Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 23, pp. 1–11. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu PRESCRIBED ENERGY SADDLE-POINT SOLUTIONS OF NONLINEAR INDEFINITE PROBLEMS YAVDAT IL’YASOV, EDCARLOS D. SILVA, MAXWELL L. SILVA Abstract. A minimax variational method for finding mountain pass-type so- lutions with prescribed energy levels is introduced. The method is based on application of the Linking Theorem to the energy-level nonlinear Rayleigh quo- tients 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. 1. Introduction Let Ω be a bounded smooth domain in RN , N ≥ 1 and consider −∆u− λu = µ|u|q−1u+ g(x, u) in Ω, u = 0 on ∂Ω, (1.1) where λ ∈ R, µ > 0, 1 < q < 2, g : Ω × R → R is a Carathéodory function with primitive G(x, u). The problem has a variational structure and under some assumptions (see below (A1)) the associated energy functional Eλ,µ ∈ C1(W̊ 1 2 (Ω),R) is Eλ,µ(u) = 1 2 (∫ |∇u|2dx− λ ∫ |u|2dx ) − µ q ∫ |u|qdx− ∫ G(x, u)dx. By definition, the critical point u ∈ W̊ 1 2 (Ω) of Eλ,µ(u) is a weak solution to (1.1). The problem with λ > λ1, where λ1 is the principal eigenvalue of the operator (−∆) in W̊ 1 2 (Ω) is called indefinite due to the fact that the linear part of (1.1) is indefinite (see [5, 25]). Equation (1.1) is related to finding the amplitude function u of the standing waves ψ = eiλtu to the nonlinear Schrödinger (NLS) equation iψt = ∆ψ + µ|ψ|q−2ψ + g(x, ψ), (t, x) ∈ R+ × Ω, (1.2) where ψ is a complex-valued function of (t, x), and it is supposed that g(x, ρeiθ) = g(x, ρ)eiθ a.e. Ω, for all ρ, θ ∈ R. The Cauchy problem for (1.2) with the initial value ψ0 ∈ W̊ 1 2 (Ω) is locally well posed and for some T (ψ0) > 0 has a unique 2020 Mathematics Subject Classification. 35G15, 35G20, 35G25, 35G30. Key words and phrases. Indefinite problems; linking theorems; Rayleigh quotient. ©2023. This work is licensed under a CC BY 4.0 license. Submitted December 5, 2022. Published March 4, 2023. 1 2 Y. IL’YASOV, E. D. SILVA, M. L. SILVA EJDE-2023/23 local solution ψ ∈ C([0, T (ψ0)), W̊ 1 2 (Ω)) ∩ C1([0, T (ψ0)), W̊−1 2 (Ω)) (see, e.g, [10]). Moreover, it holds energy and mass conservation laws: Hµ(ψ(t)) := ∫ (1 2 |∇ψ|2 − µ q |ψ|q −G(x, ψ) ) dx = const, Q(ψ(t)) := 1 2 ∫ |ψ|2dx = const. As a result, the energy functional (action) Eλ,µ(ψ(t)) := Hµ(ψ)− λQ(ψ) = const, λ ∈ R is also conserved. This article focuses on the existence of the so-called prescribed energy solution of (1.1), i.e., which for a given energy E ∈ R satisfies Eλ,µ(uE) = E, DEλ,µ(uE) = 0, where “D(·)” denotes the Fréchet derivative. In the literature, solutions to the Schrödinger equations having a prescribed fre- quency 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 en- ergy 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 use- ful (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 EJDE-2023/23 PRESCRIBED ENERGY SADDLE-POINT SOLUTIONS 3 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]: REλ (u) := 1 2 (∫ |∇u|2 dx− λ ∫ |u|2 dx ) − ∫ G(x, u) dx− E 1 q ∫ |u|qdx , for u ∈ W̊ 1 2 (Ω) \ {0} and E ∈ R. Notice that for u ∈ W̊ 1 2 (Ω) \ {0}, λ ∈ R, and E ∈ R, we have µ = REλ (u) ⇔ Eλ,µ(u) = E, µ = REλ (u), DRE(u) = 0 ⇔ Eλ,µ(u) = E, DEλ,µ(u) = 0. (1.3) We assume that (A1) there exist γ1, γ2 ∈ (2, 2∗), C > 0 such that 0 ≤ g(x, u) ≤ C(|u|γ1−1 + |u|γ2−1) a.e. Ω, u ∈ R, (A2) there exist α > 2, R0 > 0 such that αG(x, u) ≤ g(x, u)u a.e. Ω, |u| ≥ R0, where 2∗ = 2N/(N − 2) if N > 2, 2∗ = +∞ if N ≤ 2. The operator (−∆) with Dirichlet boundary conditions defines a self-adjoint op- erator in L2(Ω) (see, e.g., [14]) and its spectrum consists of an infinite sequence ordered 0 < λ1 < λ2 ≤ . . . of eigenvalues repeated according to their finite multi- plicity. Now, with the convention that λ0 = −∞, our main result is as follows. Theorem 1.1. Assume that 1 < q < 2 < γ < 2∗, λ ∈ (λk, λk+1), k = 0, . . ., and (A1)-(A2) hold. Then there exists Ekλ > 0 such that for any given E ∈ (0, Ekλ) corre- sponds µkλ(E) ∈ (0,+∞) such that (1.1) with µ = µkλ(E) possesses a non-zero weak solution uµkλ(E) ∈ C1,α(Ω), α ∈ (0, 1) with energy value E, i.e., DEµkλ(E)(uµkλ(E)) = 0, Eµkλ(E)(uµkλ(E)) = E. Furthermore, (i) µkλ(·) is a non-increasing function in (0, Ekλ); (ii) if λ < λ1, then there exists limE→0 µ 0 λ(E) = µ̄λ(0) ∈ (0,+∞) such that (1.1) possesses a non-zero weak solution uµ̄λ(0) ∈ C1,α(Ω), α ∈ (0, 1) with zero energy value E = 0 and µ = µ̄λ(0). To find solutions of (1.1), we apply the Benci-Rabinowitz Linking Theorem [5] to the energy level nonlinear Rayleigh quotient REλ (u). Note that REλ ∈ C1(W̊ 1 2 (Ω) \ {0},R), while for the application of the linking theorem, in general, it is required that the functional belongs to C1(W̊ 1 2 (Ω),R). Below we overcome this difficulty by using an appropriate truncation function for REλ 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 1.2. Condition (A2) is the well-known Ambrosetti-Rabinovich (AR) con- dition [1, 15] and implies the superquadratic behavior of G(·, s), G(x, s) ≥ C|s|α for some C > 0 and |s| large. However, the complete nonlinearity µ|u|q−1u+ g(x, u) of equation (1.1) does not satisfy the (AR) condition. Remark 1.3. The zero-energy case E = 0 is particularly interesting, since the value µ̄λ(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, 4 Y. IL’YASOV, E. D. SILVA, M. L. SILVA EJDE-2023/23 any isolated eigenvalue λn, n = 1, 2, . . ., corresponds to an eigenfunction φn of the zero-energy level, i. e., E = 1 2 〈φn, Lφn〉 − λn 1 2 〈φn, φn〉 = 0. We shall use the following notation: • W := W̊ 1 2 (Ω) denotes the standard Sobolev space with the norm ‖u‖W = ( ∫ Ω |∇u|2 dx)1/2; • |u|Lr := ( ∫ Ω |u|r dx)1/r, 1 ≤ r < +∞ denotes the norm on the Lebesgue space Lr := Lr(Ω), (·, ·) denotes the scalar product in L2; • Sp is the best Sobolev constant for the embedding W 1,2 0 (Ω) ⊂ Lp(Ω), 1 ≤ p ≤ 2∗; • ‖ · ‖∗ denotes the norm in the dual space W ∗; • d(A,B) := min{‖u − v‖W : u ∈ A, v ∈ B} denotes the distance between sets A,B ⊂W . 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 REλ and prove that the Cerami condition for REλ 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. 2. Preliminaries Let (ek) ⊂ W be the orthogonal basis in L2 of the eigenfunctions of (−∆) with zero Dirichlet conditions satisfying ‖ek‖2W = λk and |ek|2L2 = 1, for each k ∈ N. Let λ ∈ (λk, λk+1) be fixed for some k ∈ N. We write W = W+ ⊕W−, where W− = span{e1, e2, . . . , ek}, W+ = span{ek+1, ek+2, . . .}. Then one can introduce the following equivalent norm ‖ · ‖1 for ‖ · ‖W in W ‖u‖21 = ∞∑ i=k+1 (λi − λ)u2 i + k∑ i=1 (λ− λi)u2 1 := ‖u+‖21 + ‖u−‖21, where ui = (u, ei), i = 1, . . .. Then u = (u+ + u−) ∈ W, u± ∈ W±, and c0‖u‖21 ≤ ‖u‖2W ≤ c1‖u‖21, ∀u ∈ W , where 0 < c0, c1 < +∞ do not depend on u ∈ W . In addition, Hλ(u) := ‖u‖2W − λ|u|2L2 = Hλ(u+) +Hλ(u−) = ‖u+‖21 − ‖u−‖21, u ∈W. Notice that Hλ(u) = −‖u‖21 < 0 if u ∈ W− \ {0}, and Hλ(u) = ‖u‖21 > 0 if u ∈W+ \ {0}, for λ ∈ (λk, , λk+1). With this notation, we have Eλ,µ(u) = 1 2 Hλ(u)− µ q |u|qLq − ∫ G(x, u)dx, REλ (u) = 1 2Hλ(u)− ∫ G(x, u)dx− E 1 q |u| q Lq , u ∈W \ {0}. Obviously, RE ∈ C1(W \ {0},R) and DEλ,µ(u) = 0, Eλ,µ(u) = E ⇔ DREλ (u) = 0, µ = REλ (u), u ∈W \ 0. (2.1) EJDE-2023/23 PRESCRIBED ENERGY SADDLE-POINT SOLUTIONS 5 To avoid the singularity at origin of RE , we define φρ ∈ C∞(R), for ρ > 0 such that φρ(s) = { 0 if |s| < ρ/2, = 1 if |s| > ρ, and introduce REρ (u) = { φρ(‖u‖1)RE(u), u ∈W \ 0, 0, u = 0. Thus, REρ (u) ∈ C1(W ) for any ρ > 0. We define Br := {u ∈W : ‖u‖1 ≤ r}, r > 0. We need the following result. Lemma 2.1. Assume that E > 0 and λ ∈ (λk, , λk+1). Then there exists ρ(E) > 0 such that RE(u) < 0 for any u ∈ Bρ with 0 < ρ < ρ(E). Proof. Since G(x, u) ≥ 0 a.e. Ω, u ∈ R, REλ (u) < q 1 |u|qLq (1 2 Hλ(u)− E ) < q |u|qLq (1 2 ‖u‖21 − E ) , u ∈W \ {0}. Hence, setting ρ(E) := √ 2E we obtain the proof. � Corollary 2.2. Assume that ρ < ρ(E). If û is a critical point of REρ (u) such that REρ (û) > 0, then u is a critical point of RE(u) as well. Proof. By Lemma 2.1, REρ (û) > 0 implies that ‖û‖W ≥ ρ. Therefore REρ (û) = RE(û) = µ and DRE(û) = 0. � We say that (un) ⊂ W is a Cerami sequence at the level c ∈ R of RE , in short (Ce) sequence, whenever RE(un)→ c and (1+‖un‖W )‖DRE(un)‖ → 0 as n→∞. The functional RE 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 REρ are similar. Corollary 2.3. If ρ < ρ(E), then REρ (u) satisfies the (Ce) condition at level c > 0 if and only if RE(u) does. Proof. Assume RE(u) satisfies the (Ce) condition at c > 0. Let (un) be a (Ce) sequence forREρ at c, i.e., REρ (un)→ c and (1+‖un‖1)‖DREρ (un)‖∗ → 0 as n→∞. By Lemma 2.1, REρ (u) ≤ 0 as u ∈ Bρ, whereas REρ (u) = RE(u) for u ∈ W \ Bρ. Thus, REρ (u) > 0 implies RE(u) = REρ (u) and DRE(u) = DREρ (u). Therefore (un) is also (Ce) sequence for RE and consequently, (un) possesses a convergent subsequence in W . The proof of opposite statement is similar. � 3. Properties of REρ Consider the sphere S±r := {u ∈ W± : ‖u‖1 = r}, r > 0. Recall that by the assumption g(x, u) ≤ C(|u|γ1−1 + |u|γ2−1) a.e. Ω, u ∈ R, for some γ1, γ2 ∈ (2, 2∗), C > 0. Hence, for any given ε > 0, there exist C(ε) > 0 such that G(x, s) ≤ ε 2 |s|2 + C(ε)|s|γ , ∀s ∈ R, a.e. Ω, where γ := max{γ1, γ2}. This by the Sobolev inequalities implies∫ G(x, u)dx ≤ ε 2 C1‖u‖21 + C2(ε)‖u‖γ1 , u ∈W, (3.1) 6 Y. IL’YASOV, E. D. SILVA, M. L. SILVA EJDE-2023/23 where C1, C2(ε) ∈ (0,+∞) do not depend on u ∈ W and C1 does not depend on ε > 0. Proposition 3.1. For any λ ∈ (λk, λk+1), there exist Ekλ > 0 and rkλ > 0 such that infw∈S+ rk λ REρ (w) > 0, for any E ∈ [0, Ekλ), for all ρ ∈ (0, rkλ). Proof. Note that Hλ(w) = ‖w‖21, ∀w ∈ W+. Take ε ∈ (0, 1/C1). Then by (3.1) we have RE(w) ≥ q 1 2 (1− C1ε) ‖w‖21 − C2(ε)‖w‖γ1 − E |w|qLq = q f (‖w‖1)− E |w|qLq , ∀w ∈W+, where f(r) := 1 2 (1−C1ε)r 2−C2(ε)rγ . Observe that f(r) attains its global maximum Ekλ := f(rkλ) at rkλ := [(1− C1ε)/(γC2(ε))] 1/(γ−2) . Thus, for any E ∈ [0, Ekλ), inf w∈S+ rk λ RE(w) ≥ inf w∈S+ rk λ q f(rkλ)− E |w|qLq ≥ qE k λ − E Sqq (rkλ)q =: δE > 0, which implies the proof, since REρ (w) = RE(w), w ∈ S+ rkλ if ρ ∈ (0, rkλ). � Proposition 3.2. For any u ∈ W \ 0 and r > 0, it holds RE(tu + v) → −∞ as t→ +∞ uniformly for v ∈ Br. Proof. Observe that (A2) implies u|u|α d du (|u|−αG(x, u)) ≥ 0, for |u| ≥ R0. Integrating this yields G(x, u) ≥ c(x)|u|α > 0 a.e. Ω, for |u| ≥ R0, with some Lesbegue-measurable function c(x) ≥ 0. Since (A2), c(x) ∈ L∞(Ω) and G(x, s) ≥ c(x)|s|α − C0, for all s ∈ R, a.e. Ω with some constant C0 ∈ R. Note that (A2) implies α < γ < 2∗. Observe that c1/α(x) ( u(x) + v(x)/t ) → c1/α(x)u(x) in Lα(Ω) uniformly in v ∈ Br as t→ +∞. Indeed, using the Sobolev inequality we have∣∣∣ ∫ c(x) ∣∣u+ v t ∣∣α dx− ∫ c(x)|u|α dx ∣∣∣ ≤ 1 t ∫ c(x)|v|α dx ≤ 1 t Crα, v ∈ Br for some constant C which does not depend on v ∈ Br. Thus, uniformly in v ∈ Br, lim t→∞ 1 tα ∫ G(x, tu+ v) dx ≥ lim t→∞ (∫ c(x)|u+ v t |αdx− C0|Ω| tα ) = ∫ c(x)|u|αdx. This implies that lim t→∞ 1 tα (1 2 ‖tu+ v‖2W − ∫ G(x, tu+ v) dx ) ≤ − ∫ c(x)|u|αdx < 0, uniformly in v ∈ Br. Since |u + v/t|qLq ≤ C ( ‖u‖qq + Sqqr q/tq ) and α > 2 > q, we conclude that uniformly for v ∈ Br it holds lim t→∞ RE(tu+ v) ≤ lim t→∞ tα−q |u+ v t | q Lq [‖tu+ v‖2W 2tα − ∫ G(x, tu+ v) tα dx ] = −∞. � Proposition 3.3. Assume that E ∈ (0, Ekλ), 0 < ρ < rkλ. The functional REρ satisfies the (Ce) condition at any level µ > 0. EJDE-2023/23 PRESCRIBED ENERGY SADDLE-POINT SOLUTIONS 7 Proof. By Corollary 2.3, it is sufficient to prove that the functional RE satisfies the (Ce) condition at any µ > 0. Assume that (um) is a (Ce) sequence for RE , i.e., µm := RE(um)→ µ > 0 and ‖DRE(um)‖∗(1 + ‖um‖1)→ 0 as m→ +∞. Then αµm + o(1)‖um‖1(1 + ‖um‖1)−1 = αRE(um)−DRE(um)(um) = q |um|qLq (α− 2 2 Hλ(um) + ∫ (g(x, um)um − αG(x, um)) dx+ µm|um|qLq − qE ) ≥ q |um|qLq (α− 2 2 Hλ(um) + Ln(Ω)essinfx∈Ω,s∈R (g(x, s)s− αG(x, s))− qE ) By (A2), ess infx∈Ω,s∈R ( g(x, s)s−αG(x, s) ) =: c0 > −∞. Hence Hλ(um) ≤ c1 (1+ |um|qLq ), where 0 < c1 < +∞ does not depend on m = 1, 2, . . ., and therefore ‖um‖2W ≤ λ|um|2L2 + c1 (1 + |um|qLq ), m = 1, 2, . . . . (3.2) Thus, if |um|L2 is bounded, then ‖um‖W is also bounded. If |umj |L2 → ∞, for some subsequence (mj) such that mj → +∞ as j → +∞, then by (3.2) limj→∞ Hλ(umj ) |umj | 2 L2 ≤ 0, and consequently, we obtain a contradiction: 0 < µ = lim j→∞ RE(umj ) = lim j→∞ q |umj |2L2 |umj | q Lq [1 2 Hλ(umj ) |umj |2L2 − ∫ G(x, umj ) |umj |2L2 dx− E |umj |2L2 ] ≤ 0. Thus, (um) is bounded and we may assume that um ⇀ u weakly in W and um → u strongly in Lr(Ω), r ∈ [1, 2∗), as m→∞. In particular, this gives∫ G(x, um)dx→ ∫ G(x, u)dx, ‖um‖qq → ‖u‖qq as m→ +∞. (3.3) By the convergence ‖DRE(um)‖∗ → 0 we obtain DRE(um)(u− um)→ 0 as m→ +∞. Hence by (3.3), we obtain that 〈−∆um, u−um〉 → 0. Thus by the S+ property of the Laplace operator (see [13]) we derive that um → u strongly in W 1,2(Ω). � Remark 3.4. Since ‖um‖1 > ρ, for all m and REρ (um) → µ ∈ (0,+∞), it follows that |um|Lq dos not approach 0. 4. Proof of Theorem 1.1 Let λ ∈ (λk, λk+1), E ∈ (0, Ekλ) and 0 < ρ < rkλ. For T > rkλ, take ū+ ∈ S+ 1 , and define B0 = B0(T ) := { u = tū+ + sv : v ∈ S−1 , (0 < t < T, s = T ) or (t ∈ {0, T}, 0 ≤ s ≤ T ) } , B = B(T ) := {u = tū+ + sv : v ∈ S−1 , 0 < t < T, 0 ≤ s ≤ T}, Bc0 = Bc0(T ) := {u = tū+ + sv : v ∈ S−1 , (0 < t < T, s = T )}, Bd0 = Bd0 (T ) := {u = tū+ + sv : v ∈ S−1 , t ∈ {0, T}, 0 ≤ s ≤ T}. 8 Y. IL’YASOV, E. D. SILVA, M. L. SILVA EJDE-2023/23 Observe that if u = tū+ +Tv, u+ ∈ S+ 1 , v ∈ S−1 , then Hλ(u) = t2‖ū+‖21−T 2‖v‖21 = (t2 − T 2) < 0 for T > t. This implies bc(T ) := sup u∈Bc0 REρ (u) = sup u∈Bc0 φρ(‖u‖1) 1 2Hλ(u)− ∫ G(x, u) dx− E 1 q |u| q Lq ≤ 0, for ρ > 0. Note that Hλ(v) < 0, for v ∈ S−1 . This by Proposition 3.2 implies that bd(T ) := sup u∈Bd0 REρ (u) ≤ 0, for sufficiently large T . Thus, by Proposition 3.1, for ρ ∈ (0, rkλ) and sufficiently large T > rkλ, it holds b := sup u∈B0 REρ (u) ≤ 0 < inf u∈S+ rk λ REρ (u) =: a. Let λ ∈ (λk, λk+1), E ∈ (0, Ekλ) and 0 < ρ < min{rkλ, ρ(E)}. Consider µkλ(E) := inf h∈Γ max u∈B REρ (h(u)), (4.1) where Γ = {h ∈ C(B;W ) : h|B0 = idB0 }. By Propositions 3.3 and Corollary 2.3 the functionalREρ satisfies the (Ce) condition at the level c = µkλ(E) > 0. Note that, for T > rkλ, B0∩S+ rkλ = ∅, d(B0, S + rkλ ) > 0, and S+ rkλ is closed in W . Furthermore, it can be shown in a standard way (see [25, P. 156]) that {B0, B} links S+ rkλ in W . Hence, by the Benci-Rabinowitz Linking Theorem [5] for functionals satisfying (Ce) condition (see [25, Theorem 5.39] and Appendix below), there exists a nonzero critical point uµkλ(E) ∈ W \ {0} of the functional REρ such that REρ (uµkλ(E)) = µkλ(E) ≥ a > 0. Consequently, Corollary 2.2 and (1.3) yields that uµkλ(E) is a weak solution of (1.1) with µ = µkλ(E) and energy value E. Standard bootstrap arguments and Sobolev’s embedding theorem (see, e.g., [28]) entail that uµkλ(E) ∈ L∞(Ω). Therefore, by the Lp-regularity results in [16], uµkλ(E) ∈ W 2,p(Ω) for any 1 < p < ∞ and thus, by Sobolev’s embedding theorem, uµkλ(E) ∈ C1,α(Ω) for any α ∈ (0, 1). This completes the proof of the first part of the theorem. (i) Take E1 > E0 > 0. It is not hard to see that the sets {B0, B} and the path sets Γ in (4.1) can be taken the same for E1, E0 if |E1 − E0| is sufficiently small. Note that RE1 ρ (u) = RE0 ρ (u)− φρ(‖u‖1) E1 − E0∫ G(x, u) dx , ∀u ∈W \ 0, and thus max u∈B RE1 ρ (h(u)) = max u∈B ( RE0 ρ (h(u))− φρ(‖h(u)‖1) E1 − E0∫ G(x, h(u)) dx ) ≤ max u∈B RE0 ρ (h(u)), ∀h ∈ Γ, and therefore, for sufficiently small |E1 − E0|, we have µkλ(E1) = inf h∈Γ max u∈B RE1 ρ (h(u)) ≤ inf h∈Γ max u∈B RE0 ρ (h(u)) = µkλ(E0). (ii) Let E = 0. Consider R0(u) ≡ RE(u)|E=0. Using (3.1) and 1 < q < 2 it is not hard to show that R0(u)→ 0 as ‖u‖1 → 0. Consequently, by the continuation we can set that R0(0) = 0. EJDE-2023/23 PRESCRIBED ENERGY SADDLE-POINT SOLUTIONS 9 Assume that λ < λ1. Then W− = ∅ and W+ ≡W . By Proposition 3.2, one can find u1 ∈ W such that RE(u1) < 0. Let E ∈ [0, E0 λ) and 0 < ρ < min{rλ0 , ρ(E)}. Observe that (4.1) can be rewritten as follows µ0 λ(E) := inf γ∈Γ max t∈[0,1] REρ (γ(t)), (4.2) where Γ = {γ ∈ C([0, 1];W ) : γ(0) = 0, γ(1) = u1}, 0 < ρ < ρ(E). Here we set R0 ρ(u) := R0(u), ρ > 0. Note that by the above, for any E ∈ (0, E0 λ) and 0 < ρ < min{rλ0 , ρ(E)}, µ0 λ(E) > 0 and there exists a critical point uµ0 λ(E) ∈ W \ 0 of RE(u) such that DEµ0 λ(E)(uµ0 λ(E)) = 0 and Eµ0 λ(E)(uµ0 λ(E)) = E. As in the proof of (i), from (4.2) it follows that µ0 λ(E) is a non-increasing function on E ∈ [0, E0 λ). Moreover, µ0 λ(E) ≤ µ0 λ(0) < +∞, for any E ∈ (0, E0 λ). Hence there exists limE→0 µ 0 λ(E) = µ̄λ(0) ≤ µ0 λ(0). Furthermore, µ̄λ(0) > 0 since µ0 λ(E) > 0, E ∈ (0, Ekλ) and µ0 λ(E) is a non-increasing function. Since DRE(uµ0 λ(E)) = 0 and µ0 λ(E) ≡ RE(uµ0 λ(E)) → µ̄λ(0) > 0, any countable subset of (uµ0 λ(E))E∈(0,E0 λ) is a (Ce) sequence. Hence Proposition 3.3 implies that there exists a sequences uµλ(Em), m = 1, 2, . . ., such that limm→+∞Em = 0 and uµλ(Em) convergences in W to some point uµ̄λ(0) ∈ W as m → +∞. Note that uµ̄λ(0) 6= 0 (see Remark 3.4 ), and therefore uµ̄λ(0) is a weak solution of (1.1) with µ = µ̄λ(0). Moreover, 0 = lim m→+∞ Em = lim m→+∞ Eλ,µ(uµλ(Em)) = Eλ,µ(uµ̄λ(0)). Thus uµ̄λ(0) is a solution with zero energy. As above it can be shown that uµ̄λ(0) ∈ C1,α(Ω), α ∈ (0, 1). 5. 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. Fur- thermore, 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 sim- plifies 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 non- linear generalized Rayleigh quotients is limited in light of prohibitive regularity and non-degeneracy conditions for variational functionals. Indeed, the energy-level nonlinear Rayleigh quotient REλ (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 de- velop 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. 10 Y. IL’YASOV, E. D. SILVA, M. L. SILVA EJDE-2023/23 6. 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, B0 ⊂ B, C be nonempty sets in W , and idB0 is an identity map in B0. The pair {B0, B} is said to be links C in W if the following conditions hold: (a) B0 ∩ C = ∅; (b) for any h ∈ C(B;W ) with h|B0 = idB0 it holds h(B) ∩ C 6= ∅. The following result follows from [25, Theorem 5.39]. Theorem 6.1. Let {B0, B} links C in W , C closed, d(B0, C) > 0. Let Γ = {h ∈ C(B;W ) : h|B0 = idB0 } and φ ∈ C1(W,R) be such that b := supu∈B0 φ(u) ≤ infu∈S+ ρ φ(u) =: a. Let c := inf h∈Γ max u∈B φ(h(u)), (6.1) and assume that φ satisfying the (Ce)-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. Acknowledgments. Y. Il’yasov was supported by RSF grant No. 22-21-00580. References [1] A. Ambrosetti, P. H. 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Yavdat Il’yasov Institute of Mathematics with Computing Centre of Ufa, Federal Research Centre, RAS, Ufa, Russia Email address: ilyasov02@gmail.com Edcarlos Domingos da Silva Department of Mathematics, Federal University of Goiás 74001-970, Goiânia - GO, Brazil Email address: eddomingos@hotmail.com Maxwell Lizete da Silva Department of Mathematics, Federal University of Goiás 74001-970, Goiânia - GO, Brazil Email address: maxwelllizete@gmail.com 1. Introduction 2. Preliminaries 3. Properties of RE 4. Proof of Theorem ?? 5. Conclusions and discussion 6. Appendix Acknowledgments References