ROBERTA FILIPPUCCI. div(a( Du )Du) q( x )f(u) in R n,

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1 ENTIRE RADIAL SOLUTIONS OF ELLIPTIC SYSTEMS AND INEQUALITIES OF THE MEAN CURVATURE TYPE ROBERTA FILIPPUCCI Abstact. In this pape we study non existence of adial entie solutions of elliptic systems of the mean cuvatue type with a singula o degeneate diffusion depending on the solution u. In paticula we extend a pevious esult given in [4]. Moeove, in the scala case we obtain non existence of all entie solutions, adial o not, of diffeential inequalities involving again opeatos of the mean cuvatue type and a diffusion tem. We pove that in the scala case, non existence of entie solutions is due to the explosion of the deivative of evey non global adial solution in the ight extemum of the maximal inteval of existence, while in that point the solution is bounded. This behavio is qualitatively diffeent with espect to what happens fo the m-laplacian opeato, studied in [4], whee non existence of entie solutions is due, even in the vectoial case, to the explosion in nom of the solution at a finite point. Ou non existence theoems fo inequalities extend pevious esults given by Naito and Usami in [11] and Ghegu and Radulescu in [6]. Contents 1. Intoduction 1 2. Peliminay 5 3. Non existence of adial entie solutions: the vectoial case Non existence of adial entie solutions: the scala case 8 Refeences Intoduction The poblem of existence and non existence of non negative non tivial entie solutions of the inequality u f(u), f C(R + 0 ; R+ 0 ), has its oots in two papes witten independently in 1957 by Kelle [8] and Osseman [13]. Next, thei esults wee extended by Naito and Usami in [11] to the quasilinea case div(a( Du )Du) f(u) in R n, whee A C(R + ), sa(s) C(R + 0 ) C1 (R + ), A(s) > 0 and [sa(s)] > 0 fo s > 0 (see also the Intoduction of [4] and [5]). Late, Ghegu and Radulescu [6] gave a futhe extension to the case div(a( Du )Du) q( x )f(u) in R n, 1991 Mathematics Subject Classification. Pimay, 35 J 65, Seconday, 35 J 45. Key wods and phases. non existence of entie solutions, mean cuvatue opeato, elliptic systems. This eseach was suppoted by the Italian MIUR poject titled Metodi Vaiazionali ed Equazioni Diffeenziali non Lineai. 1

2 2 R. FILIPPUCCI when q C(R + 0 ; R+ ) and the function 1 n 0 sn 1 q(s)ds, = x, eithe is coecive at infinity o has deivative bounded fom below by a positive constant. Recently, we extended some of the esults (see [4] fo the non existence and [5] fo the existence) quoted above in two diections: in the vectoial case and when a diffusion tem g(u), depending on the solution u, is included in the divegence tem. Because of the fact that no maximum pinciple is available in the vectoial case, we estict ou attention to adial solutions. In paticula in this pape we ae inteested in studying non existence of adial entie solutions of singula o degeneate elliptic systems of the fom { div ( g(u)a( Du )Du ) u g(u)a( Du ) = f( x, u), x R n, u : R n R N (1.1) \ {0}, whee Du denotes the Jacobian matix of u, A(s) = s σa(σ)dσ, s > 0, is an opeato of the mean 0 cuvatue type, namely such that A( v ) v is bounded in R N, while f and g will be specified late. Futhemoe, in the scala case, we will investigate non existence of positive entie solutions, adial o not, of the elliptic inequality div ( g(u)a( Du )Du ) g (u)a( Du ) f( x, u), x R n (1.2) Fo possible geometical and physical models elated to (1.2) we efe to [1], [2] and [3]. In the standad case in which g(u) 1 the second tem in the left hand side of (1.1) is clealy zeo, othewise it comes essential fo guaanteeing the vaiational stuctue of the system (1.1). That is, the patial diffeential system (1.1) is the Eule Lagange equation of the functional J(u) = {g(u)a( Du ) + F ( x, u)}dx, R n whee F is a function of class C 1 such that f( x, u) = D u F ( x, u), namely f is of gadient type (of couse in the scala case any continuous function f is of gadient type thus the assumption is automatic). A vaiational appoach seems fa fom tivial because of the fact that g could be eithe singula o degeneate at u = 0. Fo the impotance of vaiational systems with diffusion tems in the divegence, that is the vectoial case N > 1 of (1.1), we ecall the pioneeing wok of Stuwe [20] elative to canonical second ode elliptic systems, with bounded coefficient matix possibly depending on u. Indeed the pototype fo (1.1) we have in mind is ( ) div u γ Du γ u γ 2 u ( 1 + Du 2 1 ) = f( x, u), γ R, (1.3) 1 + Du 2 thus the poblem could be eithe degeneate o singula accoding to the sign of γ, while A is the mean cuvatue opeato, that is A(s) = 1 + s 2 1, s 0. (1.4) As obseved in [4], we put in evidence that in the vectoial case thee is no obvious change of vaiable which eliminate the tem g(u) fom the divegence pat even when g is a pue powe as above. Othe main pototypes fo A, widely studied in liteatue, ae essentially given, as in [12], [11] and [6], by the m Laplacian and the mean genealized cuvatue opeato, namely espectively A(s) = ( 1 + s 2) m/2 1, s 0, 1 < m 2, (1.5) A(s) = sm, s > 0, m > 1. (1.6) m

3 ENTIRE SOLUTIONS 3 The non existence of adial entie solutions of (1.1), (1.5) and of (1.1), (1.6) has been teated in [4] by using a technique developed by Levine and Sein in [10] and concening abstact evolution equations. The main diffeence between (1.4) and (1.5) o (1.6) is due to the gowth at infinity, indeed fo (1.4) the gowth is exactly 1 while fo (1.5) and (1.6) it is stictly geate than 1. It is fo this eason that we cannot apply the technique used in [4] but we need to adapt it. In paticula a fist step in this diection has been done in Coollay 3 of [4] in which we pove non existence of adial entie solutions of (1.1), (1.4) but only when g 1 and f( x, u) = f(u) = u p 2 u. Hee we extend Coollay 3 of [4] to the geneal case (1.1). A consequence of the main theoem is given by the following Coollay 1. Let p > 1, 1 < γ < p 1. (1.7) Then the system (1.3) with f( x, u) = f(u) = u p 2 u does not admit non tivial entie adial solutions u : R n R N \ {0}, namely the odinay diffeential system ( ) u γ u + n 1 u γ u 1 + u u 2 γ u γ 2 u ( 1 + u 2 1 ) = u p 2 u, (1.8) u(0) = u 0, u (0) = 0, > 0, does not admit non tivial global solutions. Next, we investigate why solutions stop to exist at a finite value R R +. We ecall that in Coollay 4 of [4] we poved that poblem (1.1), (1.6), with f( x, u) = u p 2 u, g(u) = u γ, 1 < m < p and m p(m 1) < γ < p m, admits a one paamete family of solutions u : B(0, R) R N such that lim x R u( x ) =. Namely the non existence of adial entie solutions fo the m Laplacian case is due to the explosion of the nom of the solution at a finite point. Fo the mean cuvatue opeato the situation is quite diffeent, at least in the scala case. Indeed, when N = 1, the following example, essentially given in [15] in the case in which the non lineaity f has opposite sign, shows what happens. Pecisely, the poblem ( u 1 + u 2 ) + 1 u(0) = u 0 > 0, u (0) = 0, u = 2, > 0, 1 + u 2 which is the special case of (1.8) when γ = 0, n = 2, N = 1 but p = 1, has the positive solution u() = u , [0, 1), which clealy cannot be continued beyond = 1 but it is bounded in = 1 with deivative unbounded at the same point. In Theoem 4 of Section 4 we pove the same phenomenon happens to evey non global solution u of (1.8) when p > 1, namely the failue of adial entie solutions in the mean cuvatue case is due to the blow up of the deivative at a finite point. Moe pecisely we deduce the following

4 4 R. FILIPPUCCI Coollay 2. Assume that (1.7) holds. If q C(R + 0 ) C1 (R + ) is a non negative, non tivial and non deceasing function, then the poblem ( ) div u γ Du γu γ 1( 1 + Du 2 1 ) = q( x )u p 1, u > 0, x R n, (1.9) 1 + Du 2 with u(0) = u 0 > 0, admits a one paamete family of adial solutions u = u(), = x, defined in thei maximal inteval of existence [0, R), R > 0, such that u > 0 in (0, R), lim u() = sup u() = l (u 0, ) and lim R [0,R) R u () =. The poof of this esult stictly elies on the monotonicity of the solution and so it cannot be applied to systems. Hence fo the vectoial case the question is still open. Moeove in the scala case, by using a weak compaison pinciple due to Pucci and Sein in [17] togethe with a tecnique of Naito and Usami [11] and Ghegu and Radulescu [6], we ae able to obtain both non existence of solutions, adial o not, and to impove the lowe bound fo γ given in (1.7). In paticula Theoem 3 and Theoem 8 extend both Coollay 1 in [11] and Poposition 3.2 in [6]. Finally, a consequence of ou main esults is the following Coollay 3. Let q C(R + 0 ; R+ 0 ), q 0, and conside the elliptic inequality in Rn ( ) div u γ Du γu γ 1( 1 + Du 2 1 ) q( x )u p 1, u > 0, (1.10) 1 + Du 2 with p > 1 and γ R. If eithe (i) lim 1 n s n 1 q(s)ds =, 0 o when γ 0, (ii) q C 1 (R + ), q > 0 in R +, q 0, when 0 < γ < p 1, then (1.10) does not admit any positive entie solution. Of couse fom Coollay 3 it follows that in the case (i) the inequality ( ) div u γ Du q( x )u p 1, u > 0, γ 0, 1 + Du 2 does not admit entie positive solutions fo evey p > 1. The pape is oganized as follows: Section 2 contains some peliminaies including the statement of a cucial poposition poved in [4]; Section 3 is devoted to the vectoial case and contain the poof of the main non existence esult of adial entie solutions of the system (1.1) obtained by using a modification of a technique developed by Levine and Sein in [10] fo abstact evolution equations. In Section 4 we establish the main non existence esult of all entie solutions, adial o not, of diffeential inequalities. Hee we use both a compaison esult which is essentially given by Pucci and Sein and anothe one due to Gilbag and Tudinge and finally esults of Naito and Usami in [11] and Ghegu and Radulescu in [6]. I am vey gateful to Pofesso P. Pucci fo many helpful suggestions and he valuable help.

5 ENTIRE SOLUTIONS 5 2. Peliminay We shall be concened with adial entie solutions of elliptic systems of the fom whee Du denotes the Jacobian matix, div ( g(u)a( Du )Du ) u g(u)a( Du ) = f( x, u), x R n, (2.1) A(s) = s 0 σa(σ)dσ, s > 0, and on the functions A, f and g we conside the following assumptions: (A1) A : R + R + is continuous, s sa(s) is stictly inceasing in R + and lim sa(s) = 0; s 0 + (A2) g : P R + 0 is of class C1, whee P R N is an open set; (A3) thee exists a nonnegative function F C 1 (R + 0 RN ; R), with F (, 0) = 0 fo all 0, such that u F (, u) = f(, u) and F (, u) 0 fo all (, u) R + 0 RN. Note that A C 1 (R + 0 ; R) by (A1). Futhemoe, since the pototype fo the diffusion g we have in mind is g(u) = u γ, γ R, then P = R N if γ 0 while P = R N \ {0} if γ < 0. Since we ae inteested in adial solutions, we conside the adial vesion of (2.1), namely [ g(u)a( u )u ] n 1 + g(u)a( u )u u g(u)a( u ) = f(, u), > 0, (2.2) u(0) = u 0 0, u (0) = 0, whee u = du/d and = x. We say that u, u = (u 1,..., u N ), is a local solution of (2.2) if u is a vecto function u : I P R N of class C 1 (I) in some inteval I = [0, R), R > 0, such that A( u () )u () C 1 ((0, R); R N ) (2.3) and u satisfies the system (2.2) in (0, R). Futhemoe a local solution u of (2.2) is called global solution if R =. In this case u is an entie adial solution of (1.1). Local existence of solutions of (2.2) has been poved in [4] by using pevious esults of Pucci and Sein [16] and Leoni [9]. Fom now on, fo simplicity, the common notation whee u = u() and u = u () denote the solution and its deivative. We conside the Legende tansfom of A, namely the function defined in R N by H(ξ) = A( ξ ) ξ 2 A( ξ ), ξ R N. (2.4) Clealy, as aleady noted in [16] fo moe geneal cases, by (A1), H(0) = 0 and H(ξ) > 0 fo all ξ R N \ {0}. (2.5) As poved in [16] and [14], even in a moe geneal settings, in spite of the fact that neithe u no H need be sepaately diffeentiable, the composite function H(u ()) is diffeentiable on (0, R), povided that u neve vanishes on (0, R), and the following identity holds on (0, R) {g(u())h(u ())} = n 1 g(u)a( u ) u 2 + f(, u), u. (2.6)

6 6 R. FILIPPUCCI Consequently if we intoduce the total enegy of the vecto field u, defined along the solution by by (2.6) we get E() := g(u)h(u ) F (, u), (2.7) E () = n 1 g(u)a( u ) u 2 F (, u) (2.8) fo any (0, R), povided that u 0 fo all (0, R). Now, by (A1)-(A3), (2.5) and (2.7) and the initial conditions on u, we immediately note that Thus, fo simplicity, in what follows we conside the function E () 0 and E(0) = F (0, u 0 ). (2.9) E() = E() = g(u)h(u ) + F (, u), E () 0, E(0) = F (0, u 0 ). (2.10) Futhemoe, we emaind that, as a consequence of Theoem 1 of [4], the initial value poblem (2.2) admits a local solution u defined on [0, R), 0 < R 1, such that u() > 0 fo all (0, R). Finally we state the following cucial esult, poved in [4]. Poposition 1. Let u : [0, R) P, R, be a local solution of (2.2) in (0, R). Suppose that (A1) (A3) hold and in addition (A4) and thee exist two functions F C(R N ; R + 0 ) and φ L1 (R + ; R + 0 ) such that F (0) = 0, F (u) > 0 if u 0 0 F (, u) φ() F (u) fo all (, u) R + R N. If F (0, u 0 ) > 0, then thee exists a positive constant U 0 which depends only on F (0, u 0 ) such that u() U 0 fo all [0, R). (2.11) Remak. Note that F (0, u 0 ) > 0 implies that a solution u of (2.2) has negative initial enegy E, indeed by (2.9) we have E(0) = F (0, u 0 ) < Non existence of adial entie solutions: the vectoial case. Theoem 1. Let p > 1 and γ R. Assume that (A1) (A4) hold and that fo evey U > 0 thee exist constants c 1, c 2 > 0 such that u, f(, u) F (, u) c 1 u p, (, u) R + R N, u U, 0, (3.1) whee 0 > 0 is sufficiently lage; and fo all u R N with u U and Suppose that thee exists a constant c 3 > 0 such that If F (0, u 0 ) > 0 and then system (1.1) does not admit adial entie solutions. u g(u), u + g(u) 0 (3.2) g(u) c 2 u γ. (3.3) sa(s) c 3, s R +. (3.4) 1 < γ < p 1, (3.5)

7 ENTIRE SOLUTIONS 7 Remak. Assumptions (A4) and F (0, u 0 ) > 0 ae necessay to pove Theoem 1 above in the vectoial case. Indeed to have that evey solution of (1.1) is in nom fa fom zeo we use Poposition 1. In the scala case it is possible to eplace these assumptions with those that guaantee the monotonicity of the solution (see Lemma 1 and Lemma 2 below). Poof. The idea of poof technique is based on a method developed by Levine & Sein in [10] elative to abstact evolution equations. Assume fo contadiction that thee is an entie adial solution of (1.1), namely a global solution of (2.2). Conside the auxiliay function Z defined by Z() = E 1 α () + g(u)a( u ) u, u, (3.6) whee 0 < α < (p γ 1)/p. As noted in [10], Z is absolutely continuous in R + and a.e. it esults Z () =(1 α)e α E + u, [g(u)a( u )u ] + g(u)a( u ) u 2 n 1 g(u)a( u ) u, u + u g(u), u A( u ) + g(u)a( u ) u 2 + u, f(t, u) F (, u) + E() + g(u)h( u () ), whee we have used (2.10) and (2.2) 1. Now, by using Cauchy-Schwatz inequality, Poposition 1, (3.3) and (3.5) we obtain n 1 g(u)a( u ) u, u (n 1)c 2c 3 U γ+1 p 0 u p (3.8) fo all R + and u U 0. Futhemoe, fom (2.4), it follows that u g(u), u A( u ) + g(u)a( u ) u 2 + g(u)h( u () ) = { u g(u), u g(u)}a( u ) + 2g(u)A( u ) u 2 { u g(u), u + g(u)}a( u ), whee, in the last inequality, we have used the fact that s 2 A(s) A(s) by (2.5) 2. Consequently, thanks to (3.8), (3.9) and (3.1) with U = U 0, inequality (3.7) becomes [ ] Z () c 1 (n 1)c 2c 3 U γ+1 p 0 u p + E() + { u g(u), u + g(u)}a( u ). Finally, by (3.2), we get (3.7) (3.9) Z () C u p + E(), C > 0, (3.10) fo sufficiently lage. On the othe hand, let µ > 1, by (3.6), (3.3), (3.4) and the Cauchy-Schwaz inequality, it follows Now, if we choose µ = 1/(1 α), we have Z µ 2 µ 1 [ E µ(1 α) + c µ 2 cµ 3 u µ(γ+1)]. Z µ C 1 [E + u (γ+1)/(1 α) ] C 1 [E + U (γ+1)/(1 α) p 0 u p ] C 2 [E + u p ], C 2 > 0, whee we have used (3.5) and that α < (p γ 1)/p. Hence, fo sufficiently lage, say 0, Z C 3 Z µ, C 3 > 0.

8 8 R. FILIPPUCCI Now, integating fom 0 to we obtain [Z(t 0 )] µ+1 µ 1 [Z(t)] µ+1 1 µ [Z(t 0)] µ+1 1 µ C 3 0 ds. Consequently, the non integability of the ight hand side foces that Z cannot be defined fo lage, namely Z cannot be global. This completes the poof of the theoem. Poof of Coollay 1 of the Intoduction. It is enough to apply Theoem 1 with g(u) = u γ, A(s) = s 2, f( x, u) = f(u) = u p 2 u. Consequently (3.1), (3.3) and (3.4) hold with c 1 = 1 1/p, c 2 = c 3 = 1, while (3.2) is automatically satisfied by vitue of (3.5) Non existence of adial entie solutions: the scala case In this section we deal with non existence of entie solutions, adial o not, of the diffeential inequality in R n div ( g(u)a( Du )Du ) g (u)a( Du ) f( x, u), (4.1) whee A and g satisfy (A1) and (A2) with N = 1, while f C(R + 0 R) with f(, 0) = 0 fo all 0. In paticula, we shall teat fist the case in which the diffusion g is non inceasing. This special case does not follow fom Theoem 1, but as in [4], it is a consequence of a compaison esult togethe with an agument developed by Naito and Usami in [11] and a non existence theoem given in [6]. It is fo this eason that we can impove the lowe bound fo γ given in (3.5) and due to the technique used to obtain non existence in the vectoial case. We now give the weak compaison pinciple, mentioned above, which is essentially the weak compaison pinciple poved by Pucci, Sein and Zou in Lemma 3 of [17] (see also [18]) in which no diffeentiability assumptions ae equied on A. Theoem 2. (Weak Compaison Pinciple). Assume that q C(R + 0 ) is a non negative function, and C(R + 0 ) is such that (0) = 0, is non deceasing in [0, δ), 0 < δ. Let u and v be espective C 1 distibution solutions of the diffeential inequalities div ( A( Du )Du ) q( x ) (u) 0, u 0, (4.2) div ( A( Dv )Dv ) q( x ) (v) 0, v 0, (4.3) in a bounded domain Ω of R n, n 2. If u and v ae continuous in Ω, with u < δ in Ω and v u on Ω, then v u in Ω. Poof. Assume fo contadiction that thee exists x 1 Ω such that v(x 1 ) < u(x 1 ). Let w = v u in Ω. Then w(x 1 ) < 0 and we can fix ε > 0 sufficiently small such that w(x 1 ) + ε < 0. Consequently, by ou hypothesis and the fact that w 0 on Ω, then the function w ε = min{w + ε, 0} is non positive and has compact suppot in Ω since w ε = 0 on Ω. By the definition of distibution solutions, choosing the Lipschitzian w ε as test function, we get {A( Dv )Dv A( Du )Du} Dw ε dx q( x ){ (u) (v)}w ε dx. (4.4) Ω Ω

9 ENTIRE SOLUTIONS 9 Now, the left hand side of (4.4) is positive since in Lemma 5.5 of [18] it is poved that {A( ξ )ξ A( η )η} (ξ η) > 0 fo all ξ, η R n with ξ η, and also because Dw ε Dw = Dv Du 0 when w + ε < 0, while othewise Dw ε = 0 (a.e.). The ight side of (4.4) is non positive since q 0 and thanks to the monotonicity of. Thus, the contadiction obtained poves the theoem. Remak. In [17] the above theoem is poved when q 1 and the authos put in evidence that in Lemma 3 they eplaced the diffeentiability of A, equied in Theoem 10.7 of [7], by a stict convexity condition. Futhemoe, unde stonge egulaity assumptions on the opeato A, Theoem 2 has been poved in [11] when q 1 and in [6] in the case q 1. We now give the non existence esult fo entie solutions, adial o not, of inequality (4.1). Theoem 3. Assume that g is non inceasing in R + 0 (4.5) and that q C(R + 0 ) is non negative function and such that lim 1 n s n 1 q(s)ds =. (4.6) 0 Suppose that thee exists a non deceasing function C(R + 0 ; R+ 0 ), with (0) = 0 and such that f(, u) g(u) If lim sa(s) <, s then inequality (4.1) does not admit any entie positive solution. q() (u) fo all 0 and u > 0. (4.7) Poof. As in the poof of Theoem 7 of [4], assume by contadiction that (4.1) admits a positive entie solution u : R n R +. By (4.7), (4.5) and A(s) s 2 A(s) fo s > 0 by (A1), then u is an entie solution of div (A( Du )Du) g (u)[ A( Du ) Du 2 A( Du ) ] f( x, u) + q( x ) (u). (4.8) g(u) g(u) Now the emaining poof is simila to that of Theoem 3.3 of [6] with Poposition 3.1 of [6] eplaced with Theoem 2 above and with the weake assumptions on A given in (A1). Consequently we immediately obtain a contadiction since inequality (4.8) cannot admit entie solutions when (4.6) holds. Coollay 4. Let (4.6) hold. Then the elliptic inequality in R n, ( ) div u γ Du γu γ 1( 1 + Du 2 1 ) q( x )u p 1, u > 0, (4.9) 1 + Du 2 whee does not admit any positive entie solution. p > 1 and γ 0, Poof. It is enough to apply Theoem 3 with g(u) = u γ, γ 0, f( x, u) = q( x )u p 1 and with (u) = u p 1 γ.

10 10 R. FILIPPUCCI We now shall teat the case in which (4.5) does not hold. This case takes moe cae with espect to the pevious one since we cannot deive fom (4.1) a simplified inequality as in (4.8). Now, as noted in the Remak befoe the poof of Theoem 1, in the scala case, we can pove a monotonicity esult which allows us to obtain the same thesis of Poposition 1 but with weake assumptions. Lemma 1. Let u be a solution of (2.2) in [0, R), R. Assume that Then u 0 u > 0 in (0, R). u f(, u) > 0 fo all [0, R), u 0. (4.10) Poof. Fist note that equation (2.2) 1 can be witten as follows [ n 1 g(u)a( u )u ] = [ g (u)a( u ) + f(, u) ] n 1 := ϕ() n 1. (4.11) Without loss of geneality we assume that u 0 > 0. Now, by (A1) and (A2), we get ϕ(0) = f(0, u 0 ) > 0, whee we have used the fact that u 0 > 0. Consequently, by continuity, thee exists 1 (0, R) such that ϕ() n 1 > 0 in (0, 1 ). Thus n 1 g(u)a( u )u > 0 in (0, 1 ) (4.12) by the fact that n 1 A( u )u is stictly inceasing and takes zeo value at = 0. Hence u () > 0 in (0, 1 ). Now, let 2 be the fist value in (0, R) such that u () > 0 in (0, 2 ), u ( 2 ) = 0, (4.13) in paticula, it follows that u > 0 in [0, 2 ). Again, as above, ϕ( 2 ) > 0 and consequently [ n 1 g(u)a( u )u ] = 2 > 0, namely n 1 g(u)a( u )u is stictly inceasing nea 2. This yields, by (4.13) 2 to n 1 g(u)a( u )u < 0 fo < 2, which obviously contadicts (4.13) 1. Hence the theoem is poved. Remak. In the case in which u 0 g is non deceasing then assumption (4.10) can be weakened and the theoem above becomes the following. Lemma 2. Let u be a solution of (2.2) in [0, R), R. Assume that Then u 0 u > 0 in (0, R). u f(, u) > 0 fo all (0, R), u 0, (4.14) u 0 f(0, u) 0 fo all u R, (4.15) u 0 g (u) 0 fo all u R. (4.16) Poof. The poof is exactly that of Theoem 1 since (4.14), (4.15) and (4.16) imply that thee exists 1 (0, R) such that ϕ() > 0 in (0, 1 ). Thus (4.12) holds and consequently the poof of Theoem 1 can be epeated wod by wod. Remak. When f(, u) = q()u p 1, p > 1, q C(R + 0 ) and q is non negative, then (4.10) foces that q(0) > 0; while q(0) could be 0 when (4.14) and (4.15) holds.

11 ENTIRE SOLUTIONS 11 Theoem 4. Let p > 1 and γ such that 1 < γ < p 1. Assume that (3.1) (3.3) and (4.10) hold. Suppose that lim H(s) < (4.17) s and that g (u) = O(g(u)) as u. (4.18) Let u be a non tivial adial solution of (1.1) in its maximal domain of existence B(0, R), R <, o equivalently let u be a solution of (2.2) in [0, R), R <, then u is bounded and u admits limit when R. Moe pecisely, if u 0 > 0 in (2.2), then Remaks. lim u() = sup u() = l (u 0, ) and lim R [0,R) R u () =. (4.19) As obseved in [4], Naito and Usami in [11] showed that the following inequality holds H(s) sa(s)ds = s 2 A(s) s 1 σa(σ)dσ sa(s), s > 1. (4.20) Thus if (4.17) is valid, then automatically (3.4) is veified. Futhemoe, as showed in [18], the function H is stictly inceasing. Indeed if, fo simplicity we put Φ(s) = sa(s) when s > 0 and Φ(0) = 0, we immediately have fom (A1) that s 1 Φ(s 1 ) s o Φ(s 0 ) > (s 1 s 0 )Φ(s 1 ) > s1 s 0 Φ(σ)dσ when s 1 > s 0 0. Consequently, assumption (4.17) is equivalent to sup s>0 H(s) <. Poof. Let u be a solution of (2.2) in its maximal inteval of existence [0, R), R < by Theoem 1. Without loss of geneality we assume that u 0 > 0. Fist we pove that u is bounded. By Lemma 1 we have that u () > 0 in (0, R), namely u is stictly inceasing and admits limit as R. Now assume by contadiction that lim u() =. (4.21) R By (2.6) we get, along the solution, [ {H(u )} = g (u) g(u) H(u ) n 1 A(u )u + ] f(, u) u. g(u) Fom (3.1) and (3.3), we get [ {H(u )} g (u) g(u) H(u ) n 1 A(u )u + c ] 1 u p 1 γ u := ψ()u. c 2 By (4.17), (4.18), (3.5), (4.20) and (4.21) it esults that lim R ψ() =. Consequently fo evey K > 0 thee exists 1 < R such that {H(u )} Ku fo all ( 1, R). Now, integating fom 1 to ( 1, R), the last inequality yields to H(u ()) H(u ( 1 )) K[(u() u( 1 )] > 1. In tun we aive to a contadiction by letting R, since the left hand side is bounded by (4.17), while the ight tends to by (4.21). Thus (4.19) 1 is poved. To pove the validity of (4.19) 2, we fist show that thee exists lim R u (). Assume by contadiction that 0 lim inf u () = l 1 < lim sup u () = l 2. R R

12 12 R. FILIPPUCCI Let ( n ) n and (t n ) n be two sequences appoaching R, such that lim n u ( n ) = l 1 and lim n u (t n ) = l 2. This yields to 0 lim n A(u )u n = a 1 < lim n A(u )u tn = a 2 c 3, (4.22) by the stict monotonicity of ta(t) and thanks to (3.4). If we integate (2.2) 1 on [0, ], < R, we get n 1 A(u )u = 1 g(u) 0 { g (u)a(u ) + f(, u) } s n 1 ds, 0 < < R. (4.23) It is enough to evaluate (4.23) fist when = n and then when = t n, so that letting n we get the equied contadiction. Indeed the left hand side of (4.23) does not admit limit when R by (4.22), while the ight hand side tends to the value R{ 0 g (u)a(u ) + f(, u) } s n 1 ds / g(l). Hence the only possibility is that the limit of u at = R is exactly, othewise we could continue u to the ight beyond = R, contadicting the maximality of the inteval [0, R). Poof of Coollay 2 of the Intoduction. To pove the coollay, fist we have to apply Theoem 1 in [4] with N = 1, δ() = (n 1)/, ψ(u, v) = u γ u / 1 + v 2, Q(, u, v) = δ()ψ(u, v) in ode to obtain that (1.9) admits a local adial solution fo evey u 0 > 0. Futhemoe, to get the claim, it is enough to use Theoem 4 with g(u) = u γ and H(s) = 1 1/ 1 + s 2. Remak. Theoem 4 continue to be valid if we eplace assumption (4.10) by hypotheses (4.14) (4.16). In this setting, in the poof of Theoem 4, we deive the monotonicity of the solution thanks to Lemma 2 instead of Lemma 1. In this case, namely when (4.5) does not hold, we need to use a compaison theoem due to Gilbag Tudinge which equies much egulaity on A with espect to (A1). Theoem 5. (Theoem 10.7 (ii), [7]) Let Ω R n be a bounded domain. Let u and v be espective C 1 ( Ω) distibution solutions of the diffeential inequalities div ( A(x, Du) ) B(x, u, Du) 0, u 0, (4.24) div ( A(x, Dv) ) B(x, v, Dv) 0, v 0, (4.25) in a bounded domain Ω of R n, n 2. Assume that A(x, ξ) and B(x, z, ξ) ae continuously diffeentiable with espect to the z, ξ vaiables in Ω R R n, and that B(x, z, ξ) is non-deceasing in the vaiable z fo fixed (x, ξ) Ω R n. Then if u and v ae continuous in Ω with v u on Ω, then v u in Ω. Remak. The above theoem 5 has been ecently genealized by Pucci and Sein in [19], in paticula they equie u C 1 (Ω) athe than u C 1 (Ω) and they conside the case in which the opeato A(x, ξ), ξ R n \ {0}, can be singula (degeneate) at ξ = 0. Theoem 6. Let p > 1 and γ such that 0 < γ < p 1. Let q C(R + 0 ) be a non negative function such that q() > 0 fo all > 0. (4.26) If (4.17) holds, then thee exists a positive solution u of div ( u γ A( Du )Du ) γu γ 1 A( Du ) q( x )u p 1, x R n, (4.27) then thee exists a positive solution v of { [ n 1 v γ A( v )v ] = [ γv γ 1 A( v ) + q()v p 1] n 1, R +, v(0) > 0, v (0) = 0. (4.28)

13 ENTIRE SOLUTIONS 13 Remak. Fist, we put in evidence that Theoem 6 above still holds without assuming (4.17), fo this pupose see the poof of Theoem 7 in [4]. Anyway assumption (4.17) allows us to simplify the poof since we can apply Theoem 4. In paticula, accoding also to the Remak at the end of Theoem 4, we have that Theoem 4 can be applied with (4.10) eplaced by (4.14) (4.16), indeed (4.26) foces the validity of (4.14) and (4.15), while (4.16) holds since g(u) = u γ with γ > 0. Thus, by Lemma 2 we obtain that evey solution of (4.28) is stictly inceasing and so v() v(0) > 0 fo all 0. Poof. Let u be a positive entie solution of (4.27) such that u(0) > 0. By Theoem 1 of [4], poblem (4.28) admits a local solution v fo all a = v(0) 0. Assume by contadiction that v is a non global solution of (4.28) defined in its maximal inteval of existence [0, R), R <. Futhemoe suppose that 0 < v(0) = a < u(0). Then, by the above Remak we can apply Theoem 4 obtaining that v is bounded and lim R v () =. Futhemoe we point out that v is a adial solution of div (A( Dv )Dv) = γ v H( Dv ) + q( x )vp 1 γ in B R, (4.29) whee B R = {x R n : x < R}. As in the poof of Poposition 1 of [11] we have two possibilities. Case 1: Suppose thee exists R 1 (0, R) so that v(r 1 ) max{u(x) : x B 1 }, B 1 = {x R n : x < R 1 }. (4.30) Then, by (4.29), the function v = v( x ) is such that whee div (A( Dv )Dv) = B(x, v, Dv) in B 1, (4.31) B(x, z, ξ) = γ z H( ξ ) + q( x )zp 1 γ. Thus B is non deceasing in the vaiable z thanks to the fact that 0 < γ < p 1. Futhemoe, by the above Remak, the set {v(x) : x B 1 } is contained in a compact set which do not contain zeo, namely {v(x) : x B 1 } [v(0), max x =R 1 v(x)] = J. Consequently, Theoem 5 can be applied with A(ξ) = A( ξ )ξ and B defined in the domain B 1 J R n whee the boundness of the integals in the poof of Theoem 10.7 (ii) of [7] is ealized (fo details see the poof of [7]). Thus, since v u on B 1 by (4.30) it follows that v u in B 1 which contadicts the fact that v(0) < u(0). Case 2: If (4.30) fails fo all R 1 (0, R), then v() < max{u(x) : x = } fo all 0 < < R. Since v () as R, thee is R 2 (0, R) such that v ν (R 2) > max { u ν (x) : x B 2 whee ν is the unit oute nomal of B 2. Define now }, B 2 = {x R n : x < R 2 }, (4.32) δ := max{u(x) v(x) : x B 2 } and w(x) := v(x) + δ. Then w(x) u(x) fo all x B 2 and fo some x, with x B 2, it esults w(x ) = u(x ). We claim now that w(x 0 ) < u(x 0 ) fo some x 0, with x 0 B 2. (4.33)

14 14 R. FILIPPUCCI Assume fo contadiction that (4.33) does not hold. Consequently we have that w u 0 in B 2 and (w u)(x ) = 0 fo some x B 2. This tivially implies that necessaily that is by the definition of w w ν (x ) u ν (x ) 0, v ν (R 2) u ν (x ) fo some x B 2. Of couse this fact contadicts (4.32) and poves the claim (4.33). Finally, to finish the poof of the existence of a solution of (4.28) in Case 2, we obseve that by (4.31) and since Dw = Dv by the definition of w we have div (A( Dw )Dw) = div (A( Dv )Dv) = B(x, v, Dv) B(x, w, Dw) in B 2, whee in the last inequality we have used that B is non deceasing in the second vaiable. As in Case 1, it esults that w u on B 2. Then by Theoem 5 applied with Ω = B 2, we obtain w u in B 2 which contadicts (4.33) and completes the poof of the theoem. Fo the geneal inequality (4.1), we have the analogous esult of Theoem 6, namely Theoem 7. Assume that (4.26) holds. Suppose that the functions g (u)/g(u) and f( x, u) ae non deceasing in u fo all x R n. If thee exists a positive solution u of (4.1), then thee exists a positive solution v of { [ n 1 g(u)a( v )v ] = [ g (u)a( v ) + f(, v) ] n 1, R +, v(0) > 0, v (0) = 0. Theoem 8. Assume that p > 1 and 0 < γ < p 1. Let q C(R + 0 ) C1 (R + ) be such that (4.26) hold and in addition q () 0 fo > 0. (4.34) If (4.17) holds, then inequality (4.27) does not admit positive entie solutions. Poof. Aguing by contadiction, denote by u a positive entie solution of (4.27). Then, by Theoem 6, thee exists a positive global solution of (4.28). On the othe hand, since we ae in the scala case, Theoem 1 can be applied with (A4) and F (0, v(0)) = q(0)v p 0 /p > 0 eplaced by (4.26). Thus Theoem 1 holds with f( x, u) = q( x )u p 1, c 1 = q( 0 ) by the monotonicity of q, g(u) = u γ if u 0, c 2 = 1, c 3 = lim s sa(s) by (A1) and with (A3) veified thanks to (4.34). In tun we obtain that no global solutions of (4.28) can exist. This contadiction concludes the poof of the theoem. Poof of Coollay 3 of the Intoduction. In the case (i) non existence is given by Coollay 4, while in the case (ii) non existence follows fom Theoem 8. Futhemoe, we can extend Theoem 1 of [11] with f(u) = u p 1 fo u > 0, indeed it holds the following Coollay 5. Let p > 1 and γ such that γ < p 1. The diffeential inequality in R n div ( u γ A( Du )Du ) γu γ 1 A( Du ) u p 1, u > 0, does not admit entie positive solutions if (4.17) holds.

15 ENTIRE SOLUTIONS 15 Coollay 6. Assume that (4.17) holds. Let q satisfy (4.34). Then inequality (4.9) does not admit positive entie solutions fo evey γ < p 1. Poof. When γ 0 the conclusion of the coollay is a consequence of Coollay 4 since (4.34) foces the validity of (4.6). If 0 < γ < p 1 the esult is due to Theoem 8. Refeences [1] U. Diekes and G. Huisken, The n-dimensional analogue of the catenay: existence and non-existence, Pacific J. Math., 141 (1990), [2] U. Diekes, Minimal sufaces in singula spaces, Pog. Math., Bikh ause Velag Basel, 168 (1998), [3] U. Diekes, Singula minimal sufaces, Geometic analysis and nonlinea patial diffeential equations, Spinge, Belin, 141 (2003), [4] R. Filippucci, Nonexistence of adial entie solutions of elliptic systems, J. Diff. Equations, 188 (2003), [5] R. Filippucci, Existence of global solutions of elliptic systems, J. Math. Anal. Appl., 293 (2004), [6] M. Ghegu and V. Radulescu, Existence and nonexistence of entie solutions to the logistic diffeential equation, Abst. and Appl. Anal., 17 (2003), [7] D. Gilbag and N. Tudinge, Elliptic Patial Diffeential Equations of Second Ode, 2nd edition, Spinge-Velag, [8] J. B. Kelle, On solutions of u = f(u), Comm. Pue Appl. Math., 10 (1957), [9] G. Leoni, Existence of solutions fo stongly degeneate diffeential systems, Calc. of Va. and P.D.E., 5 (1997), [10] H. A. Levine and J. Sein, Global nonexistence theoems fo quasilinea evolutions equations with dissipation, Ach. Rat. Mech. Anal., 137 (1997), [11] Y. Naito and H. Usami, Entie solutions of the inequality div(a( Du )Du) f(u), Math. Z., 225 (1997), [12] Y. Naito and H. Usami, Nonexistence esults of positive entie solutions fo quasilinea elliptic inequalities, Canad. Math. Bull., 40 (2) (1997), [13] R. Osseman, On the inequality u f(u), Pacific J. Math., 7 (1957), [14] P. Pucci and J. Sein, Pecise damping conditions fo global asymptotic stability fo nonlinea second ode systems, Acta Math., 170 (1993), [15] P. Pucci and J. Sein, Continuation and limit behavio fo damped quasi vaiational systems, Poc. Confeence on Degeneate Diffusions, IMA Vol 47 in Math. Appl. Spinge, W. M. Ni, L.A. Peletie, J.L. Vazquez, eds (1994). [16] P. Pucci and J. Sein, On the deivation of Hamilton s equations, Achive Rat. Mech. Anal., 125 (1994), [17] P. Pucci, J. Sein and H. Zou, A stong maximum pinciple and a compact suppot pinciple fo singula elliptic inequalities, J. Math. Pues Appl., 78 (1999), [18] P. Pucci and J. Sein, The stong maximum pinciple evisited, J. Diff. Equations, 196 (2004), 1 66, eatum, J. Diff. Equations, 207 (2004), [19] P. Pucci, J. Sein, The Maximum Pinciple, book in pepaation, 172 pages. [20] M. Stuwe, Quasilinea elliptic eigenvalue poblems, Comment. Math. Helvetici, 58 (1983), [21] H. Usami, Nonexistence of positive entie solutions fo elliptic inequalities of the mean cuvatue type, J. Diff. Equations, 111 (1994), Dipatimento di Matematica e Infomatica, Univesità degli Studi di Peugia, Via Vanvitelli 1, Peugia, Italy office , fax addess: obeta@dipmat.unipg.it