Appeared in Commun. Contemp. Math. 11 (1) (2009) ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS: COMPARISON, EXISTENCE AND UNIQUENESS

Appeared in Commun. Contemp. Math. 11 1 2009 1 34. ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS: COMPARISON, EXISTENCE AND UNIQUENESS FERNANDO CHARRO AND IRENEO PERAL Abstract. We study existence of solutions to 8 >< F u, D 2 u = fλ, u in Ω, u > 0 in Ω, >: u = 0 on Ω, where F is elliptic homogeneous of degree m, either fλ, u = λu or fλ, u = λu + u r, for 0 < < m < r, λ > 0. Furthermore, in the first case we obtain that the solution is uniue as a conseuence of a comparison principle up to the boundary. Several examples, including uniformly elliptic operators the infinity laplacian are considered. 1. Introduction We study elliptic problems of the type F u, D 2 u = fλ, u in Ω, 1.1 u > 0 in Ω, u = 0 on Ω, under suitable hypotheses of ellipticity structure on F. In general we will consider a function F : R n S n R, verifying: F1 Degenerate Ellipticity: For every p R n, Fp, X Fp, Y whenever Y X, with X, Y S n. F2 H omogeneity of degree m: Ftp, tx = t m Fp, X for all t > 0. We further assume F0, 0 = 0. Our main concern is with the existence of nontrivial solutions in the viscosity sense for right h sides f 1 λ, u = λu f 2 λ, u = λu + u r, for 0 < < m < r, with m the degree of homogeneity of F. Our main result, Theorem 2.1, is a comparison result up to the boundary for problems with a right h side satisfying ft 1.2 t > 0 is non-increasing for all t > 0 some 0 < < m, which can be interpreted as a viscosity counterpart for fully nonlinear euations of the comparison result by Brezis-Oswald in the variational setting see [9] also [8]. Notice that f 1 verifies 1.2. In particular the comparison result up to the boundary yields uniueness of positive viscosity solutions of 1.1 if f fulfills 1.2. Date: 9th January, 2008. 2000 Mathematics Subject Classification. 35J15, 35J60, 35J70. Key words phrases. Elliptic euations, Fully nonlinear, Comparison principles, Viscosity solutions, Existence, Uniueness. Work partially supported by projects MTM2007-65018, MEC, CCG06-UAM/ESP-0340, CAM-UAM, Spain. The first author was also supported by a FPU grant of MEC, Spain. 1

2 FERNANDO CHARRO AND IRENEO PERAL Since f 1 m 1 is a concave power, we will refer to this problem as a concave problem. To avoid any ambiguity, it is worth to emphasize that we are not assuming any concavity property on F. Similarly, the case with f 2 will be referred to as a concave-convex problem. The main result in this case shows that there exists Λ > 0 such that 1.1 has at least one nontrivial solution for λ 0, Λ does not have any nontrivial viscosity solution for λ > Λ. The uniueness of solutions in the concave case has been studied in some particular cases, such as the p-laplacian in the variational framework see for instance [1] the references therein, in the viscosity framework, fully nonlinear euations arising as a limit of p-laplacian type euations see [14] k-hessian euations see [20]. Our main goal is to show that this is a general fact for euations given by a homogeneous F. For the proof of existence in both cases, with f 1 f 2, we follow the elementary ideas in [7]. The paper is organized as follows. In Section 2, the main comparison result Theorem 2.1 is established. As an application, if F satisfies F1 F2 above f is under hypothesis 1.2, we obtain uniueness of solutions for problem 1.1. In Section 3, we prove the existence of solutions for both f 1 f 2 when F is uniformly elliptic, which implies m = 1. In the case with f 1, our result is included in Theorem 6 in [6]. However, we provide an alternative proof based on the comparison result in Theorem 2.1 the simple construction in [7]. Again by using the monotonicity argument in [7] we are able to prove the existence for a concave-convex right h side f 2. We can find a related existence result for Pucci s operators in [10], where it is considered a right h side of the type fu = λu + gλ, u with g continuous gλ, s = o s as s 0, bifurcation from the eigenvalues techniues are used. However, in our case the solutions branch off from 0, 0 since the concave part is not differentiable at 0. In the p-laplacian case this kind of behavior was studied in [2]. Finally, in Section 4, we extend the simple techniues in Section 3 to some nonuniformly elliptic problems which have degree of homogeneity greater than one such as the infinity laplacian with both normalizations, a context in which these results are new, Monge-Ampere euations, for which we obtain some extension of known results. See for instance [20] where different methods are used. Other examples considered range from the linear problem with variable coefficients to the p-laplacian, or the euation already considered in [14]. min{ u λu, u} = 0, 2. Some comparison results The following comparison principle is the main result of this section. Theorem 2.1. Let Ω R n be a bounded domain consider a viscosity subsolution u a supersolution v of 2.1 F w, D 2 w = fw in Ω, where F : R n S n R satisfies F1, F2 for some m, f satisfy ft 2.2 t > 0 is non-increasing for all t > 0 some 0 < < m. Suppose that both, u v are strictly positive in Ω, continuous up to the boundary satisfy u v on Ω. Then, u v in Ω.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 3 As a conseuence, we have the following uniueness result for the Dirichlet problem. Corollary 2.2. The problem F u, D 2 u = fu in Ω, 2.3 u > 0 in Ω, u = 0 on Ω, with F verifying F1 F2 for some m, f verifying 2.2, has at most one positive viscosity solution. Notice that our euation, written in the form Gw, w, D 2 w = 0 with G : R R n S n R r, p, X Fp, X fr, is not proper in the sense of [16] since f may fail to be non-increasing in r, it is conseuently out of the scope of the general theory in [16]. Let us point out that a logarithmic change of variables w = logw, can be used to transform the euation into F w, D 2 w + w w = e m w f e w in Ω, which is proper, either under the hypothesis ft t is strictly decreasing t > 0 some 0 < < m, or ft t m is strictly decreasing t > 0, the precise assumption in [9]. Then it is possible to show comparison using stard arguments see [16], [21], [23]. However, strict positivity of the supersolution is needed in the whole Ω in order to carry out the proof. Hence the above argument is not sufficient to conclude uniueness of nontrivial solutions of 2.3. Thus, we instead follow the change of variables subseuent ideas in [14] to prove comparison up to the boundary even under zero boundary data. Lemma 2.3. Let w > 0 be a supersolution subsolution of problem 2.1 in Ω consider some < m for which 2.2 holds. Then, wx = 1 1 m is a viscosity supersolution subsolution of 2.4 F w, D 2 w + m in every Ω such that Ω Ω. w w f = w w 1 m x [1 ] 1 1 m wx m [ 1 ] 1 m wx m Proof. Let us denote = /m for simplicity. Consider φ C 2 Ω, a function touching wx from below at x 0 Ω define φx = 1 φx [ ] 1 1, which touches wx from below at x 0. Notice that φx is C 2 in a neighborhood of x 0 since w > 0,

4 FERNANDO CHARRO AND IRENEO PERAL in Ω implies φx > 0 near x 0. We can compute the derivatives of φx in terms of those of φx [ φx 0 = 1 φx ] 1 0 φx 0, D 2 φx 0 = [ 1 φx ] 1 0 D 2 φx0 + 1 As w is a viscosity solution of 2.1, we have that [1 f wx0 ] 1 1 = f wx 0 F φx 0, D 2 φx 0 = [ 1 φx 0 ] m 1 F φx 0, D 2 φx0 + 1 φx 0 φx 0. φx 0 φx 0 φx 0, φx 0 by homogeneity. Since wx 0 = φx 0, we conclude that ũ is a viscosity supersolution of 2.4. The subsolution case is analogous. The function giving rise to euation 2.4, namely, G : R + R n S n R r, p, X F p, X + m [1 f p p m r ] 1 1 r m [ 1 ] 1 m r m is both degenerate elliptic proper. In the following lemma, we show that it is possible to construct strict supersolutions starting from any positive supersolution. Lemma 2.4. Let ṽx > 0 be a viscosity supersolution of 2.4 in Ω such that Ω Ω. Then, for any ǫ > 0, 2.5 ṽ ǫ x = 1 + ǫ ṽx + ǫ, is a strict supersolution of the same euation; indeed, 2.6 F ṽ ǫ, D 2 ṽ ǫ + ṽ ǫ ṽ f ǫ 1 + ǫ m m ṽ ǫ Moreover, ṽ ǫ v uniformly in Ω as ǫ 0. [ 1 m ṽǫ x [ 1 mṽǫ x, ] 1 1 m ] 1 m Proof. Let φ C 2 be a function touching ṽ ǫ x from below in some x 0 Ω. We define Φx = 1 φx ǫ, 1 + ǫ which clearly touches ṽx from below in x 0. We can compute the derivatives of Φx in terms of those of φx, this is 2.7 Φx 0 = 1 + ǫ 1 φx 0 D 2 Φx 0 = 1 + ǫ 1 D 2 φx 0. Since ṽx is a viscosity supersolution of 2.4 in Ω, we deduce [ 1 ] 1 f 1 m ṽx0 m [ 1 ] F Φx 0, D 1 m 2 Φx 0 + ṽx0 m m = 1 1 + ǫ m F φx 0, D 2 φx 0 + m Φx 0 Φx 0 ṽx 0 φx 0 φx 0 ṽǫ x 0 ǫ1 + ǫ..

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 5 Notice that our hypothesis 2.2 implies [ 1 ] 1 f 1 m ṽx0 m [ 1 ] 1 m ṽx0 m Since D 2 φx 0 + m [ 1 f m ṽǫ x 0 [ 1 m ṽǫ x 0 φx 0 φx 0 ṽǫ x 0 ǫ1 + ǫ D2 φx 0 + ] 1 1 m ] 1 m φx 0 φx 0 m ṽ ǫ x 0 in the matrix sense, we get 2.6 by degenerate ellipticity. For the second statement, notice that ṽ ǫ v L Ω ǫ ṽ L Ω + ǫ 1 + ǫ. Proof of Theorem 2.1. Since u v CΩ Ω is compact, u v attains a maximum at Ω. In order to arrive at a contradiction, we suppose that max Ω u v > 0. Consider 2.8 ũx = ux1 m 1 m define 2.9 ṽ ǫ x = 1 + ǫ ṽx + ǫ. Notice that, since u v Ω 0, we have ũ ṽ ǫ = ũ 1 + ǫṽ 1 + ǫ ǫ < 0 ṽx = vx1 m 1 m, on Ω. Furthermore, by uniform convergence, we have max Ω ũṽ ǫ > 0 for ǫ small enough, hence, we can fix ǫ > 0 suppose that there exists Ω with Ω Ω containing all the maximum points of ũ ṽ ǫ. We have proved in Lemmas 2.3 2.4 that ũ ṽ ǫ are respectively a subsolution a strict supersolution of 2.4 in Ω. Now, for each τ > 0, let x τ, y τ be a maximum point of ũx ṽ ǫ y τ 2 x y 2 in Ω Ω. By the compactness of Ω, we can suppose that x τ ˆx as τ for some ˆx Ω notice that also y τ ˆx. Proposition 3.7 in [16] implies that ˆx is a maximum point of ũ ṽ ǫ, conseuently, it is an interior point of Ω. We also have lim τ ũx τ ṽ ǫ y τ τ 2 x τ y τ 2 = ũˆx ṽ ǫ ˆx > 0, then, for τ large enough, both x τ y τ are interior points of Ω 2.10 ũx τ ṽ ǫ y τ τ 2 x τ y τ 2 > 0. Applying the Maximum Principle for semicontinuous functions see for instance, [15] [16], there exist two symmetric matrices X τ, Y τ such that τxτ y τ, X τ J 2,+ũxτ, τx τ y τ, Y τ J 2,ṽǫ y τ, 2.11 X τ ξ, ξ Y τ η, η 3τ ξ η 2 ξ, η R n. Where, following [16], J 2,+ wˆx = { p, X R n S n : φx = wˆx + p, x ˆx + 1 2 Xx ˆx, x ˆx touches w from above in ˆx }, J 2, wˆx = { p, X R n S n : ψx = wˆx + p, x ˆx + 1 2 Xx ˆx, x ˆx touches w from below in ˆx },.

6 FERNANDO CHARRO AND IRENEO PERAL their closures, J 2,+ wˆx = { p, X R n S n : x n B r ˆx, p n, X n J 2,+ wx n s.t. x n, p n, X n ˆx, p, X as n }, J 2, wˆx = { p, X R n S n : x n B r ˆx, p n, X n J 2, wx n s.t. x n, p n, X n ˆx, p, X as n }. By definition of viscosity sub- supersolution see [16], we have F τx τ y τ, X τ + τ 2 x τ y τ x τ y τ m ũx τ [ 1 ] 1 2.12 f 1 m ũxτ m [ 1 ], 1 m ũxτ m 2.13 F τx τ y τ, Y τ + m τ 2 x τ y τ x τ y τ ṽ ǫ y τ [ 1 f m ṽǫ y τ 1 + ǫ m [ 1 m ṽǫ y τ ] 1 1 m ] 1 m Since ṽ ǫ y τ < ũx τ X τ Y τ, from 2.10 2.11 respectively, we have X τ + τ 2 x τ y τ x τ y τ Y τ + τ 2 x τ y τ x τ y τ m ũx τ m ṽ ǫ y τ in the sense of matrices. Subtracting 2.12 from 2.13, by hypothesis 2.2 degenerate ellipticity, we have [ 1 ] 1 0 < [ 1 + ǫ m 1 ] f 1 m ṽǫ y τ m [ 1 ] 1 m ṽǫ y τ m [ 1 ] 1 [ f 1 m ṽǫ y τ 1 ] 1 m f 1 m ũxτ m 1 + ǫ m [ 1 ] [ 1 1 ] m ṽǫ y τ m 1 mũxτ m F τx τ y τ, Y τ + τ 2 x τ y τ x τ y τ m ṽ ǫ y τ F τx τ y τ, X τ + τ 2 x τ y τ x τ y τ 0, m ũx τ a contradiction. We also provide the following simple result, which turns out to be very useful when proving existence of solutions. Theorem 2.5. Let Ω R n be a bounded domain consider f CΩ with f > 0 in Ω, F : R n S n R satisfying F1 F2 for some m. Let u, v CΩ, respectively a viscosity sub- supersolution of Assume u v on Ω, then u v in Ω. F w, D 2 w = fx in Ω..

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 7 Proof. Without loss of generality, we can assume v 0 in Ω, since adding a constant to both u v does not affect the problem. Now, let v ǫ x = 1 + ǫvx. By homogeneity, v ǫ is a strict supersolution, indeed, 2.14 F v ǫ, D 2 v ǫ 1 + ǫ m fx > fx u v ǫ 0 on Ω. Now, we argue by contradiction. Suppose that there exists x 0 Ω such that u v ǫ x 0 = maxu v ǫ > 0. Ω Then, 2.14 implies x 0 / Ω. As in the proof of Theorem 2.1, one finds x τ, y τ x 0 symmetric matrices X τ, Y τ such that τxτ y τ, X τ J 2,+ uxτ τx τ y τ, Y τ J 2, vǫ y τ, with X τ Y τ. Then, by definition of viscosity sub- supersolution, we have F τx τ y τ, X τ fxτ, F τx τ y τ, Y τ 1 + ǫ m fy τ. Then, by degenerate ellipticity, we have 1 + ǫ m fy τ fx τ F τx τ y τ, Y τ F τxτ y τ, X τ 0. Letting τ, we have, by continuity, 0 < 1 + ǫ m 1 fx 0 0, a contradiction. Thus, u v ǫ in Ω,, letting ǫ 0, we get u v in Ω. 3. Existence of solutions to uniformly elliptic euations For simplicity, let us assume that Ω is a smooth domain. In some results we will indicate a relaxation of such regularity hypothesis without looking for the optimal hypotheses. Assume that F : R n S n R satisfies F2 for m = 1 upgrade F1 to the following uniform ellipticity condition: F1 There exist constants 0 < θ Θ such that for all X, Y S n with Y 0, Θ tracey Fp, X + Y Fp, X θ tracey for every p R n. Notice that the uniform ellipticity forces m = 1. We further reuire the following structure condition: F3 There exists γ > 0 such that, for all X, Y S n, p, R n, P θ,θ X Y γ p Fp, X F, Y P+ θ,θ X Y + γ p, where P ± θ,θ are the extremal Pucci s operators defined as P + θ,θ M = θ λ i>0 λ i M Θ λ i<0λ i M, P θ,θ M = Θ λ i>0 λ i M θ λ i<0λ i M, with λ i M, i = 1,...n, the eigenvalues of M. Indeed, for P θ,θ M = inf {traceam}, A A θ,θ P + θ,θ M = sup {traceam} A A θ,θ A θ,θ = { A S n : θ ξ 2 Aξ, ξ Θ ξ 2 ξ R n}.

8 FERNANDO CHARRO AND IRENEO PERAL We will uote several results from [11] along this work. Notice that in [11], Pucci s operators are defined with a different sign convention. Both definitions are related through the following expressions M M, θ, Θ = P + θ,θ M, M+ M, θ, Θ = P θ,θ M, where M ± M, θ, Θ is the notation used in [11]. Remark 3.1. As it is pointed out in [13], the structure condition F3 when p = is nothing but uniform ellipticity. We state the existence uniueness of nontrivial solutions for every λ > 0 of F u, D 2 u = λu in Ω, u > 0 in Ω, u = 0 on Ω, where 0 < < 1. Then, we consider 0 < < 1 < r the problem F u, D 2 u = λu + u r in Ω, u > 0 in Ω, u = 0 on Ω. We prove the existence of a value Λ > 0 such that there exists a nontrivial solution for λ 0, Λ, any nontrivial solution otherwise. The mentioned results rely on a monotone iteration method the analysis of simpler auxiliary problems for which showing the existence of solutions is easier. We point out that our results can be extended to m-homogeneous degenerate elliptic euations, merely satisfying F1 F2. We will give examples of this issue in the next section. 3.1. Existence of solutions for the problem with a concave power. Assume that F : R n S n R satisfies the above hypotheses. We are interested in studying the existence of nontrivial solutions to F u λ, D 2 u λ = λu λ in Ω, 3.1 u λ > 0 in Ω, u λ = 0 on Ω, where 0 < < 1 λ > 0. The following is our main result in this section. Theorem 3.2. Let Ω R n be a bounded smooth domain, F : R n S n R satisfies F1, F2 for m = 1 F3 with 0 < < 1. Then, there exists a uniue solution to 3.1 for every λ > 0 given by 3.2 u λ x = λ 1 1 u1 x, where u 1 is the solution with λ = 1. Notice that 3.2 is a smooth curve of solutions with respect to λ. By homogeneity, it is enough to prove the result for λ = 1, namely, F u 1, D 2 u 1 = u 1 in Ω, 3.3 u 1 > 0 in Ω, u 1 = 0 on Ω.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 9 In order to obtain positive sub supersolutions to problem 3.3, we introduce the following auxiliary problems, F v, D 2 v = 1 in Ω, F w, D 2 w = dx in Ω, 3.4 v > 0 in Ω, w > 0 in Ω, v = 0 on Ω, w = 0 on Ω, where dx is the normalized distance to the boundary, namely, distx, Ω dx =. dist, Ω Our construction rely on the following fact. Proposition 3.3. Both auxiliary problems in 3.4 have a uniue viscosity solution. Proof. We show the existence of w, since the proof for v is similar. First, notice that 0 is a subsolution to our problem. Next, we are going to construct a supersolution following Proposition 3.2 in [17]. Let U be the uniue viscosity solution to { P θ,θ D2 U γ U = 1 in Ω, U = 0 on Ω. We point out that here it is enough to have an exterior cone condition on Ω. The structure condition F3 implies F U, D 2 U P θ,θ D2 U γ U = 1 dx in the viscosity sense. Hence 0 w U by comparison, Theorem 2.5, we can invoke the Perron method see for instance, Theorem 4.1 in [16] to get that there exists a uniue w such that { F w, D 2 w = dx in Ω, w = 0 on Ω. For the strict positivity of w, notice that w 0 by construction w 0 we get a contradiction with the euation otherwise. Since P + θ,θ D2 w + γ w F w, D 2 w = dx > 0 in the viscosity sense, the weak Harnack ineuality an adaptation of Theorem 4.8 in [11] using the ABP estimate in Proposition 2.12 in [13] implies w > 0 in Ω. Next, we use v, w, the solutions to the auxiliary problems, to construct a subsolution a supersolution to 3.3. 1 Lemma 3.4. ux = v vx is a viscosity supersolution of 3.3. 1 Proof. We denote T = v for brevity. By homogeneity the definition of v, we have F u, D 2 u = T F v, D 2 v = T, in the viscosity sense while, by definition of T, we have u T v = T. Now, we are going to construct a subsolution to 3.3. Lemma 3.5. There exists δ > 0 small enough, such that ux = t wx is a viscosity subsolution of 3.3 for every t 0, δ.

10 FERNANDO CHARRO AND IRENEO PERAL Proof. By definition, we have F u, D 2 u = t F w, D 2 w = t dx. Indeed, t dx u x since this is euivalent to t 1 dx w x, which holds for all t < δ small enough by the Hopf boundary lemma Proposition A.1 in the Appendix below. At this stage, the comparison principle Theorem 2.1, Lemmas 3.4 3.5 allow us to invoke the Perron method Theorem 4.1 in [16], which completes the proof of Theorem 3.2. Remark 3.6. Theorem 3.2 remains true for domains for which Hopf s Lemma holds it is possible to construct a barrier in each point of the boundary in problems 3.4 for instance if Ω satisfies both an interior sphere an uniform exterior cone condition. 3.2. Study of a concave-convex problem. Consider F : R n S n R as in the previous section, namely, satisfying F1, F2 the structure condition F3. Now, our goal is to study the existence non-existence of viscosity solutions of the problem, F u, D 2 u = λu + u r in Ω, 3.5 u > 0 in Ω, u = 0 on Ω, where 0 < < 1 < r λ > 0. 3.2.1. Existence of solutions for small λ. The present section is devoted to the proof of the following result, which extends the arguments in [7] to the viscosity framework. Theorem 3.7. Consider a smooth bounded domain Ω R n let F : R n S n R satisfy F1, F2 with m = 1, F3. Then, there exists a constant λ 0 > 0 such that, for every λ 0, λ 0 ], problem 3.5 has at least one nontrivial viscosity solution. Since the scaling property 3.2 is not available for problem 3.5, let us consider for every λ > 0 the following variant of the auxiliary problems in Section 3 F v λ, D 2 v λ = λ in Ω, F w λ, D 2 w λ = λdx in Ω, 3.6 v λ > 0 in Ω, w λ > 0 in Ω, v λ = 0 on Ω, w λ = 0 on Ω, where dx is the normalized distance to the boundary, namely, distx, Ω dx =. dist, Ω Notice that, by homogeneity, 3.7 v λ x = λ v 1 x, where v 1 is the solution to 3.4. A similar relation holds for w λ. Then, Proposition 3.3 gives existence of a uniue viscosity solution to both auxiliary problems in 3.6. Now, we are able to construct a supersolution to 3.5.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 11 Lemma 3.8. There exists λ 0 > 0 such that λ 0, λ 0 ] we can find a constant Tλ such that u λ x = Tλ v λ x is a viscosity supersolution of 3.5. In fact, 3.8 λ 0 = r 1 Tλ = 1 v 1 1 1 [ v1 r ] r 1 r1, 1 1 r r+1 λ r. r 1 Proof. On the one h, by homogeneity the definition of v λ, we have F u λ, D 2 u λ = TλF v λ, D 2 v λ = Tλλ. On the other h, using 3.7, one has λ u λ + uλ r λ 1+ Tλ v 1 + λ r Tλ r v 1 r. Thus, we will be done whenever 3.9 λ 1+ Tλ v 1 + λ r Tλ r v 1 r Tλλ. To prove 3.9 is euivalent to demonstrate that Φ λ T 1, with Φ λ T = c λ T 1 + c r λ r1 T r1, where c = v 1 for brevity. Indeed, we are going to show that, for λ small enough, the minimum of Φ λ is smaller than 1. It is easy to check that d dt Φ λt = 0 Tλ = 1 1 1 r r+1 λ r, c r 1 which, indeed, is a minimum of Φ λ. Since we want Φ λ Tλ = c λ r1 r r r 1 we get, after a simple calculation, that where λ 0 is given by 3.8. λ λ 0, 1 r 1 Now, we are going to construct a subsolution of 3.5. 1 r 1, Lemma 3.9. Let 0 < < 1 < r. Then, for every λ > 0, there exists δλ > 0 small enough such that u λ x = t w λ x is a viscosity subsolution of 3.5 for every t 0, δλ. Proof. By definition, we have F u λ, D 2 u λ = t F w λ, D 2 w λ = t λdx where distx, Ω dx =. dist, Ω If it is true that t λdx λu λ + ur λ, we are done. This is euivalent to t 1 dx w λ + 1 λ tr w r λ.

12 FERNANDO CHARRO AND IRENEO PERAL Hopf s lemma Proposition A.1 implies that, fixed λ > 0, we can find δλ small enough such that the above is true for all t < δ. Finally, we have the following ordering result. Lemma 3.10. For all λ 0, λ 0 ], u λ, u λ as in Lemmas 3.8 3.9, there is a small enough t as in Lemma 3.9 such that u λ u λ in Ω. Proof. On the boundary u λ = u λ = 0, while F u λ, D 2 u λ = t λdx Tλλ F u λ, D 2 u λ in Ω, by definition of u λ, u λ. Then, Theorem 2.5 gives the desired result. Now, we can finish the proof of the main Theorem. Proof of Theorem 3.7. Let λ 0 be given by Lemma 3.8, fix λ 0, λ 0 ]. Applying Lemmas 3.8, 3.9, 3.10, we get that there exist constants Tλ t 0, δλ such that u λ x = Tλv λ x u λ x = t w λ x are, respectively, a viscosity super- subsolution moreover u λ u λ in Ω. For simplicity, we will drop the λ write u u hereafter. Now, define w 1 x, the solution to { F w1, D 2 w 1 = λu + u r in Ω, w 1 = 0 on Ω, in the viscosity sense. For brevity, all the euations ineualities in the seuel will st in the viscosity sense without any further reference. Hence, by definition, F u, D 2 u λu + u r F w 1, D 2 w 1 = λu + u r, with w 1 = u = 0 on Ω. Theorem 2.5 implies w 1 u in Ω. Moreover, since u u, we get F u, D 2 u λu + u r λu + u r F w 1, D 2 w 1 = λu + u r, with u = w 1 = 0 on Ω. Hence, by comparison Theorem 2.5, we get u w 1 in Ω, combining both estimates, u w 1 u in Ω. Next, define w 2 as the solution to { F w2, D 2 w 2 = λw 1 + wr 1 in Ω, Then one has w 2 = 0 on Ω. F w 2, D 2 w 2 = λw 1 + wr 1 λu + u r F w 1, D 2 w 1 = λu + u r, where w 2 = w 1 = 0 on Ω. By comparison Theorem 2.5, w 2 w 1 in Ω. On the other h, u w 1 implies F u, D 2 u λu + u r λw 1 + wr 1 F w 2, D 2 w 2 = λw 1 + wr 1, u = w 2 = 0 on Ω. Hence, u w 2 in Ω then u w 2 w 1 u in Ω. We can iterate the above procedure construct a seuence {w k } k 1 of solutions to { F wk, D 2 w k = λw k1 + wr k1 in Ω, 3.10 w k = 0 on Ω.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 13 such that u... w k w k1... w 2 w 1 u in Ω. In particular, for every x Ω, the seuence {w k x} k 1 is bounded nonincreasing hence convergent. We denote ux this pointwise limit of the w k. By the structure condition F3 the definition of w k on the boundary, we have { P + θ,θ D2 w k + γ w k λw k1 + wr k1, w k = 0 on Ω, By construction, { P θ,θ D2 w k γ w k λw k1 + wr k1, w k = 0 on Ω. λw k1 + wr k1 L n Ω λu + u r Ln Ω, k 1. Thus, we can adapt Proposition 4.14 Remark 6 in [11] using the ABP estimate in [13] then, deduce that there exists a modulus of continuity ρ independent of k such that w k x w k y ρ x y, x, y Ω. Hence, the seuence {w k } is uniformly bounded euicontinuous, as a conseuence of the Arzela-Ascoli compactness criterion, there exists a subseuence w kj such that w kj converges uniformly to u in Ω. Since the seuence is monotone, not only the subseuence but the whole seuence {w k } converges to u. Finally, it is easy to prove that the uniform limit u is a viscosity solution of 3.1. First of all, notice that, ux = lim k w kx = 0, x Ω. Then, consider φ C 2 x 0 Ω such that u φ has a strict local maximum at x 0, this is, u φx < u φx 0, for all x x 0 in a neighborhood of x 0. By uniform convergence, we deduce that w k φ has a local maximum at some x k, this is, w k φx w k φx k, for all x x k near x k. In addition, x k x 0 as k. Since v k is a viscosity solution of 3.10, we have Taking limits as k, we get F φx k, D 2 φx k λw k1 x k + w r k1 x k. The supersolution case is analogous. F φx 0, D 2 φx 0 λu x 0 + u r x 0. 3.2.2. Non existence for large λ. Under the hypotheses of this section, F1, F2 F3, Theorem 8 in [6] holds. Hence, we know that there exists a principal eigenvalue λ 1 for F defined as λ 1 = sup{λ v > 0 in Ω s.t. F v, D 2 v λv}. in the sense that λ 1 < there exists a nontrivial solution eigenfunction to F v, D 2 v = λv in Ω, 3.11 v > 0 in Ω, v = 0 on Ω.

14 FERNANDO CHARRO AND IRENEO PERAL Moreover, by definition of λ 1, we know that for every λ > λ 1, problem 3.11 does not have strictly positive solutions. Other references for the existence of eigenvalues in the fully nonlinear setting are [30], [10] the references therein. On the other h, in [21] the existence of a principal eigenvalue for the 1-homogeneous -laplacian is studied. Notice that the 1-homogeneous -laplacian is outside of the scope of the results in [6]. The existence of such principal eigenvalue eigenfuntion is needed in the proof of the following result. Theorem 3.11. For λ large enough, problem 3.5 has no solution in the viscosity sense. Proof. Fix µ > λ 1 consider 1 λ 0 = µ r r1 1 1 r1 r 1. r r In order to reach a contradiction, suppose that there exists λ > λ 0 such that the problem F u λ, D 2 u λ = λu λ + ur λ in Ω, 3.12 u λ > 0 in Ω, u λ = 0 on Ω, has a solution u λ. Then, we have 3.13 F u λ, D 2 u λ = λu λ + ur λ > µu λ in Ω, in the viscosity sense. In fact, it is enough to demonstrate that It is easy to check that min t R + Φ λt > µ where Φ λ t = λt 1 + t r1. d dt Φ λt = 0 t λ = 1 λ1 r, r 1 which, indeed, is a minimum. Since Φ λ t both as t 0 t, it is a global minimum. Then, 1 Φ λ t λ = λ r1 r 1 r r r 1 r1 r by our election of λ. On the other h, define ψ = δϕ 1, where ϕ 1 is a solution of 3.11. Hopf s Lemma Proposition A.1 implies that there exists δ > 0 such that ψ u λ. Then, > µ 3.14 F ψ, D 2 ψ = λ 1 ψ < µ u λ in Ω, by definition of µ. By construction, we have 0 < ψ u λ, where ψ u λ satisfy 3.13 3.14. Hence, we can apply the iteration method as in the proof of existence to get v, satisfying ψ v u, a viscosity solution of F v, D 2 v = µv. Hence, v is a positive solution of 3.11, which is a contradiction with the definition of λ 1. Corollary 3.12. There exists Λ R + with 0 < Λ < such that 3.5 has a positive viscosity solution for every λ 0, Λ.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 15 Proof. Let us define Λ = sup { λ R + : problem 3.5 has a solution }. Theorem 3.7 implies Λ > 0, while Theorem 3.11 gives Λ <. Inded, there exists λ M near Λ u M, such that F u M, D 2 u M = λ M u M + ur M in Ω, u M > 0 in Ω, u M = 0 on Ω. Fix 0 < µ < λ M take λ m such that 0 < λ m < µ. We have proved that there exists a uniue u m solution to F u m, D 2 u m = λ m u m in Ω, 3.15 u m > 0 in Ω, u m = 0 on Ω. Obviously, u M is a supersolution of 3.15 Theorem 2.1 implies u m u M. Since, F u m, D 2 u m = λ m u m < µ u m + ur m, F u M, D 2 u M = λ M u M + ur M > µ u M + ur M, we can apply the iteration method described in the proof of Theorem 3.7 to get the existence of a solution u µ > 0. 4. Examples further results In this section, we show several operators for which Theorem 3.2 still applies even though the hypotheses of Section 3 do not hold. We emphasize that the fundamental ingredients of the method are the Harnack ineuality the Hopf boundary Lemma; such properties are still available in the examples considered below. 4.1. The linear problem with variable coefficients. Consider the linear problem trace AxD 2 u λ + bx, uλ = λu λ in Ω, 4.1 u λ > 0 in Ω, u λ = 0 on Ω, with λ > 0 0 < < 1. We assume that Ax is a symmetric uniformly elliptic matrix, i.e., there exist constants 0 < θ < Θ such that θ ξ 2 Axξ, ξ Θ ξ 2 ξ R n, that the coefficients Ax, bx are, say, Lipschitz continuous. Then, it is possible to adapt the arguments in the proof of Theorem 2.1. Notice that the hypotheses are not optimal see for instance Section 5.A in [16]; with this example, we intend to illustrate the technical difficulties arising in the proof of comparison, since the argument remains essentially the same. These technical difficulties are hled using the regularity of the coefficients. Theorem 4.1. Let 0 < < 1, λ = 1 Ω R n be a bounded domain. Consider u, v CΩ a viscosity sub- supersolution of 4.1 with the coefficients Ax, bx under the hypotheses above. Finally, suppose that both, u v are strictly positive in Ω, that one of them is in C α Ω for α > 1/2, u v on Ω. Then, u v in Ω. The proof is an adaptation of that of Theorem 2.1.

16 FERNANDO CHARRO AND IRENEO PERAL Proof. We argue by contradiction. As u v CΩ Ω is compact, u v attains a maximum at Ω. In order to arrive at a contradiction, let us suppose that max Ω u v > 0. Consider the functions ũ ṽ ǫ as in the proof of Theorem 2.1. Arguing as before, we can fix ǫ > 0 small enough suppose that there exists Ω with Ω Ω which contains all the maximum points of ũ ṽ ǫ trace AxD 2 ũ trace AxD 2 ṽ ǫ Ax ũ, ũ + bx, ũ 1 in Ω, 1 ũ Ax ṽ ǫ, ṽ ǫ + bx, ṽ ǫ 1 + ǫ > 1 in Ω, 1 ṽ ǫ in the viscosity sense. Consider wx, y = ũx ṽ ǫ y τ 2 x y 2 for each τ > 0, l let x τ, y τ be a maximum point of w in Ω Ω. The arguments in the proof of Theorem 2.1 show that, for τ large enough, x τ, y τ Ω ũx τ > ṽ ǫ y τ. Moreover, the Maximum Principle for semicontinuous functions implies the existence of two symmetric matrices X τ, Y τ such that τxτ y τ, X τ J 2,+ũxτ τx τ y τ, Y τ J 2,ṽǫ y τ, 4.2 3τ Hence, we have I 0 0 I traceax τ X τ traceay τ Y τ Xτ 0 3τ 0 Y τ I I I I. τ 2 Ax τ x τ y τ, x τ y τ 1 ũx τ + τ bx τ, x τ y τ 1, τ 2 Ay τ x τ y τ, x τ y τ 1 ṽ ǫ y τ + τ by τ, x τ y τ 1 + ǫ,, subtracting the first euation from the second one, we have 0 < ǫ traceay τ Y τ Ax τ X τ { τ 2 Ay τ x τ y τ, x τ y τ 1 ṽ ǫ y τ } + τ2 Ax τ x τ y τ, x τ y τ τ bx τ by τ, x τ y τ ũx τ trace Ay τ Y τ Ax τ X τ + τ 2 Ax τ Ay τ x τ y τ, x τ y τ 1 ũx τ τ bx τ by τ, x τ y τ, since ṽ ǫ y τ ũx τ. Now, we estimate each term in the right h side of the above expression separately. We follow Example 3.6 in [16]. For the first term, multiply the right part of 4.2 by the nonnegative, symmetric matrix Axτ 1/2 Ax τ 1/2 Ay τ 1/2 Ax τ 1/2 Ax τ 1/2 Ay τ 1/2 Ay τ 1/2 Ay τ 1/2, where Ax 1/2 is well defined since Ax is uniformly elliptic. Taking traces we arrive at traceay τ Y τ Ax τ X τ 3τ trace[ Ayτ 1/2 Ax τ 1/2 2 ].

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 17 As the matrix Ax is Lipschitz continuous uniformly bounded away from 0, Ax 1/2 is Lipschitz as well. Then, traceay τ Y τ Ax τ X τ C τ x τ y τ 2. Next, for the second third terms, we use that A, b are Lipschitz, indeed τ 2 Ax τ Ay τ x τ y τ, x τ y τ C τ 2 x τ y τ 3, τ bx τ by τ, x τ y τ c τ xτ y τ 2. Collecting all the estimates above, we get 4.3 0 < ǫ C τ x τ y τ 2 + τ 2 x τ y τ 3. It remains to show that the right h side of 4.3 tends to 0 as τ. We follow the observations in Section 5.A in [16]. By definition of x τ, y τ, we have ũx ṽ ǫ y τ 2 x y 2 ũx τ ṽ ǫ y τ τ 2 x τ y τ 2, for all x, y near x τ, y τ. Since either u or v are Holder continuous, say u C α Ω, we have that ũ C α Ω as well then, setting x = y = y τ, we get τ 2 x τ y τ 2 ũx τ ũy τ C x τ y τ α. Hence τ x τ y τ σ 0 as τ 0 σ > 2 α. Hence, since α > 1/2, the right h side of 4.3 tends to 0 as τ, which is a contradiction. Remark 4.2. Notice the fundamental role that 4.2 plays in the final estimate of the above proof. For the existence, notice that the operator is uniformly elliptic hence both the Harnack ineuality the Hopf Lemma in Section 3 are still available. Hence, Theorem 3.2 applies in this context. Moreover once a viscosity solution is found, by elliptic regularity, it is C 2,α. 4.2. The p-laplacian, p <. The p-laplacian operator is given by p u = div u p2 u u u = trace Id + p 2 u 2 D 2 u u p2. The divergence-form is useful for variational problems, while the exped version, is preferred in the viscosity setting. It is immediate to check that it verifies hypotheses F1 F2, it is degenerate elliptic homogeneous of degree p1. Take 0 < < p 1 < r λ > 0, consider the problems p u λ = λu λ, in Ω, 4.4 u λ > 0 in Ω, u λ = 0 on Ω, 4.5 p u λ = λu λ + ur λ in Ω, u λ > 0 in Ω, u λ = 0 on Ω. It is well-known that continuous variational weak solutions to this kind of problems are viscosity solutions, see [5] [23].

18 FERNANDO CHARRO AND IRENEO PERAL The following already known result see [1] [7] the references therein can be achieved following the ideas in Section 3 as a conseuence of Theorem 2.1, the Harnack ineuality the Hopf boundary lemma. Theorem 4.3. Let Ω R n be a smooth bounded domain, 0 < < p 1 < r, λ > 0. Then, 1 There exists a uniue solution to 4.4 for every λ > 0 satisfying u λ x = λ 1 p1 u1 x, where u 1 is the solution with λ = 1. 2 Suppose in addition that Ω satisfies an interior sphere condition. Then exists 0 < Λ < such that there is at least one positive viscosity solution of 4.5 for every λ < Λ, no nontrivial solution for λ > Λ. In this case the subsolution can be taken as a positive eigenfunction conveniently scaled. For variational eigenvalues of the p-laplacian see for instance [18] [28]. 4.3. The 3-homogeneous infinity-laplacian. Consider the infinity laplacian operator, n u = D 2 u u, u = u xi,x j u xi u xj. i,j=1 Clearly, it is homogeneous of degree 3 degenerate elliptic hence, not in the framework of Theorem 3.2. Nevertheless, the methods in Section 3 can be adapted to produce the following result. Theorem 4.4. Let Ω R n be a smooth bounded domain, 0 < < 3 < r λ > 0. Then, 1 There exists a uniue viscosity solution of u λ = λu λ in Ω, 4.6 u λ > 0 in Ω, u λ = 0 on Ω. Indeed, u λ x = λ 1 3 u1 x, where u 1 is the solution for λ = 1. 2 There exists at least one solution of u λ = λu λ + ur λ in Ω, 4.7 u λ > 0 in Ω, u λ = 0 on Ω, for λ small enough. The uniueness assertion in the first part of Theorem 4.4 is a conseuence of Theorem 2.1. For the existence part, we follow the construction in Section 3. We start studying the auxiliary problems. Proposition 4.5. The auxiliary problem v λ = λ in Ω, 4.8 v λ > 0 in Ω, v λ = 0 on Ω, has a uniue viscosity solution for every λ.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 19 Proof. As 0 is a subsolution, we only have to construct a suitable supersolution apply Perron s method. Consider x 0 / Ω let Φx = α β 1/3 2 x x 0 2 λ, with β distx 0, Ω 2 α βc2 2, where C = max x x 0. By direct computation, x Ω in Ω, while w Φ on Ω, since Φx = β 3 x x 0 2 λ, Φx = α β 2 x x 0 2 0, by construction. In remains to construct barriers in order to force Φ to have the correct boundary value. Following [16] [21], for every z Ω, we define where Ψ z x = D x z 1/2, A straightforward computation shows D 3 16 λdiamω 5/2. x Ω Ψ z x = D3 16 x z 5/2 λ, therefore v λ x = 0 on Ω. Furthermore, 0 v λ Φ by comparison, Theorem 2.5. Since v λ 0 we get a contradiction with the euation otherwise, the Harnack ineuality see [26], [25] [4] implies v λ > 0. Corollary 4.6. The problem w λ = λdx in Ω 4.9 w λ > 0 in Ω w λ = 0 on Ω, where dx is the normalized distance to the boundary, namely, distx, Ω dx =, dist, Ω has a viscosity solution. Proof. It is sufficient to consider v λ as a supersolution proceed as in the previous proposition. Now, we can construct a supersolution of the concave problem 4.6 as in Lemma 3.4. For the construction of the subsolution we argue as in Lemma 3.5 using the Hopf Lemma in [4]. Then, the Comparison Principle Theorem 2.1 yields existence uniueness of solutions to 4.6 via the Perron method for λ = 1. By homogeneity, we extend the result for every λ. In the concave-convex case, problem 4.7, we follow Lemmas 3.8 3.9. Hence, we construct a sub- supersolution u λ, u λ from v λ, w λ in 4.8 for λ < λ 0, the latter given by 3.8. Following the iteration procedure in Subsection 3.2 we get a seuence where u λ... w k w k1... w 2 w 1 u λ in Ω, { w k = λw k1 + wr k1 in Ω, w k = 0 on Ω.

20 FERNANDO CHARRO AND IRENEO PERAL for k 1 by convention w 0 = u λ. Notice that, since w k > 0 in Ω, we have see [25] [26] that w k x w kx distx, Ω u λx distx, Ω a.e. x Ω, for every k > 1. Hence, both w k w k are uniformly bounded on compact subsets of Ω. Since w k = 0 for all k, by the Ascoli-Arzela compactness criterion the monotonicity of the {w k }, the whole seuence converges uniformly in Ω to some u λ CΩ which is a solution of 4.7 in the viscosity sense. We have now finished the proof of Theorem 4.4. 4.4. The 1-homogeneous infinity-laplacian. In the previous section we have considered the classical infinity laplacian with degree of homogeneity 1. Other definition can be considered, namely, u = D 2 u u u, u u This operator naturally arises in many euations obtained as a limit of p-laplacian type euations; it is easy to adapt the arguments in [23] [14] to get an example of this issue. Furthermore, when studying evolution euations governed by the infinity laplacian see [22] the references therein, it is natural to consider the 1-homogeneous infinity laplacian rather than its 3-homogeneous version since then, the homogeneity of the parabolic part matches that of the infinity laplacian. Finally, the Poisson problem for this infinity laplacian has been recently studied in [29] using Tug-of-War games, in [21], an eigenvalue-type euation is studied. Our aim here is to study the existence uniueness of solutions to u λ = λu λ in Ω, 4.10 u λ > 0 in Ω, u λ = 0 on Ω, 4.11. u λ = λu λ + ur λ in Ω, u λ > 0 in Ω, u λ = 0 on Ω, with λ > 0 0 < < 1 < r. It would be necessary to make precise the definition of viscosity solution respectively sub- supersolution in this context, since the operator is singular when u = 0. Our definition follows the one in [21] for the eigenvalue problem. First, given a matrix A S n, we denote its largest smallest eigenvalues by MA ma, respectively, this is, MA = max Aξ, ξ, ξ =1 Then, we have the following definition. ma = min Aξ, ξ. ξ =1 Definition 4.7. Let Ω R n be a bounded domain. An upper semicontinuous function v : Ω R is a viscosity subsolution of 4.10 in Ω if, whenever ˆx Ω φ C 2 are such that v φx < v φˆx = 0 for all x ˆx in a neighborhood of ˆx, then { φˆx λφ ˆx, if φˆx 0, MD 2 φˆx λφ ˆx, if φˆx = 0.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 21 A lower semicontinuous function w : Ω R is a viscosity supersolution of 4.10 in Ω if, whenever ˆx Ω φ C 2 are such that w φx > w φˆx = 0 for all x ˆx in a neighborhood of ˆx, then { φˆx λφ ˆx, if φˆx 0, md 2 φˆx λφ ˆx, if φˆx = 0. Finally, a continuous function u : Ω R is a viscosity solution of 4.10 in Ω if it is both a viscosity subsolution viscosity supersolution. Notice that this definition is slightly different to the one in [29]; however, it is easy to see that both definitions are euivalent. If u satisfies u 0 in Ω in the viscosity sense, then u 0 in Ω as well. Hence the Harnack ineuality Hopf Lemma used in the previous section see [4], [25] [26] also apply to the operator. Then, the results in [29] the Harnack ineuality imply the existence of a uniue solution to the auxiliary problem v = λ in Ω, 4.12 v > 0 in Ω, v = 0 on Ω. According to the results in [21], there exists a solution for the eigenvalue problem w = λ 1 w in Ω, w > 0 in Ω, w = 0 on Ω, with w = 1 for general Ω. Then we can use as a subsolution t w, t small, then avoid the use of the Hopf boundary lemma. Moreover, the existence of a first eigenvalue allows us to prove the non existence of solutions for large λ in the concave-convex case. Following the ideas in Section 3 completing the details as in the case of the 3-homogeneous infinity laplacian, we get the following result. Theorem 4.8. Let Ω R n be a bounded domain, 0 < < 1 < r λ > 0. Then, 1 There exists a uniue positive viscosity solution of 4.10 for every λ > 0. In fact, u λ x = λ 1 1 u1 x, where u 1 is the solution for λ = 1. 2 If Ω satisfies an interior sphere condition, then there exists 0 < Λ < such that there is at least one positive viscosity solution of 4.11 for every λ < Λ no nontrivial solution for λ > Λ. 4.5. Monge-Ampere euations. The above theory also applies to euations of the Monge-Ampere type. It is well known see for example [11] [16] that the Monge-Ampere operator F : S n R M detm, is elliptic only for positive definite matrices. Hence, it is natural to look for strictly convex solutions. Let Ω R n be a bounded, smooth, strictly convex set, 0 < < n < r λ > 0. Our model concave problem reads detd 2 u = λ u in Ω, 4.13 u convex in Ω, u = 0 on Ω,

22 FERNANDO CHARRO AND IRENEO PERAL, in the concave-convex case, detd 2 u = λ u + u r in Ω, 4.14 u convex in Ω, u = 0 on Ω. Notice that Ω is the zero level set of the strictly convex function u hence it is natural to assume that Ω is strictly convex. Since det is increasing in the set of positive definite-matrices, the euation should be written in the form detd 2 u + λ u = 0 in Ω in order to follow the same convention as in the rest of the paper. As we are looking for negative solutions, we have to adapt the arguments in Theorem 2.1 which in this context reads as follows. Theorem 4.9. Let Ω R n be a bounded strictly convex domain consider a viscosity subsolution u a supersolution v of 4.15 detd 2 w + λ w = 0 in Ω, where 0 < < n. Suppose that both, u v are strictly negative in Ω, continuous up to the boundary satisfy u v on Ω. Then, u v in Ω. For the proof, as before, we can take λ = 1 by homogeneity. Replace 2.8 by ũx = 1 1 ux 1 n ṽx = 1 1 vx n n 1, n 2.9 by Then, since ũ ṽ on Ω, we have ṽ ǫ x = 1 ǫ ṽx + ǫ. ũ ṽ ǫ = ũ 1 ǫṽ 1 ǫǫ < 0 on Ω. Then, following the arguments in the proof of Theorem 2.1, we can suppose that there exists Ω such that Ω Ω contains all the maximum points of ũ ṽ ǫ for ǫ small enough. Then, ũ, ṽ ǫ satisfy det D 2 ũ + det D 2 ṽ ǫ + n n we can complete the proof as in Theorem 2.1. ũ ũ + 1 0 in Ω ũ ṽ ǫ ṽ ǫ + 1 ǫ n 0 in Ω, ṽ ǫ The Comparison Principle in Theorem 2.5 can also be adapted to the present setting in a similar way. For the existence of solutions of 4.13 4.14, we have the following result. Theorem 4.10. Let Ω R n be a bounded, strictly convex, smooth domain, consider 0 < < n < r λ > 0. Then, 1 There exists a uniue solution to 4.13 for every λ > 0 satisfying u λ x = λ 1 n u1 x, where u 1 is the solution with λ = 1. 2 There exists 0 < Λ < such that there is at least one positive viscosity solution of 4.14 for every λ < Λ, no nontrivial solution for λ > Λ.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 23 The above result was already known, see [20] for example where the Leray- Schauder degree is used. Our proof is different constructive. For the proof of both statements, we adapt the monotone iteration scheme in Section 3. Hence, fix ǫ 0, 1 consider the auxiliary problem detd 2 v λ = λ in Ω, v λ convex in Ω, v λ = ǫ on Ω. Since λ > 0, it is known see [12] that there exists a uniue strictly convex solution v λ C Ω. Indeed, v λ < 0 in Ω by strict convexity. Furthermore, consider the Monge-Ampere eigenvalue problem detd 2 ψ 1 = λ 1 ψ 1 n in Ω, ψ 1 convex in Ω, ψ 1 = 0 on Ω. In [27], it is shown that there exists ψ 1 C 1,1 Ω C Ω such that ψ 1 < 0 in Ω that λ 1 > 0 is isolated. Then, we can construct a subsolution a supersolution of 4.13 with λ = 1. Lemma 4.11. Let T = v n t λ 1 n 1. Then ux = T vx ux = t ψ 1 x are, respectively, classical sub- supersolution of 4.13 with λ = 1. The proof is similar to that of Lemmas 3.4 3.5 but, since now we are looking for negative solutions, the details are slightly different. In a similar fashion to Lemmas 3.8 3.9 we get strictly convex, classical subsolutions supersolutions. Lemma 4.12. There exists λ 0 > 0 such that for all λ 0, λ 0 ], we can find constants Tλ, t > 0 such that u λ x = Tλv λ x u λ x = t ψ 1 x, are respectively a classical sub- supersolution of 4.14. Indeed, t is chosen such that u λ u λ in Ω for all λ 0, λ 0. Then, consider w 1, the solution of detd 2 w 1 = λ u λ + u λ r w 1 convex in Ω w 1 = Tλ ǫ 2 on Ω. in Ω Theorem 1.1 in [12] imply that exists a uniue w 1 C Ω it is strictly convex. By comparison, u λ w 1 u λ in Ω. Then, consider detd 2 w 2 = λ w 1 + w 1 r in Ω, w 2 convex in Ω, w 2 = Tλ ǫ 3 on Ω. Again, Theorem 1.1 in [12] implies the existence of a uniue w 2 C Ω, which is strictly convex. Then, by comparison, u λ w 1 w 2 u λ in Ω.

24 FERNANDO CHARRO AND IRENEO PERAL We can iterate this procedure. In general, we get detd 2 w k = λ w k1 + w k1 r in Ω, w k convex in Ω, w k = Tλ ǫ k+1 on Ω. Each w k is uniue, strictly convex, C Ω. Furthermore, u λ w 1 w 2... w k1 w k... u λ in Ω. Since every w k C Ω, it is a weak solution in the sense of [32], the Holder estimate in [32, Theorem 4.1] gives the necessary relative compactness. Hence, the Ascoli-Arzela compactness criterion the monotonicity of the seuence, gives uniform convergence of the whole seuence to some u CΩ. Then, for every compact subset Ω of Ω, we can pass to the limit in the viscosity sense get detd 2 u = λ u + u r in Ω. Furthermore, u is convex is an uniform limit of strictly convex functions verifies u = 0 on Ω by construction. 4.6. The euation min{ u λu, u} = 0. We finally consider the euation min{ u λ λu λ, u λ } = 0 on Ω, 4.16 u λ > 0 on Ω, u λ = 0 on Ω, for λ > 0 0 < < 1. The existence uniueness of positive solutions to 4.16 for every λ > 0 has already been studied in [14]. In fact, it is proven that there exists a smooth curve of solutions given by u λ x = λ 1 1 u1 x, where u 1 is the solution to 4.16 with λ = 1. Clearly, 4.16 does not fit the general shape considered in this work, namely 3.1. However, both the comparison principle in [14] our Theorem 2.1 for m=1 are particular cases of the following slightly more general result. Theorem 4.13. Let Ω R n be a bounded domain F : R R n S n R such that 1 Ft s, t ξ, t X = t Fs, ξ, X for every t > 0, F0, 0, 0 = 0. 2 Fs, ξ, X Fs, ξ, Y whenever Y X. 3 For fixed ξ R n X S n, F, ξ, X is strictly decreasing. Let u, v CΩ be a viscosity subsolution a viscosity supersolution to 4.17 F fw, w, D 2 w = 0 in Ω, where f satisfies hypothesis 2.2 with m = 1. Assume that u, v are strictly positive in Ω u v on Ω. Then, u v in Ω. Let us point out that we are not under the scope of [16] since we are not assuming the properness of F. The proof is an adaptation of the one to Theorem 2.1. In this context, Lemma 2.3 reads as follows. Lemma 4.14. Let w > 0 be a supersolution subsolution of problem 4.17 in Ω as in 2.2. Then, wx = 1 1 w1 x

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS 25 is a viscosity supersolution subsolution of [1 ] 1 1 f wx 4.18 F [ 1 ] 1 wx, w, D 2 w + 1 w w w = 0, in every Ω such that Ω Ω. The rest of the proof of Theorem 3.2 applies almost unchanged. Appendix A. Hopf s Lemma for uniformly elliptic euations We recall the Hopf boundary lemma, used in the proof of the analogous to Lemma 3.5. For further refinements, see [24] see also [3] [31]. Proposition A.1 Hopf s Lemma. Let Ω be a bounded domain u a viscosity solution of F u, D 2 u 0 in Ω, where F satisfies F1, F2, F3. In addition, let x 0 Ω satisfy i ux 0 > ux for all x Ω. ii Ω satisfies an interior sphere condition at x 0. Then, for every nontangential direction ξ pointing into Ω, ux 0 + tξ ux 0 lim < 0. t 0 + t Proof. Our proof follows [19, Section 3.2]. Since Ω satisfies an interior sphere condition at x 0, there exists a ball B = B R y Ω with x 0 B. For 0 < ρ < R, we consider A = {x Ω : ρ < x y < R} define vx = e α xy 2 2 e αr2 2, wx = ux ux 0 + ǫ vx, for x A, where α, ǫ > 0 are constants yet to be determined. Then, 1. P θ,θ D2 w γ w 0 in A for α large enough. Let φ C 2 ˆx A such that w φ has a local maximum at ˆx. It is easy to see that u Φ has a local maximum at ˆx, with Φx = φx ǫ vx. Since v C 2, so it is Φ, the definition of u the structure condition F3 imply 0 F Φˆx, D 2 Φˆx = F φˆx ǫ vˆx, D 2 φˆx ǫ D 2 vˆx F φˆx, D 2 φˆx + P θ,θ ǫ D2 vˆx γ ǫ vˆx F0, 0 + P θ,θ D2 φˆx γ φˆx ǫ P + θ,θ D2 vˆx γ ǫ vˆx. By direct computation, P + θ,θ D2 vˆx = e α xy 2 2 P + θ,θ α 2 x y x y α I e α xy 2 2 α 2 ρ 2 θ + α n Θ, vˆx α R e α xy 2 2. Combining the expressions above, P θ,θ D2 φˆx γ φˆx ǫ e α xy 2 2 for α large enough. α 2 ρ 2 θ + α n Θ γ R 0,

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