Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 149–167. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu OSCILLATORY SOLUTIONS AND CRITICAL EXPONENTS FOR FULLY NONLINEAR EQUATIONS FILOMENA PACELLA, DAVID STOLNICKI Abstract. We study existence and nonexistence of radial positive solutions for a class of fully nonlinear equations involving Pucci’s extremal operators. By analyzing the periodic orbits of an associated dynamical system we are able to give estimates on the range of the exponents for which entire oscillating solutions exist. This, in turn, allows to improve previous bounds on the critical exponents defined in [2]. 1. Introduction In this article we study positive radial solutions of the fully nonlinear Lane Emden equation −M±λ,Λ(D2u) = up in Ω ⊆ RN , N ≥ 3 (1.1) where Ω is either the whole RN or the ball BR of radius R > 0, centered at the origin. In the last case, we assume homogeneous Dirichlet boundary condition. We recall that for a C2 function in RN , N ≥ 2, the Pucci’s operators are defined by M+ λ,Λ(D2u) = Λ ∑ µi>0 µi + λ ∑ µi<0 µi M−λ,Λ(D2u) = λ ∑ µi>0 µi + Λ ∑ µi<0 µi where 0 < λ ≤ Λ are the ellipticity constants and µi = µi(D 2u), i = 1, . . . , N are the eigenvalues of the hessian matrix D2u. Associated with M±λ,Λ are dimension like numbers Ñ+ and Ñ− defined as Ñ+ = λ Λ (N − 1) + 1, Ñ− = Λ λ (N − 1) + 1 (1.2) These numbers allow us to give estimates for the exponent p for which existence or nonexistence of radial solutions of (1.1) in BR or RN holds when N ≥ 3 and Ñ+ > 2, (note that Ñ− is always larger than two if N ≥ 3). Indeed a first result obtained in [1] shows that if N ≥ 3 and p ≤ ѱ ѱ−2 , then no nontrivial positive viscosity supersolutions of (1.1) exist in RN . Using this result, 2010 Mathematics Subject Classification. 35J15, 35J60, 35B09, 34C25. Key words and phrases. Fully nonlinear equations; radial solutions; critical exponents; dynamical systems. c©2021 This work is licensed under a CC BY 4.0 license. Published October 6, 2021. 149 150 F. PACELLA, D. STOLNICKI EJDE/SI/01 the existence of positive solutions in bounded domains, not necessarily radial, was proved in [6] for the same range of exponents. In the radial setting Felmer and Quaas [2] (see also [3]) provided, for N ≥ 3, the existence of critical exponents p∗+ forM+ λ,Λ and p∗− forM−λ,Λ satisfying, for λ < Λ, max { Ñ+ Ñ+ − 2 , N + 2 N − 2 } < p∗+ < Ñ+ + 2 Ñ+ − 2 , Ñ− + 2 Ñ− − 2 < p∗− < N + 2 N − 2 (1.3) which are thresholds for the existence of radial positive solutions of (1.1) in the ball or in RN . To state precisely their result, let us recall some definitions. Definition 1.1. Let u be a radial solution of (1.1) in RN and α = 2 p−1 . Then u is said to be: (i) fast decaying if there exists c > 0 such that limr→∞ rÑ−−2u(r) = c (or limr→∞ rÑ+−2u(r) = c), (ii) slow decaying if there exists c > 0 such that limr→∞ rαu(r) = c. (iii) pseudo slow decaying if there exist constants 0 < c1 < c2 such that c1 = lim inf r→∞ rαu(r) < lim sup r→∞ rαu(r) = c2 . The definitions (i) and (ii) are standard in the theory of classical semilinear Lane Emden equations. Instead (iii) was introduced in [2] and corresponds to solutions oscillating at +∞ by changing concavity infinitely many times. Theorem 1.2 ([2]). Assume tildeN+ > 2 and λ < Λ, then (1) if p ∈ (1, p∗±) there is no positive solution of (1.1) in RN ; (2) if p = p∗± there exists a unique positive fast decaying solution of (1.1) in RN ; (3) if p > p∗± there exists a unique positive solution of (1.1) in RN , which is either slow decaying or pseudo-slow decaying; (4) a positive solution in BR exists (and is unique) if and only if p ∈ (1, p∗±). In (2) and (3) uniqueness is meant up to scaling. In addition, in the case of M+ λ,Λ, the precise range of exponents p for which oscillating pseudo-slow decaying solutions exist, is provided in [2]. The proof of Theorem 1.2 given in [2] is quite long and technically complicated. It is a combination of the Emden-Fowler phase plane analysis and the Coffman Kolodner technique. An alternative simpler proof, which also extends to weighted equations and singular solutions is given in [4]. It is entirely based on the study of the orbits of an associated quadratic dynamical system. The same approach has been used in [5] to define and give bounds for a critical exponent that can be defined for the operator M−λ,Λ in dimension N = 2. While analyzing the dynamical system (defined in Section 2) it is important to understand for which values of p periodic orbits exist. Indeed, this enters not only in detecting the presence of pseudo-slow decaying solutions of (1.1) but also to study the existence of fast decaying solutions. This idea was exploited in [5] to estimate the critical exponent in dimension two. EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 151 In this article we refine the results obtained in [2] and [4] by extending the approach of [5] to higher dimensions. In this way, we obtain, on the one hand, information on the exponents p for which oscillating pseudo-slow decaying solutions do not exist; on the other hand, a better estimate of the critical exponents p∗±, as compared with [2]. More precisely we prove the following theorems. Theorem 1.3 (M+ λ,Λ case). Let â+ = ( N N−1 ) N−2 2 − 1 1− (Λ(Ñ+−2) λ(N−1) ) Ñ+−2 2 . (1.4) If p ≤ â+(N + 2) + (Ñ+ + 2) â+(N − 2) + (Ñ+ − 2) = h(â+), (1.5) then problem (1.1) for M+ λ,Λ does not have any pseudo-slow decaying solution. Moreover p∗+ ≥ h(â+) > N + 2 N − 2 . (1.6) Theorem 1.4 (M−λ,Λ case). Let â− = ( N N−1 ) N−2 2 − 1 1− (λ(Ñ−−2) Λ(N−1) ) Ñ−−2 2 . (1.7) If p ≥ â−(N + 2) + (Ñ− + 2) â−(N − 2) + (Ñ− − 2) = h(â−) , (1.8) then problem (1.1) for M−λ,Λ does not have any pseudo-slow decaying solution. Moreover p∗− ≤ h(â−) < N + 2 N − 2 . (1.9) Although not so simple to write, the numbers h(â+) and h(â−) give estimates for the critical exponents substantially better than (1.3). This can be easily seen by assigning values to N and to the ellipticity constant λ,Λ (see Section 5). Even better estimates could be obtained with different choices of certain sets related to the periodic orbits of the dynamical systems ( see Corollary 3.3, Corollary 3.6 and Section 5). The article is organized as follows. In Section 2 we study the dynamical system associated to (1.1), while in Section 3, we prove results about its periodic orbits. In Section 4 we prove Theorem 1.3 and Theorem 1.4. In Section 5 we show some comparisons between our estimates for the critical exponents and the one obtained in [2] and [4]. 2. Associated dynamical systems Since we consider radial solutions to (1.1) we write u = u(r) = u(|x|) as an expression of a function u in radial coordinates. Then, the eigenvalues of D2u are the simple eigenvalue u′′(r) and, with multiplicity (N − 1), u′(r) r (see [2]). 152 F. PACELLA, D. STOLNICKI EJDE/SI/01 Thus u satisfies a corresponding ODE from which it is easy to deduce that u is decreasing as long as it is positive, concave in an interval (0, τ0) and changes concavity at least once (see [2] or [4]): u′′ = M(−λ(N − 1)r−1u′ − up), for M+ λ,Λ u′′ = m(−Λ(N − 1)r−1u′ − up) for M−λ,Λ, (2.1) where M(s) = { s/Λ if s ≤ 0 s/λ if s > 0, (2.2) and m(s) = { s/λ if s ≤ 0 s/Λ if s > 0, (2.3) Next we introduce the auxiliary functions X(t) = −ru ′(r) u(r) , Z(t) = −ru(r)p u′(r) for t = ln r (2.4) whenever u(r) 6= 0, u′(r) 6= 0. Since u > 0, and u′ < 0, the above quantities belong to the first quadrant of the (X,Z) plane. In the new variables, the equations (2.1) become the following autonomous dynamical system (Ẋ, Ż) = F (X,Z) = ( f(X,Z), g(X,Z) ) , (2.5) where the dot ˙ stands for derivation with respect to t, and f, g are given by for M+ λ,Λ: f(X,Z) = { X(X − (N − 2) + Z λ ) if (X,Z) ∈ R+ λ X(X − (Ñ+ − 2) + Z Λ ) if (X,Z) ∈ R−λ , (2.6a) g(X,Z) = { Z(N − pX − Z λ ) if (X,Z) ∈ R+ λ Z(Ñ+ − pX − Z Λ ) if (X,Z) ∈ R−λ , (2.6b) (2.6c) and for M−λ,Λ: f(X,Z) = { X(X − (N − 2) + Z Λ ) if (X,Z) ∈ R+ Λ X(X − (Ñ− − 2) + Z λ ) if (X,Z) ∈ R−Λ , (2.6d) g(X,Z) = { Z(N − pX − Z Λ ) if (X,Z) ∈ R+ Λ Z(Ñ− − pX − Z λ ) if (X,Z) ∈ R−Λ (2.6e) where the regions R+ λ , R − λ , R+ Λ and R−Λ are: R+ λ = {(X,Z) : Z > λ(N − 1)}, (2.7) R−λ = {(X,Z) : 0 < Z < λ(N − 1)}, (2.8) with R+ Λ and R−Λ defined analogously. Note that whenever (X(t), Z(t)) belongs to the lines `+ = {(X,Z) : Z = λ(N − 1)} for M+ λ,Λ (2.9) `− = {(X,Z) : Z = Λ(N − 1)} for M−λ,Λ. (2.10) EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 153 Then the corresponding solution u of (2.1) satisfies u′′(r) = 0, r = et. Hence R+ λ , R + Λ and R−λ , R − Λ represent, in terms of the new variables (X,Z), the regions of concavity and convexity of u, respectively. The dynamical system (2.5) was studied in [4] with full details. We recall here the main proprieties which are needed in the sequel, referring the reader to [4, Section 2] for the proofs and further results. Other important sets which are relevant to study the dynamics induced by (2.5) are: `+1 = {(X,Z) : Z = Λ(Ñ+ − 2−X)} for M+ λ,Λ, (2.11) `−1 = {(X,Z) : Z = λ(Ñ− − 2−X)} for M−λ,Λ, (2.12) which is the set where Ẋ = 0. Notice that `+1 (analogously `−1 ) is a segment entirely contained in R−λ (or R−Λ in the case of `−1 ), since there are no other points in 1Q, where Ẋ = 0 in the interior of the region R+ λ (R+ Λ for `−1 ). We also introduce `+2 = {(X,Z) ∈ R+ λ : Z = λ(N − pX)} ∪ {(X,Z) ∈ R−λ : Z = Λ(Ñ+ − pX)} for M+ λ,Λ , and `−2 = {(X,Z) ∈ R+ Λ : Z = Λ(N − pX)} ∪ {(X,Z) ∈ R−Λ : Z = λ(Ñ− − pX)} for M−λ,Λ , which is the set where Ż = 0. Since our aim is to improve the bounds for the critical exponents obtained in [2] we restrict our attention to p greater than Ñ+ Ñ+−2 or Ñ− Ñ−−2 . Proposition 2.1. The ODE system (2.5) admits the following stationary points. For M+ λ,Λ: (i) N0 = (0, Nλ) which is a saddle point, (ii) A0 = (Ñ+ − 2, 0) which is a saddle point for p > Ñ+ Ñ+−2 , (iii) M0 = ( 2 p−1 ,Λ(Ñ+ − 2− 2 p−1 ) ) which is a source for Ñ+ Ñ+−2 < p < Ñ++2 Ñ+−2 , a center for p = Ñ++2 Ñ+−2 , a sink for p > Ñ++2 Ñ+−2 , (iv) O = (0, 0) which is a saddle point. For M−λ,Λ: (i) N0 = (0, NΛ) which is a saddle point, (ii) A0 = (Ñ− − 2, 0) which is a saddle point for p > Ñ− Ñ−−2 , (iii) M0 = ( 2 p−1 , λ(Ñ− − 2− 2 p−1 ) ) which is a source for Ñ− Ñ−−2 < p < Ñ−+2 Ñ−−2 , a center for p = Ñ−+2 Ñ−−2 , a sink for p > Ñ−+2 Ñ−−2 , (iv) O = (0, 0) which is a saddle point. The stationary points and the direction of the vector field F on the relevant sets for (2.5) are displayed in Figure 1. The set of limit points of a trajectory τ(t) of the system (2.5), as t → −∞, is usually called its α-limit and denoted by α(τ). Analogously it is defined the ω-limit ω(τ) at +∞. The precise correspondence between the solutions of (2.1) and the trajectories of (2.5) has been analyzed in [4, Section 3]. We recall here what is relevant to study the entire positive solutions or the ones in a ball of (2.1). 154 F. PACELLA, D. STOLNICKI EJDE/SI/01 M0 A0 `+ `+2 `+2 `+1 →→→→→→→→→→→→→→→→→→→→→→ → → → → → → → → → → → → ←← ←← ←← ←← ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗↗↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ+ − 2 Z = Nλ PP1 P2 L2 L1 M0 A0 `− `−2 `−1 ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ− − 2 Z = NΛ P2 P1 P ↓ ↓ ↓ →→→→→→ →→ →→→→ →→→→→→→→→ ←←←←←←←← L1 L2 M+ λ,Λ M−λ,Λ Figure 1. Vector field F and the stationary points. Proposition 2.2. For every p > 1, any regular solution up of (2.1) satisfying the initial conditions: up(0) = γ > 0 , u′p(0) = 0, corresponds to the unique trajectory Γp of (2.5) whose α-limit is the stationary point N0. Moreover: (i) if up = up(r) is positive ∀r ∈ (0,+∞) then Γp is bounded and remains in the rectangle (0, Ñ+ − 2)× (0, Nλ), for all time. ((0, Ñ− − 2)× (0, NΛ) in case of M−λ,Λ) (ii) if there exists Rp such that up(Rp) = 0 and up > 0 in (0, Rp), then the corresponding trajectory Γp = (Xp, Zp) blows up in finite time T , in partic- ular, lim t→T Xp(t) = +∞ lim t→T Zp(t) = 0 and there exists t1 < T such that Xp(t1) = Ñ+ − 2. (Xp(t1) = Ñ− − 2 in the case of M−λ,Λ) Concerning the entire positive solution of (2.1), the classification, recalled in Definition 1.1, induces an analogous classification for the orbits of (2.5), which is the following one, (see [4, Section 3] for details and proofs). Proposition 2.3. If up is a regular solution of (2.1), positive in (0,∞), then for the corresponding trajectory Γp of (2.5) it holds: (i) up is fast decaying ⇐⇒ ω(Γp) = A0 (ii) up is slow decaying ⇐⇒ ω(Γp) = M0 (iii) up is pseudo-slow decaying ⇐⇒ ω(Γp) is a periodic orbit around M0. Considering also the positive solutions of the Dirichlet problem in a ball, given by Proposition 2.2(ii), we define the sets: C = { p > 1 : for Γp = (Xp, Zp) it holds that lim t→T Xp(t) = +∞ and lim t→T Zp(t) = 0 for some T > 0 } (2.13) F = {p > 1 : ω(Γp) = A0} (2.14) P = {p > 1 : ω(Γp) is a periodic orbit around M0} (2.15) EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 155 S = {p > 1 : ω(Γp) = M0} (2.16) 3. Nonexistence of periodic orbits 3.1. M+ λ,Λ case. Here we analyze the dynamical system (2.5) in the case of the operator M+ λ,Λ, with the aim of providing estimates on the values of p for which periodic orbits do not exist. We follow the approach of [4] which was used in [5] in the bi-dimensional case. By [4, Proposition 2.10], we already know that there are no periodic orbits for 1 < p ≤ N+2 N−2 or p > Ñ++2 Ñ+−2 . Moreover for p = Ñ++2 Ñ+−2 we know that M0 is a center which means that there exist infinitely many periodic orbits around M0. Therefore we assume that N + 2 N − 2 < p < Ñ+ + 2 Ñ+ − 2 . (3.1) Moreover, again by [4, Proposition 2.10], we know that for p 6= Ñ++2 Ñ+−2 any periodic orbit O must intersect both R+ λ and R−λ and cannot be tangent to the line `+. For a given periodic orbit O we denote by P1 = (x1, λ(N − 1)) and P2 = (x2, λ(N − 1)), x1 < x2, (3.2) the two points in the intersection O ∩ `+ and by D the bounded region inside the orbit O. Both sets D1 = D ∩ R+ λ and D2 = D ∩ R−λ have positive measure, and by the direction of the flow induced by (2.5), it is easy to see that D1 ⊂ Q1 and D2 ⊃ Q2 , (3.3) where Q1 = (x1, x2)× (λ(N − 1), λN), Q2 = (x1, x2)× (Λ(Ñ+ − 2), λ(N − 1)) (see Figure 2). Then we define the function ϕ(X,Z) = XαZβ , with α = 2 p− 1 , β = 3− p p− 1 , (3.4) and denote a′ = ∫ D1 ϕdX dZ∫ D2 ϕdX dZ , a′′ = ∫ Q1 ϕdX dZ∫ Q2 ϕdX dZ . (3.5) Finally we consider the families of subsets Σ1 = {S ⊂ R2 : |S| > 0, D1 ⊂ S̄ ⊆ Q1}, Σ2 = {S ⊂ R2 : |S| > 0, Q2 ⊆ S̄ ⊂ D2} . Obviously, for any couple of sets (S1, S2) ∈ Σ1 × Σ2 we have that the number a = a(S1, S2) = ∫ S1 ϕdX dZ∫ S2 ϕdX dZ (3.6) belongs to the interval (a′, a′′]. We have the following condition for the existence of a periodic orbit. 156 F. PACELLA, D. STOLNICKI EJDE/SI/01 M0 A0 `+ `+2 `+2 `+1 →→→→→→→→→→→→→→→→→→→→→→ → → → → → → → → → → → → ←← ←← ←← ←← ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗↗↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ+ − 2 Z = Nλ PP1 P2 Q1 Q2 D L1 L2 Figure 2. Periodic orbit and the sets Q1, Q2 Proposition 3.1. Let p ∈ ( N+2 N−2 , Ñ++2 Ñ+−2 ) and assume a periodic orbit Op for (2.5) exists. Then, for every a = a(S1, S2) as in (3.6) it holds p > h(a), h(a) = a(N + 2) + (Ñ+ + 2) a(N − 2) + (Ñ+ − 2) , h(a) ∈ (N + 2 N − 2 , Ñ+ + 2 Ñ+ − 2 ) . (3.7) Proof. Since p satisfies (3.1), it follows that − p(Ñ+ − 2) > −Ñ+ − 2 and − p(N − 2) < −N − 2 (3.8) We consider the function ϕ as in (3.5) and define Φ(X,Z) = ∂X(ϕf) + ∂Z(ϕg) (3.9) with f and g as in (2.6a),(2.6b). By computations we obtain Φ(X,Z) = { XαZβ −p(N−2)+(N+2) p−1 if (X,Z) ∈ R+ λ XαZβ −p(Ñ+−2)+(Ñ++2) p−1 if (X,Z) ∈ R−λ . (3.10) Then, using the above notation we have 0 = ∫ ∂D ϕ{fdZ − gdX} = ∫ D Φ dX dZ = ∫ D1 Φ dX dZ + ∫ D2 Φ dX dZ , where the first equality holds because dX = fdt and dZ = gdt. Hence, by (3.10), 0 = −p(N − 2) + (N + 2) p− 1 ∫ D1 XαZβ dX dZ EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 157 + −p(Ñ+ − 2) + (Ñ+ + 2) p− 1 ∫ D2 XαZβ dX dZ. By (3.8) the coefficient of the first term is negative, while the second one is positive. Hence, since ϕ > 0, we obtain that for every (S1, S2) ∈ Σ1 × Σ2: 0 > −p(N − 2) + (N + 2) p− 1 ∫ S1 ϕdX dZ + −p(Ñ+ − 2) + (Ñ+ + 2) p− 1 ∫ S2 ϕdX dZ . Thus for any a = a(S1, S2) defined as in (3.6) we have 0 > −p(N − 2)a− p(Ñ+ − 2) + (N + 2)a+ (Ñ+ + 2) p− 1 ∫ S2 ϕdX dZ . Since ϕ > 0, the above inequality implies −p(N − 2)a− p(Ñ+ − 2) + (N + 2)a+ (Ñ+ + 2) p− 1 < 0 which is equivalent to (3.7). Finally a straightforward computation gives that h(a) belongs to the interval (N+2 N−2 , Ñ++2 Ñ+−2 ). � Note that the function h(a) in (3.7) is decreasing with respect to a. The above result will be used to obtain a bound for the values of p for which the corresponding dynamical system (2.5) cannot admit periodic orbits. This can be achieved by taking a couple of sets (S1, S2) ∈ Σ1 × Σ2 such that the number a = a(S1, S2) can be estimated by a constant which depends only on the dimension N and the ellipticity constants λ and Λ. This is the content of the next two corollaries in which we choose particular sets (S1, S2). Corollary 3.2. Let â+ = ( N N−1 ) N−2 2 − 1 1− (Λ(Ñ+−2) λ(N−1) ) Ñ+−2 2 . (3.11) Then for p ≤ h(â+) = â+(N + 2) + (Ñ+ + 2) â+(N − 2) + (Ñ+ − 2) (3.12) system (2.5) does not admit periodic orbits. Proof. Let us assume that (3.12) holds and that there exists a periodic orbit for (2.5). Then we choose the couple of rectangles (Q1, Q2) defined as in (3.3) to compute and estimate a′′ = a(Q1, Q2) (see (3.6)). Integrating we obtain∫ Q1 ϕdX dZ = ∫ Q1 XαZβ dX dZ = (λ(N))β+1 − (λ(N − 1))β+1 β + 1 ∫ x2 x1 XαdX (3.13) and analogously for Q2,∫ Q2 ϕdX dZ = ∫ Q2 XαZβ dX dZ = (λ(N − 1))β+1 − (Λ(Ñ+ − 2))β+1 β + 1 ∫ x2 x1 XαdX. (3.14) 158 F. PACELLA, D. STOLNICKI EJDE/SI/01 Thus we obtain a′′ = a(Q1, Q2) = ( N N−1 )β+1 − 1 1− (Λ(Ñ+−2) λ(N−1) )β+1 < ( N N−1 ) N−2 2 − 1 1− (Λ(Ñ+−2) λ(N−1) ) Ñ+−2 2 = â+ Since the function h(a) is decreasing, by Proposition 3.1 we deduce that p > h(a′′) > h(â+) which contradicts assumption (3.12). � To improve the estimate (3.12) it would be better to take a couple (S1, S2) with S1 contained in Q1 and S2 containing Q2. For the next result we consider the couple (Q1, Q2 ∪ T ) where T = co(A1, A2,M0) which is the convex hull of the points A1 = (x1,Λ(Ñ+ − 2)), A2 = (x2,Λ(Ñ+ − 2)), M0 , where M0 is the stationary point of Proposition 2.1 (see Figure 3). M0 A0 `+ `+2 `+2 `+1 →→→→→→→→→→→→→→→→→→→→→→ → → → → → → → → → → → → ←← ←← ←← ←← ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗↗↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ+ − 2 Z = Nλ PP1 P2 Q1 Q2 D T L1 L2 Figure 3. Periodic orbit and relevant sets Corollary 3.3. Let ã+ = ã+(p) = (λ(N))β+1−(λ(N−1))β+1 β+1 (λ(N−1))β+1−(Λ(Ñ+−2))β+1 β+1 + Λ p−1 min{(Λ(Ñ+ − 2− 2 p−1 )β , (Λ(Ñ+ − 2)β} . (3.15) EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 159 Then for p ≤ h(ã+) = ã+(N + 2) + (Ñ+ + 2) ã+(N − 2) + (Ñ+ − 2) (3.16) system (2.5) does not admit any periodic orbit. Proof. Integrating the function ϕ(X,Z) = XαZβ on the triangle T we obtain∫ T XαZβ dX dZ ≥ inf T Zβ ∫ T Xα dX dZ = 1 2 ∫ Q3 Xα dX dZ inf T Zβ = ∫ x2 x1 XαdX Λ p− 1 inf T Zβ since ∫ T Xα dX dZ is half of the integral on the rectangle Q3 = (x1, x2)× ( Λ(Ñ+ − 2− 2 p− 1 ),Λ(Ñ+ − 2) ) . Using (3.13) and (3.14) we obtain a(Q1, Q2 ∪ T ) ≤ ã+ = (λ(N))β+1−(λ(N−1))β+1 β+1 (λ(N−1))β+1−(Λ(Ñ+−2))β+1 β+1 + Λ p−1 min{(Λ(Ñ+ − 2− 2 p−1 )β , (Λ(Ñ+ − 2)β} Since the function h is decreasing, we deduce from Proposition 3.1 that if a periodic orbit exists then p > h(a(Q1, Q2 ∪ T )) ≥ h(ã+). Hence we obtain the assertion by the assumption (3.16). � Note that while the right hand side of (3.12) does not depend on p, in the estimate (3.16) we have that ã+ depends on p, so that inequality (3.16) is more difficult to use to have explicit bounds on p. Nevertheless we will use it in Section 5 in some particular cases to show sharper estimates. 3.2. M−λ,Λ case. Here we analyze the dynamical system (2.5) in the case of the operator M−λ,Λ. Though the statements and proofs are similar to those for M+ λ,Λ, we repeat the main steps for the reader’s convenience. Based on the result of [4, Proposition 2.10], we restrict our attention only to a certain interval of values of p in order to obtain better estimates for the critical exponent, hence we assume Ñ− + 2 Ñ− − 2 < p < N + 2 N − 2 . (3.17) Moreover, again by [4, Proposition 2.10], we know that for p 6= N+2 N−2 any periodic orbit O must intersect both R+ Λ and R−Λ and cannot be tangent to the line `−. For a given periodic orbit O we denote by: P1 = (x1,Λ(N − 1)) and P2 = (x2,Λ(N − 1)) , x1 < x2 (3.18) the two points in the intersection O ∩ `− and by D the bounded region inside the orbit O. 160 F. PACELLA, D. STOLNICKI EJDE/SI/01 The sets D1 = D ∩ R+ Λ and D2 = D ∩ R−Λ have both positive measure and, by the direction of the flow induced by (2.5), it is easy to see that D1 ⊂ Q1 and D2 ⊃ Q2 , (3.19) where Q1 = (x1, x2)× (Λ(N − 1),ΛN), Q2 = (x1, x2)× (λ(Ñ−− 2),Λ(N − 1)) (see Figure 4). M0 A0 `− `−2 `−1 ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ− − 2 Z = NΛ P2 P1 P ↓ ↓ ↓ →→→→→→ →→ →→→→ →→→→→→→→→ ←←←←←←←← L1 L2 D Q1 Q2 Figure 4. Periodic orbit and auxiliary sets Now we recall the definition of the function ϕ defined in (3.5) ϕ(X,Z) = XαZβ , with α = 2 p− 1 , β = 3− p p− 1 (3.20) and as done in the M+ λ,Λ case, we denote a′ = ∫ D1 ϕdX dZ∫ D2 ϕdX dZ , a′′ = ∫ Q1 ϕdX dZ∫ Q2 ϕdX dZ , (3.21) and the families of subsets: Σ1 = {S ⊂ R2 : |S| > 0 , D1 ⊂ S̄ ⊆ Q1}, Σ2 = {S ⊂ R2 : |S| > 0 , Q2 ⊆ S̄ ⊂ D2} Obviously, for any couple of sets (S1, S2) ∈ Σ1 × Σ2 we have that the number a = a(S1, S2) = ∫ S1 ϕdX dZ∫ S2 ϕdX dZ (3.22) belongs to the interval (a′, a′′]. We have the following condition for the existence of a periodic orbit. EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 161 Proposition 3.4. Let p ∈ ( Ñ−+2 Ñ−−2 , N+2 N−2 ) and assume a periodic orbit Op for (2.5) exists. Then, for every a = a(S1, S2) as in (3.22) it holds p < h(a), h(a) = a(N + 2) + (Ñ− + 2) a(N − 2) + (Ñ− − 2) , h(a) ∈ (Ñ− + 2 Ñ− − 2 , N + 2 N − 2 ) . (3.23) Proof. The proof is similar to the one for the operator M+ λ,Λ, given in Proposition 3.1. We write the computations for the reader’s convenience. Since p satisfies (3.17), − p(Ñ− − 2) < −Ñ− − 2 and − p(N − 2) > −N − 2 . (3.24) Again as done in the M+ λ,Λ case we define Φ(X,Z) = ∂X(ϕf) + ∂Z(ϕg) , (3.25) with ϕ as in (3.5), f and g as in (2.6a),(2.6b). By computations we obtain Φ(X,Z) = { XαZβ −p(N−2)+(N+2) p−1 if (X,Z) ∈ R+ XαZβ −p(Ñ+−2)+(Ñ++2) p−1 if (X,Z) ∈ R− . (3.26) It follows from the above computation that 0 = ∫ ∂D ϕ{fdZ − gdX} = ∫ D Φ dX dZ = ∫ D1 Φ dX dZ + ∫ D2 Φ dX dZ , where the first equality holds because dX = fdt and dZ = gdt. Hence, by (3.26), 0 = −p(N − 2) + (N + 2) p− 1 ∫ D1 XαZβ dX dZ + −p(Ñ− − 2) + (Ñ− + 2) p− 1 ∫ D2 XαZβ dX dZ. By (3.24) the first right term coefficient is positive, while the second one is negative. Hence, since ϕ > 0, we obtain that for every (S1, S2) ∈ Σ1 × Σ2, 0 < −p(N − 2) + (N + 2) p− 1 ∫ S1 ϕdX dZ + −p(Ñ− − 2) + (Ñ− + 2) p− 1 ∫ S2 ϕdX dZ . Thus for any a = a(S1, S2) defined as in (3.6) we have 0 < −p(N − 2)a− p(Ñ− − 2) + (N + 2)a+ (Ñ− + 2) p− 1 ∫ S2 ϕdX dZ . Since ϕ > 0, the above inequality implies −p(N − 2)a− p(Ñ− − 2) + (N + 2)a+ (Ñ− + 2) p− 1 > 0 which is equivalent to (3.23). Finally a straightforward computation gives that h(a) belongs to the interval ( Ñ−+2 Ñ−−2 , N+2 N−2 ). � Note that the function h(a) in (3.23) is increasing with respect to a. Taking particular couples of sets in Σ1 × Σ2, as in the case of M+ λ,Λ , we obtain the following results. 162 F. PACELLA, D. STOLNICKI EJDE/SI/01 Corollary 3.5. Let â− = ( N N−1 ) N−2 2 − 1 1− (λ(Ñ−−2) Λ(N−1) ) Ñ+−2 2 . (3.27) Then for p ≥ h(â−) = â−(N + 2) + (Ñ− + 2) â−(N − 2) + (Ñ− − 2) (3.28) system (2.5) does not admit periodic orbits. Proof. It is similar to the one of Corollary 3.2 choosing the couple of rectangles (Q1, Q2) defined in (3.19) to compute and estimatea′′ = a(Q1, Q2). � To improve the estimate (3.23) we consider the couple (Q1, Q2 ∪ T ) where T = co(A1, A2,M0) which is the convex hull of the points A1 = (x1, λ(Ñ− − 2)), A2 = (x2, λ(Ñ− − 2)), M0, where M0 is the stationary point of Proposition 2.1 (see Figure 5). M0 A0 `− `−2 `−1 ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ− − 2 Z = NΛ P2 P1 P ↓ ↓ ↓ →→→→→→ →→ →→→→ →→→→→→→→→ ←←←←←←←← L1 L2 D Q1 Q2 T Figure 5. Auxiliary sets Analogously to Corollary 3.3 and with a similar proof obtain the following result. EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 163 Corollary 3.6. Let ã− = ã−(p) = (Λ(N))β+1−(Λ(N−1))β+1 β+1 (Λ(N−1))β+1−(λ(Ñ−−2))β+1 β+1 + λ p−1 min{(λ(Ñ− − 2− 2 p−1 )β , (λ(Ñ− − 2)β} . (3.29) Then for p ≥ h(ã−) = ã−(N + 2) + (Ñ− + 2) ã−(N − 2) + (Ñ− − 2) (3.30) system (2.5) does not admit any periodic orbit. 4. Pseudo-slow decaying solutions and critical exponents We first consider the operatorM+ λ,Λ . To prove Theorem 1.3, we start by showing the connection between the critical exponent p∗+ and the existence of periodic orbits. As recalled in the introduction the critical exponent p∗+ is defined by the propriety of being the only exponent p for which (1.1) admits an entire fast decaying solution. In [4] it is proved that this is equivalent to define p∗+ = sup C , (4.1) where the set C is defined in (2.13). The following result is important for deriving the estimate for p∗+. Proposition 4.1. Let p ∈ (N+2 N−2 , Ñ++2 Ñ+−2 ) and assume that no periodic orbits exist for the system (2.5) (corresponding to such p). Then p ∈ C. Proof. Considering the classification of the sets C, F , P, S, given in (2.13)-(2.16), we have that obviously p /∈ P. Moreover, since M0 is a source for p < Ñ++2 Ñ+−2 , then p /∈ S. Finally if p would belong to F , then the regular trajectory Γp, together with the X and Z axis, would enclose an invariant set for the flow which would not allow any ω-limit for the trajectories starting from M0. Thus, p ∈ C (see Figure 6). � Proof of Theorem 1.3. The first assertion follows by Corollary 3.2 because if there are no periodic orbits for the dynamical system (2.5) there cannot be pseudo-slow decaying solutions for (1.1). The estimate (1.5) on the critical exponent p∗+ is a consequence of (4.1), Propo- sition 4.1, and Corollary 3.2. � One could get a sharper bound for the critical exponent by considering the number h(ã+) = ã+(N + 2) + (Ñ+ + 2) ã+(N − 2) + (Ñ+ − 2) (4.2) for ã+ = ã+(p) defined in (3.16) and applying Corollary 3.3 Now we study the case of the operatorM−λ,Λ . To prove Theorem 1.4 we use the following result, where h(â−) is defined as in (3.27). Proposition 4.2. If p ≥ h(â−), then p /∈ C. 164 F. PACELLA, D. STOLNICKI EJDE/SI/01 M0 A0 `+ `+2 `+2 `+1 →→→→→→→→→→→→→→→→→→→→→→ → → → → → → → → → → → → ←← ←← ←← ←← ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗↗↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ+ − 2 Z = Nλ Z L1 L2 Γp Figure 6. The case when there are no periodic orbits Proof. Since p ≥ h(â−) > Ñ−+2 Ñ−−2 , M0 is a sink while N0 and A0 are saddle points. Moreover there are no periodic orbits by Corollary 3.5. Suppose p ∈ C, then the trajectory Γp starting from N0 blows up forwardly in finite time after crossing the line L1 = {(X,Z) : X = Ñ− − 2}. This implies that the domain bounded by the X and Z axes, the orbit Γp and the line L1 is such that any orbit inside it can only exit in forward time (through L1), but not in backward time. Thus, the unique trajectory τp whose ω-limit is A0 cannot go anywhere backward in time and this provides a contradiction (see Figure 7). � Proof of Theorem 1.4. It is similar to that of Theorem 1.3 using Corollary 3.5, Proposition 4.2 and the fact that p∗− = sup C as shown in [4]. � As in the case for M+ λ,Λ , we can obtain a sharper estimate by considering the number h(ã−) = ã−(N + 2) + (Ñ− + 2) ã−(N − 2) + (Ñ− − 2) (4.3) for ã− defined in (3.30) and applying Corollary 3.6. 5. Concluding remarks To better appreciate the newer bound for both operators we would like to present a few examples of how they compare to the one previously known. To this aim let EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 165 M0 A0 `− `−2 `−1 ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ− − 2 Z = NΛ P2 P1 P ↓ ↓ ↓ →→→→→→ →→ →→→→ →→→→→→→→→ ←←←←←←←← L1 L2 Γp Figure 7. Contradiction arising in Proposition 4.2 us define p̃+ = sup{p solutions of (3.16)}, (5.1) p̃− = inf {p solutions of (3.30)}. (5.2) Note that ã+ and ã− depend on p. Then, based on what was shown in Section 4 we obtain the following ordering. For M+ λ,Λ : Ñ+ + 2 Ñ+ − 2 > p∗+ > p̃+ > h(â+) > N + 2 N − 2 , where p̃+ and h(â+) are defined in (5.1) and (3.12) respectively, while the other bounds are from [2] and Proposition 3.1. For M−λ,Λ : Ñ+ + 2 Ñ+ − 2 < p∗− < p̃− < h(â−) < N + 2 N − 2 , where p̃− and h(â−) are defined in (5.2) and (3.28) respectively. In the case of M+ λ,Λ we can appreciate the difference by computing each one of the bounds in low dimensions. For that we will fix λ = 1,Λ = 2 and vary N from 4 to 8. Then we can observe the values of p̃+,h(â+) and N+2 N−2 on the following table Another interesting case is when we consider a class of problems driven by M+ λ,N−1 (i.e. Λ = N − 1) in RN for λ > 1. It follows from [2] that for every dimension N greater than 2, the critical exponent p∗+ is bounded uniformly from above by the constant Ñ++2 Ñ+−2 = λ+3 λ−1 . On the other hand, as we may observe from the graph in Figure 8, while the classical lower bound presented in [2], i.e. N+2 N−2 converges to 1 as N goes to +∞, 166 F. PACELLA, D. STOLNICKI EJDE/SI/01 Table 1. N N+2 N−2 h(â+) p̃+ Ñ++2 Ñ+−2 4 3 3.92 6.47 9 5 2.3 2.85 3.93 5 6 2 2.36 3 3.67 7 1.8 2.07 2.44 3 8 1.67 1.88 2.11 2.6 Figure 8. Graphs of p̃+, h(â+) and N+2 N−2 , for λ = 2,Λ = N − 1 the bound obtained in this paper by considering more refined set, namely the one obtained in Corollary 3.3, converges to λ+3 λ−1 by (4.2), since by (3.15): lim N→+∞ ã+ = 0 Hence this proves that asymptotically the critical exponent of the problem driven by M+ λ,N−1 in RN converges to λ+3 λ−1 = Ñ++2 Ñ+−2 . References [1] A. Cutri, F. Leoni; On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 17, 2000. [2] P. L. Felmer, A. Quaas; On critical exponents for the Pucci’s extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 20, 2003. [3] P. L. Felmer, A. Quaas; Positive radial solutions to a semilinear equation involving the Pucci’s operator, J. Differential Equations, vol. 199, 2004. EJDE-2021/SI/01 OSCILLATORY SOLUTIONS 167 [4] L. Maia, G. Nornberg, F. Pacella; A dynamical system approach to a class of radial weighted fully nonlinear equations, Comm. Partial Differential Equations, vol. 46, 2021. [5] F. Pacella, D. Stolnicki; On a class of fully nonlinear elliptic equation in dimension two, J. Differential Equations (to appear), [6] A. Quaas, B. Sirakov; Existence results for nonproper elliptic equations involving the Pucci operator, Comm. Partial Differential Equations, vol. 31, 2006. 23 Filomena Pacella Dipartimento di Matematica, Universityà di Roma Sapienza, Italy Email address: pacella@mat.uniroma1.it David Stolnicki Dipartimento di Matematica, Universityà di Roma Sapienza, Italy Email address: david.stolnicki@uniroma1.it 1. Introduction 2. Associated dynamical systems 3. Nonexistence of periodic orbits 3.1. M,+ case 3.2. M,- case 4. Pseudo-slow decaying solutions and critical exponents 5. Concluding remarks References