Ω), r ∈ [1, 2 * ), as m → ∞.In particular, this gives By the convergence DR E (u m ) * → 0 we obtain DR E (u m )(u − u m ) → 0 as m → +∞.Hence by (3.3), we obtain that −∆u m , u−u m → 0. Thus by the S + property of the Laplace operator (see [13]) we derive that u m → u strongly in W 1,2 (Ω).

Conclusions and discussion
In this paper, we develop the mountain pass methods applicable to a new class of problems.In particular, an approach to finding mountain pass-type solutions with prescribed energy for indefinite elliptic problems with nonlinearities which does not satisfy the Ambrosetti-Rabinowitz growth conditions is introduced.Furthermore, the method of nonlinear Rayleigh quotients is used for the first time to solve indefinite elliptic equations with general forms of non-linearities.
A valuable property of the nonlinear Rayleigh quotients method is that it simplifies the complexity problem in a sense by reducing degree of degeneracy of the system (see [3,8]).However, applicability of general theories like the Mountain Pass Theorem, Index Theory, and Ljusternik-Schnirelman's Theory, etc. to nonlinear generalized Rayleigh quotients is limited in light of prohibitive regularity and non-degeneracy conditions for variational functionals.Indeed, the energy-level nonlinear Rayleigh quotient R E λ (u) corresponding to problem (1.1) is not regular at zero.Hence, direct applying the Mountain Pass Theorem in this case is impossible.We have overcome this difficulty in the present work by introducing an appropriate truncation function.We believe, however, there are other ways for overcoming this obstacle.For instance, one might try to answer the question: Is it possible to develop general methods, like Mountain Pass Theorem, etc, applicable to the Rayleigh quotient type function?The answer to this question would help apparently resolve a number of open problems.

Appendix
We use a generalized version of the Benci-Rabinowitz Linking Theorem [5] for functionals satisfying (Ce) c condition.this was developed by D. Motreanu, V. Motreanu, N. Papageorgiou [25].Let (W, • W ) be a Banach space, B 0 ⊂ B, C be nonempty sets in W , and id B0 is an identity map in B 0 .The pair {B 0 , B} is said to be links C in W if the following conditions hold: and assume that φ satisfying the (C e )-condition at c. Then c ≥ a and c is a critical value of φ, i.e., there exists u ∈ W \ 0 such that Dφ(u) = 0 and φ(u) = c.