On the inequality $$F(x,D^2u)\ge f(u) +g(u)\, |Du|^q$$F(x,D2u)≥f(u)+g(u)|Du|q

We consider fully nonlinear degenerate elliptic equations with zero and first order terms. We provide a priori upper bounds and characterize the existence of entire subsolutions under growth conditions on the lower order coefficients which extend the classical Keller–Osserman condition for semilinear equations.


Introduction
In 1957 J.B. Keller [22] and R. Osserman [28] simultaneously and independently proved that, for a given positive, continuous and nondecreasing function f , the semilinear differential inequality Carrying on the study started in [12], the aim of the present paper is to establish analogous results for the more general fully nonlinear differential inequality Here, f and g are assumed to be continuous and monotone increasing, with f positive, q belongs to (0, 2] and F is a second order degenerate elliptic operator, that is a continuous function F : R n × S n → R satisfying F (x, O) = 0 and the (normalized) ellipticity condition S n being the space of symmetric n × n real matrices equipped with the usual partial ordering.
The model cases for F that we have in mind are the degenerate maximal Pucci operator M + where µ 1 (X) ≤ µ 2 (X) ≤ . . . ≤ µ n (X) are the ordered eigenvalues of the matrix X.
The Pucci extremal operators, in the uniformly elliptic case, have been extensively studied in a monograph by L. Caffarelli and X. Cabré , see [10]. Let us recall here that the operator (1.4) is maximal in the class of degenerate elliptic operators vanishing at X = O. In particular, for any 1 ≤ k ≤ n and for all X ∈ S n , one has P + k (X) ≤ M + 0,1 (X) .
As for the operators P + k , we refer to the recent works of R. Harvey and B. Lawson Jr [20] and L. Caffarelli, Y.Y. Li and L. Nirenberg [11], see also M.E. Amendola, G. Galise, A. Vitolo [3] and G. Galise [18] and the references therein for further results. We just point out here that such degenerate operators arise in several frameworks, e.g. the geometric problem of mean curvature evolution of manifolds with co-dimension greater than one, as in L. Ambrosio, H.M. Soner [2], as well as in the PDE approach to the convex envelope problem, see A. Oberman, L. Silvestre [27].
Since the considered operators are in non-divergence form, a natural approach to the analysis of the partial differential inequality (1.3) is that of viscosity solutions. So, by solution of (1.3) or (1. 7) or (1.6) we always mean an upper semicontinuous subsolution in the viscosity sense.
Let us present now our results in a rather informal way. A key point is that while we assume f to be positive we will not make a sign assumption on the function g, so that the first order terms can be of either "absorbing" or "reaction" type.
Let us discuss first the case where lim t→+∞ g(t) > 0. In this case, the necessary and sufficient "sublinearity" condition (1.2) which rules the semilinear case (1.1) should be of course generalized in order to take in proper account the first order terms. We prove indeed that inequality (1.7) possesses an entire viscosity solution (see Theorem 3.3 below) if and only if (1.8) q ≤ 1 and The same condition is also proved, in Theorem 3.5, to be necessary and sufficient for the existence of an entire viscosity solution of (1.6), provided that g(t) ≥ 0 for all t ∈ R. In particular, we see that if q > 1 then no entire subsolutions can exist, independently of how slow the growth of f and g is, whereas growth restrictions both on f and g are needed in the case q ≤ 1. Moreover, in Theorem 3.6 we show that if condition (1.8) is violated and u satisfies inequality (1.3) in a proper open subset Ω ⊂ R n , then u satisfies the universal pointwise upper bound where d(x) = dist(x, ∂Ω) and R is given (assuming for simplicity g ≥ 0) by the formula Assume now that lim t→+∞ g(t) ≤ 0. In this case, the zero and the first order terms in inequality (1.7) are competing with opposite signs. Our analysis shows that in this case entire viscosity subsolutions exist if and only if a relaxed version of condition (1. 2) involving f , g and q holds true, namely In particular, if lim t→+∞ g(t) < 0, then (1.9) is proved to be equivalent to Note that the above condition becomes, for q = 2, the "subquadratic" growth condition Moreover, also in this case as in the previously discussed one, if condition (1.9) fails and u satisfies (1.3) in any open subset Ω ⊂ R n , then u is universally estimated from above by an explicit function of the distance from the boundary determined by f , g and q, see Theorem 3.6.
As in the original papers [22], [28] our strategy for proving the above results, see Theorem 3.3 and Theorem 3.5, is based on comparison with radially symmetric solutions of (1.6) and (1.7), after a detailed analysis of the existence of entire maximal solutions of the associated ordinary differential equation. As a matter of fact, entire solutions of (1.7) exist if and only if entire radially symmetric solutions exist. Remarkably, this fact is proved by a comparison argument which works also in the currently considered degenerate cases, without any a priori growth assumptions at infinity on u.
Let us point out finally that, as a general fact, when the Keller-Osserman type conditions do not hold true and the zero order term f (u) is an odd function satisfying the sign condition f (u) u ≥ 0, one can obtain universal local bounds from above and from below. This property has been largely used in the literature to obtain existence results for entire solutions as well as existence of "large" solutions in bounded domains, that is solutions blowing up on the boundary. We did not investigate in this direction in the present paper and we just recall in this respect the result of H. Brezis [9] about existence and uniqueness of entire solutions of semilinear equations with f (u) = |u| p−1 u, p > 1. For subsequent extensions to more general divergence form principal parts and zero order terms we refer to L. Boccardo, T. Gallouet and J.L. Vazquez [7], [8], F. Leoni [25], F. Leoni, B. Pellacci [26] and L. D'Ambrosio, E. Mitidieri [14]. In the fully nonlinear framework, analogous results have been more recently obtained by M.J. Esteban, P. Felmer, A. Quaas [16], G. Diaz [15] and G. Galise, A. Vitolo [19], M. E. Amendola, G. Galise, A. Vitolo [4], and for Hessian equations, involving the k-th elementary symmetric function of the eigenvalues µ 1 (D 2 u), . . . , µ n (D 2 u) by J. Bao, X. Ji [5], J. Bao, X. Ji, H. Li [6], Q. Jin, Y.Y. Li, H. Xu [21]. For application to removable singularities results see also D. Labutin [23].
As far as equations with gradient terms are concerned, the analogous "absorbing" property of superlinear first order terms in semilinear elliptic equations was singled out first by J.M. Lasry and P.L. Lions [24] and then extensively studied, see e.g. S. Alarcón, J. García-Melián and A. Quaas [1] and P. Felmer, A. Quaas and B. Sirakov [17] for fully nonlinear uniformly elliptic equations with purely first oder terms of the form h(|Du|). Moreover, some results obtained by A. Porretta [29] on the existence of entire solutions of the equation are closely related to the present paper. Let us observe that they are obtained by performing the change of unknown v = +∞ u e − t 0 g(s) ds dt which can be used only when the power q of the gradient term is 2.

On the associated ODE
In this section we perform a fairly complete qualitative analysis of the Cauchy problem where f, g are continuous non decreasing functions and c, q are positive real numbers. As indicated in the Introduction this analysis is one of the basic tools in our approach to the study of entire solutions of the elliptic PDE By solution of (2.1) in [0, R) with 0 < R ≤ +∞, we mean here and in the sequel a function ϕ ∈ C 2 ((0, R)) ∩ C([0, R)) satisfying, moreover, Therefore the ordinary differential equation in (2.1) has to be satisfied for r = 0, too. Let us observe that the existence of local solutions of (2.1) follows from the standard theory of ordinary differential equations with continuous data. (i) if f is positive, then ϕ is strictly increasing; (ii) if f is positive and f, g are non decreasing, and c ≥ 1, then ϕ is convex and (iii) if f is positive, f, g are non decreasing, g is non negative and c ≥ 1, then Proof. From (2.1) it follows that so that ϕ ′ is increasing, hence positive, in some interval (0, r 0 ). Actually, one has ϕ ′ (r) > 0 in the whole interval (0, R), since, if not, there should be a point r * ∈ (0, R) satisfying ϕ ′ (r * ) = 0 and ϕ ′′ (r * ) ≤ 0 on the one hand, and ϕ ′′ (r * ) = f (ϕ(r * )) > 0 on the other hand. Hence (i) is proved. Next, let us prove (ii). Since ϕ ′′ (0) > 0, there exists some r 1 > 0 such that ϕ ′ (r) > 0 and ϕ ′′ (r) > 0 for r ∈ (0, r 1 ]. By contradiction, let us assume that there exists τ > r 1 such that ϕ ′′ (τ ) < 0. Then, the function ϕ ′ has a strict local maximum point r 0 ∈ (0, τ ) and the set On the other hand, equation (2.1) tested at σ and τ yields In view of the monotonicity of f , g and ϕ and the assumption c ≥ 1, this contradicts (2.4). Therefore, ϕ is convex in [0, R) and from equation (2.1) we immediately obtain which yields (2.2). This proves (ii). Finally, let us prove (iii). Multiplying equation (2.1) by r c−1 and integrating between 0 and r yields since ϕ, ϕ ′ , f and g are non decreasing. Hence (2.3) is proved.
For the next results we focus on the case in which the data of problem (2.1) satisfy (2.5) q ∈ (0, 2], c ≥ 1, f, g continuous and non decreasing, f positive and we obtain sharp estimates from above and from below on the function ϕ ′ . and also, if g(a) ≥ 0, Furthermore, Proof. By Lemma 2.1, ϕ is convex and increasing in [0, R), so that, from the equation in (2.1) we immediately deduce In particular, one has Consider first the case q < 2. Multiplying (2.12) by ϕ ′ and integrating in (0, r), jointly with the increasing monotonicity of ϕ ′ , then yields By Young inequality with exponent 2/q > 1, we then obtain , which immediately gives (2.6) in the case q < 2. If q = 2, we multiply (2.12) by 2e −2 ϕ(r) a g + (t) dt ϕ ′ (r) and deduce Integrating in (0, r) then yields (2.6) in the case q = 2.
On the other hand, the equation in (2.1) may be written as and integration in (0, r), by the monotonicity of ϕ, ϕ ′ , f and g + , yields The above inequality inserted in (2.1) then implies Assume now that g(a) ≥ 0; in this case g(ϕ(r)) ≥ 0 for every r ∈ [0, R) and the above inequality becomes If q = 2, inequality (2.14) can be easily integrated as before yielding inequality (2.7) for q = 2.
If q < 2, (2.14) can be split into the two From the former it easily follows that whereas the integration of the latter yields .
Adding term by term we obtain (2.7) also for q < 2.

By multiplying both sides by 2ϕ
and by integrating in (0, r) we obtain On the other hand, if g is nonpositive, inequality (2.11) reads which, on the account of (2.2), implies and integration in (0, r) yields, as before, and (2.9) is proved.
Finally, let us prove (2.10). Since ϕ is convex and increasing in [0, R), the limits lim r→R − ϕ(r) and lim r→R − ϕ ′ (r) exist. Now, if R = +∞, then (2.10) follows by convexity and strictly increasing monotonicity. On the other hand, from estimate (2.6) it follows that if ϕ is bounded in [0, R), then ϕ ′ is bounded as well, and this contradicts the maximality of R, if R < +∞ . Hence, (2.10) holds true in any case.
For the sequel, it is convenient to rewrite the estimates of Lemma 2.2 in the following equivalent formulations.
as well as, if g(a) ≥ 0, Furthermore, for any q ∈ (0, 2] and r ∈ (0, R), one has Proof. We rewrite estimate (2.6) in the form The proof of this Lemma will be detailed below for the convenience of the reader.
By observing that, for all a ∈ R and t ≥ 2|a| one has as well as the above condition is then equivalent to q ≤ 1 and Now, let (2.21), and therefore (2.24), be satisfied. If ϕ ∈ C 2 ([0, R)) is a maximal solution of (2.1) for some a ∈ R, then, by letting r → R − in (2.16) with q ≤ 1, and by using (2.10) and (2.24) we immediately infer R = +∞.
Conversely, assume that for any a ∈ R any maximal solution of the Cauchy problem (2.1) belongs to C 2 ([0, +∞)), and let us select a maximal solution ϕ satisfying the initial condition ϕ(0) = a > 0 such that g(a) > 0. By inequality (2.17) it follows that, if q > 1, then which is a contradiction to the unboundedness of r. Hence, one has q ≤ 1, and letting r → +∞ in (2.17), we obtain, by (2.10), that (2.24) is satisfied for sufficiently large a, and, therefore, (2.21) holds true.
(ii) Assume that lim t→+∞ g(t) ≤ 0, so that g(t) ≤ 0 for all t ∈ R, and let ϕ ∈ C 2 ([0, R)) be any maximal solution of the Cauchy problem (2.1). By inequalities ( Now, we observe that, for t ≥ a, on the one hand one has Therefore, for t ≥ a, one has  By applying further Young inequality with exponent 2/q we then obtain Hence, Since for t sufficiently large we have t a f (s) ds ≥ t/2 f (t/2) and, moreover, which is the solution of the linear first order initial value problem ψ a is a C 1 (R) increasing function, since, by the monotonicity of the functions f and h, for t ≥ a, one has Therefore, the condition only depends on the growth of ψ a (t) for t → +∞. We observe that, for any a , a ′ ∈ R, one has Now, assume there exists a ∈ R such that R(a) < +∞. Then, it follows that lim t→+∞ ψ a (t) = +∞, and, by the above identity, Hence, R(a ′ ) < +∞ for all a ′ ∈ R, and the first assertion in the statement follows.
Next, let us assume R(a) < +∞ for all a ∈ R. For a 1 < a 2 , we easily obtain being, by monotonicity, f (σ + a 1 − a 2 ) ≤ f (σ) and h(ρ + a 1 − a 2 ) ≥ h(ρ) . (We also notice that the above inequality is strict if either f or h is strictly monotone) Moreover, for any natural number n, we have {ξ n } is a non increasing sequence of integrable functions and, by the monotone convergence theorem, it follows that Remark 2.7. As a consequence of the nondecreasing monotonicity of the functions t f (t) and t g + (t), condition (2.21) can be easily proved to be equivalent to the two conditions +∞ 0 dt (tf (t)) 1/2 = +∞ and +∞ t0 dt (t g + (t)) 1/(2−q) = +∞ , for every t 0 > 0 such that g + (t 0 ) > 0. If 0 < lim t→+∞ g(t) < +∞, the second integral is infinite because q ≤ 1 and therefore in this case (2.21) becomes the usual Keller-Osserman condition If lim t→+∞ g(t) = +∞, then (2.21) restricts the growth at infinity both for f and g; for instance, for power like blowing up functions g(t) ≃ t α for t → +∞, then (2.21) requires α ≤ 1 − q, and for q = 1 at most logarithmic growth g(t) ≃ (ln t) α with α ≤ 1 is allowed.
Remark 2.9. Let us consider the case lim t→+∞ g(t) = 0. By inequality (2.27), it follows that is a sufficient condition in order to have that all maximal solutions are globally defined in [0, +∞).
On the other hand, inequality (2.28) shows that if there exists a global maximal solution ϕ ∈ C 2 ([0, +∞)), then Conditions (2.31) and (2.32) in general are not equivalent (also for q < 2). An easy example occurs for q = 2, f (t) ≃ t (ln t) α and g(t) ≃ −1/t for t → +∞. In this case, (2.31) is satisfied if and only if α ≤ 2 whereas (2.32) holds true up to α ≤ 3. We observe that, in this case, (2.22) actually requires α ≤ 2. On the other hand, if q < 2 and g − (t) decays to 0 as a power function of 1/t, that is c 1 /t β ≤ g − (t) ≤ c 2 /t β for positive constants c 1 , c 2 and β and for t sufficiently large, then (2.31) and (2.32) can be easily proved to be equivalent and, in such a case, they both are a more easy to read equivalent formulation of (2.22). Remark 2.10. Let us explicitly remark that for g ≡ 0, condition (2.22) reduces to the classical Keller-Osserman condition (2.30). An analogous condition is also recovered when g > 0 in the limit q → 0. Indeed, for q → 0 condition (2.21) becomes (2.30) applied to the positive non decreasing nonlinearity f (t) + g(t).

Viscosity subsolutions of fully nonlinear degenerate elliptic equations
In this section we apply the previous results on the Cauchy problem (2.1) to derive a priori upper estimates and necessary and sufficient conditions for the existence of entire viscosity solutions of inequalities of the form where F : R n × S n → R is a continuous functions which we assume always to satisfy F (x, O) = 0 and the (normalized) degenerate ellipticity condition S n being the space of symmetryic n × n real matrices equipped with the usual ordering.
Special attention will be devoted to study in particular subsolutions of the equation |Du| q and of the equation Let us immediately observe that, by the maximality of operator M + 0,1 in the class of second order degenerate elliptic operators, if u is a viscosity solution of (3.1) then u is a subsolution of (3.2).
In the next result we recall that form of comparison principle that will be needed in the sequel. It is an immediate consequence of the definition of sub/supersolution when one of the functions to be compared is smooth. For the general regularizing argument needed to compare merely viscosity sub and super solutions we refer to [13].
Proposition 3.2. Let f, g be continuous functions, with f strictly increasing and g non decreasing, and let further u ∈ U SC(B R ) and Φ ∈ C 2 (B R ) satisfy Proof. By contradiction, suppose u − Φ has a positive maximum achieved at some interior point x 0 ∈ B R . By using Φ(x) + u(x 0 ) − Φ(x 0 ) as test function at x 0 in the definition of viscosity subsolution for u it follows that which, by the strict monotonicity of f and the monotonicity of g, contradicts the fact that Φ is a supersolution.
Our first main result provides necessary and sufficient conditions for the existence of entire subsolutions of equation (3.2). In the same way, statements (ii) and (iii) follow by Theorem 2.5 (ii) and (iii) respectively. The same proof of Theorem 3.3 can be applied to subsolutions of equation (3.3). But, in this case, the extra assumption g(t) ≥ 0 is needed in order to have correspondence between solutions of (2.1) and radially symmetric solutions of (3.3). The comparison argument used in the proof of Theorem 3.3 actually can be applied in order to estimate from above viscosity solutions of inequality (3.1) in any open subset Ω ⊂ R n . If ∂Ω = ∅, for x ∈ R n , we set d(x) = dist(x, ∂Ω) to denote the distance function from the boundary ∂Ω. In order to state our universal upper bounds we need to invert strictly decreasing functions R : R → (0, +∞) such that lim a→+∞ R(a) = 0.
For b > 0, we will denote by R −1 (b) the unique real number a such that R(a) = b if it exists, and where t 0 = inf{t ∈ R : g(t) ≥ 0} and R : R → (0, +∞) is defined as where R : R → (0, +∞) is defined as Proof. (i) As already observed, u is also a subsolution in Ω of equation (3.2). Let x 0 ∈ Ω be fixed, and let us set d 0 = d(x 0 ). In particular, u ∈ U SC B R (x 0 ) is a subsolution of (3.2) in B R (x 0 ) for every 0 < R < d 0 . On the other hand, we know from Theorem 2.5 (i) that, for any a ∈ R, any maximal solution ϕ a of (2.1) is defined on a maximal bounded interval [0, R(a)), with 0 < R(a) < +∞. Moreover, by (2.10) and by estimate (2.17) of Lemma 2.3, it follows that, if a ≥ t 0 , then R(a) ≤ R(a), with R(a) defined by (3.5). Also, Lemma 3.1 implies that the function Φ a (x) = ϕ a (|x − x 0 |) solves equation (3.2) in B R(a) (x 0 ). Therefore, by Proposition 3.2, it follows that provided that a ≥ t 0 is such that R(a) < d 0 . Now, by arguing as in the proof of Lemma 2.4, the function R, which is well defined by definition if q > 1 and by assumption if q ≤ 1, is easily seen to be strictly decreasing. If R(t 0 ) ≤ d 0 , then the conclusion follows by (3.8). If R(t 0 ) > d 0 , in order to conclude it is enough to show that lim a→+∞ R(a) = 0.
In the former case, we simply observe that R(a) ≤ 2 n 2 − q 1/(2−q) +∞ a dt t a f (s) ds 1/2 and that the right hand side goes to 0 as a → +∞ by applying Lemma 2.4 with h ≡ 0.
In the latter case, we notice that necessarily one has Then, by a similar argument as above, for every δ > 0 sufficiently small, we can estimate, up to the constant factor 2 By letting first a → +∞ and then δ → 0 we get the conclusion also for every q ≤ 1.
(ii) We repeat the same comparison argument as above, and conclude by using estimate (2.18) of Lemma 2.3 and by applying Lemma 2.4.
(iii) We argue again as in (i) and (ii), and we further apply inequality (2.28). The function R is easily proved also in this case to be strictly decreasing and to tend to 0 as a → +∞ by analogous arguments as in (i). To show it necessity, we can use the same counterexample exhibited in [29]. Indeed, let us assume q = 2 and g ≡ 1, and for any a ∈ R let ϕ ∈ C 2 ([0, R(a))) be a maximal solution of