Existence of entire positive k -convex radial solutions to Hessian equations and systems with weights

Linear Algebra and its Applications 466 (2015) 102–116 Applied Mathematics Letters 50 (2015) 48–55

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Linear Algebra and its Applications Applied Mathematics Letters www.elsevier.com/locate/laa www.elsevier.com/locate/aml

Inverse eigenvalue problem Jacobi matrix Existence of entire positive k-convex radialof solutions to Hessian mixed data equations and with systems with weights YingZhou Wei 1 Zhijun Zhang ∗ , Song School of Mathematics Department and Information Science, Yantai University, Yantai 264005, Shandong, PR China of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

article

abstract i n f o

info a r t i c l e

a b s t r a c t

Article history: Under the simple conditions on f and g, we show that entire positive k-convex radial 2 u)) = problem Received 5 April 2015 Article history: this paper, the inverse eigenvalue of reconstructing solutions exist for theInHessian equation σk (λ(D p(|x|)f (u), x ∈ RN and Received in revised form 15 May 16 January system 2 v))its N , where Received 2014 σ (λ(D 2 u)) =ap(|x|)f Jacobi matrix from eigenvalues, its leading principal (v), σ (λ(D = q(|x|)g(u), x ∈ R p, q : k k 2015 Accepted 20 September 2014 submatrix and part of the eigenvalues of its submatrix [0, ∞) → (0, ∞) are continuous. Accepted 15 May 2015 Available online 22 October 2014 is considered. The necessary and sufficient conditions for © 2015 Elsevier Ltd. All rights reserved. Available online 12 JuneSubmitted 2015 by Y. Wei

the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given. © 2014 Published by Elsevier Inc.

MSC: 15A18 15A57

Keywords: Hessian equations Systems Entire solutions k-convex radial solutions Keywords: Existence Jacobi matrix Eigenvalue Inverse problem Submatrix

1. Introduction 2

∂ u(x) For any N × N real symmetric matrix A, we let λ(A) denote the eigenvalues of A, D2 u(x) = ( ∂x ) i ∂xj 2 N denotes the Hessian of u ∈ C (R ). The purpose of this paper is to investigate the existence of entire positive k-convex radial solutions to the following Hessian equation

σk (λ(D2 u)) = p(|x|)f (u(x)),

x ∈ RN ,

σk (λ(D2 u)) = p(|x|)f (v),

x ∈ RN ,

σk (λ(D2 v)) = q(|x|)g(u),

x ∈ RN ,

(1.1)

and system 

E-mail address: [email protected].

1 where, k = 1, 2, . . . , N ,Tel.: and+86 13914485239.  http://dx.doi.org/10.1016/j.laa.2014.09.031 σk (λ) = λi1 · · · λik ,

0024-3795/© 2014 Published by Elsevier Inc. 1≤i1 <···
λ = (λ1 , . . . , λN ) ∈ RN ,

denotes the kth elementary symmetric function. ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Zhang).

http://dx.doi.org/10.1016/j.aml.2015.05.018 0893-9659/© 2015 Elsevier Ltd. All rights reserved.

(1.2)

(1.3)

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Z. Zhang, S. Zhou / Applied Mathematics Letters 50 (2015) 48–55

Denote Γk := {λ ∈ RN : σj (λ) > 0, 1 ≤ j ≤ k}.

(1.4)

It is easy to see that σ1 (λ(D2 u(x))) =

N 

λi = ∆u;

i=1

σN (λ(D2 u(x))) =

N 

λi = det (D2 u).

i=1 2

We call a function u ∈ C (R ) k-convex in R if λ(D2 u(x)) ∈ Γk for all x ∈ RN . We assume that p, q, f and g satisfy the following hypotheses. N

N

(S1 ) p, q : [0, ∞) → (0, ∞) are continuous; (S2 ) f, g : [0, ∞) → [0, ∞) are continuous and increasing. Denote C0 =

(N − 1)! , k!(N − k)! r  k−N

 P (∞) := lim P (r), r→∞

P (r) :=

 Q(∞) := lim Q(r), r→∞

Q(r) :=

C0

0

0

t



t

0

r  k−N



t

C0

0

t

1/k sN −1 p(s)ds dt,

r ≥ 0;

(1.5)

1/k sN −1 q(s)ds dt,

r ≥ 0,

(1.6)

and, for an arbitrary a > 0,  H1a (∞) := lim H1a (r),

r

H1a (r) :=

r→∞

a

 H2a (∞) := lim H2a (r),

H2a (r) :=

r→∞

a

r

dτ , (f (τ ))1/k

r ≥ a;

dτ , (f (τ ) + g(τ ))1/k

(1.7) r ≥ a.

(1.8)

We see that ′ H1a (r) =

1 > 0, (f (r))1/k

′ H2a (r) =

1 > 0, (f (r) + g(r))1/k

∀r > a,

−1 −1 and H1a , H2a have the inverse functions H1a and H2a on [0, H1a (∞)) and [0, H2a (∞)), respectively. First, let us review the following model

△u = p(|x|)f (u),

x ∈ RN .

(1.9)

For p(|x|) ≡ 1 on RN : when f satisfies (S2 ), Keller [1] and Osserman [2] first supplied a necessary and sufficient condition  ∞  t dt  = ∞, F (t) = f (s)ds, (1.10) 2F (t) 1 0 for the existence of entire large positive radial solutions (or subsolutions) to problem (1.9). For N ≥ 3, f (u) = uγ , γ ∈ (0, 1], and p : [0, ∞) → [0, ∞) is continuous, Lair and Wood [3] first showed that (1.9) has infinitely many entire large positive radial solutions if and only if  ∞ rp(r)dr = ∞. (1.11) 0

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Z. Zhang, S. Zhou / Applied Mathematics Letters 50 (2015) 48–55

The result has been extended by many authors and in many contexts, see, for instance, [4–8] and the references therein. Next let us review the system  ∆u = p(|x|)f (v), x ∈ RN , (1.12) ∆v = q(|x|)g(u), x ∈ RN . When N ≥ 3, f (v) = v α , g(u) = uγ , 0 < α ≤ γ, Lair and Wood [9] have considered the existence and nonexistence of entire positive radial solutions to (1.12). For the further results, see, for instance, [10–14,7,15] and the references therein. Subsequently, let us review the following boundary blow-up problem det (D2 u(x)) = b(x)f (u(x)),

x ∈ Ω , u|∂Ω = +∞,

(1.13)

where the last condition means that u(x) → +∞ as d(x) = dist(x, ∂Ω ) → 0, Ω is a strictly convex, bounded smooth domain in RN with N ≥ 2. Such problems arise in Riemannian geometry and have been considered by Cheng and Yau [16,17], and ¯ ) with b > 0 on Ω ¯ and f (u) = exp(u) or f (u) = up with p > N , they Lazer and McKenna [18] for b ∈ C ∞ (Ω showed that problem (1.13) has a unique solution u ∈ C ∞ (Ω ). Moreover, Lazer and McKenna obtained that (i) when f (u) = up with p > N , it holds c1 (d(x))−(N +1)/(p−N ) ≤ u(x) ≤ c2 (d(x))−(N +1)/(p−N ) ,

x ∈ Ω,

where c1 and c2 are positive constants; (ii) if f (u) = up with p ∈ (0, N ], then there does not exist a solution to problem (1.13); (iii) when f (u) = exp(u), u satisfies |u(x) − ln((d(x))−(N +1) )| is bounded on Ω .

The results have been extended by many authors and in many contexts, see, for instance, [19–25] and the references therein. Now let us return to (1.1). For p ≡ 1 on [0, ∞): when f (u) = uγk , γ > 1, Jin, Li and Xu [26] proved that (1.1) has no entire k-convex positive solutions. Then X. Ji and J. Bao [27] showed the following results. Lemma 1.1. If f is a continuous function defined on R and satisfies (S02 ) f (s) > 0 is monotone non-decreasing in (0, ∞), and f (s) = 0 in (−∞, 0],

then (1.1) has an entire positive k-convex radial solution if and only if  ∞ dt = ∞. 1/(k+1) (F (t)) 1

(1.14)

Z. Zhang, S. Zhou / Applied Mathematics Letters 50 (2015) 48–55

51

With regard to the other works of entire solutions to the Monge–Amp´ere equation or Hessian equation, see, for instance, [28–31] and the references therein. For the following system  2  det (D u) = f (−v), in B, (1.15) det (D2 v) = g(−u), in B,  u = v = 0, on ∂B, where B is a unit ball in RN . H. Wang [32] first showed the existence of convex radial solutions under some proper conditions on f and g. Then F. Wang and Y. An [33] proved the existence of at least three convex radial solutions to (1.15). Inspired by the above works, in this paper, by using a monotone iterative method and Arzela–Ascoli theorem, we show existence of entire positive k-convex radial solutions to (1.1) and (1.2) under simple conditions on p, q, f and g. Our main results are as follows. Theorem 1.1. Under the hypotheses (S1 )–(S2 ) and (S3 ) H1a (∞) = ∞, (1.1) has one entire positive k-convex radial solution u ∈ C 2 (RN ). Moreover, when P (∞) < ∞, u is bounded, and when P (∞) = ∞, limr→∞ u(r) = ∞. Theorem 1.2. Under the hypotheses (S1 )–(S2 ) and (S4 ) P (∞) < H1a (∞) < ∞, (1.1) has one entire positive bounded k-convex radial solution u ∈ C 2 (RN ) satisfying −1 a + (f (a))1/k P (r) ≤ u(r) ≤ H1a (P (r)) ,

∀r ≥ 0,

where a is given in (1.7). 1 Remark 1.1. When 0 (f (τdτ))1/k = ∞, one can see that there is a > 0 sufficiently small such that (S4 ) holds provided P (∞) < ∞ and H1a (∞) < ∞, where a is given as in (1.7). Theorem 1.3. Under the hypotheses (S1 )–(S2 ) and (S5 ) H2a (∞) = ∞, system (1.2) has one entire positive k-convex radial solution (u, v) in C 2 (RN ) × C 2 (RN ). Moreover, when P (∞) + Q(∞) < ∞, u and v are bounded; when P (∞) = ∞ = Q(∞), limr→∞ u(r) = limr→∞ v(r) = ∞. Theorem 1.4. Under the hypotheses (S1 )–(S2 ) and (S6 ) P (∞) + Q(∞) < H2a (∞) < ∞,

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system (1.2) has one entire positive bounded k-convex radial solution (u, v) in C 2 (RN ) × C 2 (RN ) satisfying −1 a/2 + (f (a/2))1/k P (r) ≤ u(r) ≤ H2a (P (r) + Q(r)) , 1/k

a/2 + (g(a/2))

Q(r) ≤ v(r) ≤

−1 H2a

(P (r) + Q(r)) ,

∀r ≥ 0; ∀r ≥ 0,

where a is given in (1.8). Remark 1.2. By using a similar proof, one can see that Theorems 1.3 and 1.4 can be extended to the more general system  σk (λ(D2 u)) = p(|x|)f1 (v)f2 (u), x ∈ RN , σk (λ(D2 v)) = q(|x|)g1 (v)g2 (u),

x ∈ RN ,

where fi , gi (i = 1, 2) satisfy (S2 ).

2. Proof of Theorems 1.1 and 1.2 In the section we prove Theorems 1.1–1.2. For R ∈ (0, ∞], let BR := {x ∈ RN : |x| < R}. Lemma 2.1 (Lemma 2.1, [27]). Assume ψ ∈ C 2 [0, R) with ψ(0) = 0. Then for u(x) = ψ(r), we have that u ∈ C 2 (BR ), and  ′ ′ (ψ ′′ (r), ψ (r) , . . . , ψ (r) ), r ∈ (0, R), r r λ(D2 u) =  ′′ (ψ (0), ψ ′′ (0), . . . , ψ ′′ (0)), r = 0,  ′ ′ C k−1 ψ ′′ (r) ψ (r) k−1 + C k  ψ (r) k , r ∈ (0, R), N −1 2 N −1 r r σk ((D u)) =  k ′′ CN (ψ (0))k , r = 0, k = where CN

N! k!(N −k)! .

Lemma 2.2 (Lemma 2.2, [27]). Under the hypothesis (S2 ), for any positive number a, assume ψ ∈ C[0, R) ∩ C 1 (0, R) is a solution to the Cauchy problem   rk−N r N −1 1/k ψ ′ (r) = s f (ψ(s))ds , r > 0, ψ(0) = a > 0. (2.1) C0 0 Then ψ ∈ C 2 [0, R), and it satisfies the following equation  ψ ′ (r) k−1  ψ ′ (r) k k−1 ′′ k CN ψ (r) + C = f (ψ), N −1 −1 r r

r > 0,

(2.2)

with ψ ′ (0) = 0 and  λr :=

ψ ′′ (r),

ψ ′ (r) ψ ′ (r) ,..., r r

 ∈ Γk

for r ∈ [0, R).

Since p is continuous with p(|x|) > 0, ∀x ∈ RN , by using the analogous proof of Lemma 2.2, we have Lemma 2.3. Under the hypotheses (S1 ) and (S2 ), Lemma 2.2 continues to hold if f (u) is replaced by p(|x|)f (u).

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Thus we consider the equations  u′ (r) k  u′ (r) k−1 k + CN = p(r)f (u), r > 0, −1 r r 1/k  k−N  r r sN −1 p(s)f (u(s))ds , r > 0, u(0) = a, u′ (r) = C0 0 k−1 ′′ CN −1 u (r)

(2.3) (2.4)

and r

 u(r) = a +



0

tk−N C0



t

0

1/k sN −1 p(s)f (u(s))ds dt,

r ≥ 0.

(2.5)

Note that solutions in C[0, ∞) to (2.5) are solutions in C[0, ∞) ∩ C 1 (0, ∞) to (2.4). Let {um }m≥1 be the sequences of positive continuous functions defined on [0, ∞) by r

 u0 (r) = a, um (r) = a +



0

tk−N C0

 0

t

1/k dt, sN −1 p(s)f (um−1 (s))ds

r ≥ 0.

(2.6)

Obviously, for all r ≥ 0 and m ∈ N, um (r) ≥ a, and u0 ≤ u1 . Then (S2 ) yields u1 (r) ≤ u2 (r), ∀r ≥ 0. Continuing this line of reasoning, we obtain that the sequences {um } is non-decreasing on [0, ∞). Moreover, we obtain by (S1 ) and (S2 ) that for each r > 0  k−N  r 1/k r ′ N −1 um (r) = s p(s)f (um−1 (s))ds ≤ (f (um (r)))1/k P ′ (r), C0 0 and 

um (r)

a

dτ ≤ P (r). (f (τ ))1/k

Consequently, H1a (um (r)) ≤ P (r),

∀r ≥ 0.

(2.7)

(i) When (S3 ) holds, we see that −1 H1a (∞) = ∞

−1 and um (r) ≤ H1a (P (r)) ,

∀r ≥ 0.

(2.8)

It follows that the sequences {um } and {u′m } are bounded on [0, R0 ] for an arbitrary R0 > 0. By Arzela–Ascoli theorem, {um } has subsequences converging uniformly to u on [0, R0 ]. Since {um } is non-decreasing on [0, ∞), we see that {um } itself converges uniformly to u on [0, R0 ]. By the arbitrariness of R0 and Lemma 2.3, we see that u is entire positive k-convex solution to (1.1). Moreover, when P (∞) < ∞, we see by (2.8) that −1 u(r) ≤ H1a (P (∞)) ,

∀r ≥ 0;

when P (∞) = ∞, by (S2 ) and the monotone of {um } u(r) ≥ a + (f (a))1/k P (r),

∀r ≥ 0.

Thus limr→∞ u(r) = ∞. (ii) When (S4 ) holds, we see by (2.7) that H1a (um (r)) ≤ P (∞) < H1a (∞) < ∞.

(2.9)

−1 Since H1a is strictly increasing on [0, H1a (∞)), we have −1 um (r) ≤ H1a (P (∞)) < ∞,

∀r ≥ 0.

The last part of the proof follows from (i). The proof is finished.

(2.10)

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Z. Zhang, S. Zhou / Applied Mathematics Letters 50 (2015) 48–55

3. Proof of Theorems 1.3 and 1.4 In the section we prove Theorems 1.3–1.4. Consider the systems   u′ (r) k  u′ (r) k−1 k−1 ′′ k  CN + CN = p(r)f (v), r > 0, −1 −1 u (r) r r ′ ′  C k−1 v ′′ (r) v (r) k−1 + C k  v (r) k = q(r)g(u), r > 0, N −1 N −1 r r and  r k−N  t  t 1/k   dt, sN −1 p(s)f (v(s))ds u(r) = a/2 + C 0  0r k−N  0t t 1/k   v(r) = a/2 + sN −1 q(s)g(u(s))ds dt, C0 0 0

r ≥ 0, r ≥ 0.

Let {um }m≥1 and {vm }m≥0 be the sequences of positive continuous functions defined on [0, ∞) by  v0 (r) = a/2,    r k−N  t   t 1/k  um (r) = a/2 + sN −1 p(s)f (vm−1 (s))ds dt, r ≥ 0, C0 0 0   r k−N  t   1/k t  vm (t) = a/2 + sN −1 q(s)g(um (s))ds dt, r ≥ 0. C0 0 0 Obviously, for all r ≥ 0 and m ∈ N, um (r) ≥ a/2, vm (r) ≥ a/2 and v0 ≤ v1 . (S2 ) yields u1 (r) ≤ u2 (r), ∀r ≥ 0, then v1 (r) ≤ v2 (r), ∀r ≥ 0. Continuing this line of reasoning, we obtain that the sequences {um } and {vm } are increasing on [0, ∞). Moreover, we obtain by (S1 ) and (S2 ) that for each r > 0   rk−N r N −1 1/k u′m (r) = s p(s)f (vm−1 (s))ds C0 0 ≤ (f (vm (r)))1/k P ′ (r)  1/k ′ ≤ f (vm (r) + um (r)) + g(vm (r) + um (r)) P (r);  r k−N 1/k r ′ sN −1 q(s)g(um (s))ds vm (r) = C0 0 ≤ (g(um (r)))1/k Q′ (r)  1/k ′ ≤ f (vm (r) + um (r)) + g(vm (r) + um (r)) Q (r), and  1/k ′ ′ u′m (r) + vm (r) ≤ f (vm (r) + um (r)) + g(vm (r) + um (r)) (P (r) + Q′ (r)). Consequently,  a

um (r)+vm (r)

dτ ≤ Q(r) + P (r), (f (τ ) + g(τ ))1/k

r > 0,

and H2a (um (r) + vm (r)) ≤ P (r) + Q(r),

∀r ≥ 0.

The remaining proof is similar to that for Theorems 1.1 and 1.2. Here we omit their proof.

(3.1)

Z. Zhang, S. Zhou / Applied Mathematics Letters 50 (2015) 48–55

55

Acknowledgments This work is supported in part by NNSF of P. R. China under grant 11301301. The authors are greatly indebted to the anonymous referees for the very valuable suggestions and comments which surely improved the quality of the presentation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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