Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms

(1)

Volume 2012, Article ID 697565,15pages doi:10.1155/2012/697565

Research Article

Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms

Yongju Yang

¹

and Xinguang Zhang

²

1School of Mathematics and Statistics, Nanyang Normal University, Henan, Nanyang 473061, China

2School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China

Correspondence should be addressed to Xinguang Zhang,[email protected] Received 1 September 2012; Accepted 8 November 2012

Academic Editor: Yong Hong Wu

Copyrightq2012 Y. Yang and X. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms, Δui |∇ui|² pi|x|fiu1, u2, . . . , ud for i 1,2, . . . , d on R^N, N ≥ 3 and d∈ {1,2,3, . . .}. We establish the conditions onpithat ensure the existence of nonnegative radial solutions blowing up at infinity and also the conditions for bounded solutions on the entire space.

The condition onfiis simple and diﬀerent to the Keller-Osserman condition.

1. Introduction

We study the existence of entire blow-up positive solutions of the following elliptic system with quadratic gradient terms:

Δu1|∇u1|²p₁|x|f1u1, u₂, . . . , u_d, x∈R^N, ...

...

Δud|∇ud|²pd|x|fdu1, u2, . . . , ud, x∈R^N,

1.1

whered≥1,N≥3,pi i1,2, . . . , darec-positive functions andfi :0,∞^d → 0,∞are nonnegative, continuous, and nondecreasing functions for each variable.

(2)

For convenience we recall the definitions aboutc-positive functions and entire blow- up positive solutions.

iA functionpisc-positiveor circumferentially positivein a domainΩ⊆ R^Nifp is nonnegative onΩand satisfies the following condition: ifx₀ ∈Ωandpx0 0, then there exists a domainΩ0such thatx0∈Ω0⊂Ωandpx>0 for allx∈∂Ω0. iiA solutionu1, u2, . . . , udof the system1.1is called an entire blow-up solution

or explosive solutionif it is a classical solution of the above problem onR^Nand u_ix → ∞,i1,2, . . . , das|x| → ∞.

Existence and nonexistence of blow-up solutions of semilinear elliptic equations and systems have received much attention worldwide. Bieberbach1is the first to study blow-up solutions to the semilinear elliptic problem

Δufu, x∈Ω, 1.2

wherefu e^u. Following Bieberbach’s work, many authors have studied related problems for single equations and systems. In 1957, Keller 2 and Osserman 3 established the necessary and suﬃcient conditions for the existence of solutions to1.2on bounded domains inRⁿ. They showed that blow-up solutions exist onΩif and only iffsatisfies the following Keller-Osserman condition:

_∞

1

_t

0

fsds _−1/2

dt <∞. 1.3

Bandle and Marcus4later examined the equation

Δupxfu 1.4

withf is nondecreasing on0,∞and proved the existence of positive blow-up solutions under the condition that the functionf satisfies the Keller-Osserman condition1.3andp is continuous and strictly positive onΩ. Lair5showed that the results also hold for1.4 whenpis allowed to vanish on a large part ofΩ, including its boundary. In addition, many authors have examined some more specific forms of1.4. The equation

Δupxu^γ 1.5

has been of particular interest. Cheng and Ni6considered the superlinear caseγ >1 and proved that for this case 1.5 has blow-up solutions on bounded domains provided p is strictly positive on∂Ω. Lair and Wood7 generalized this to allow p to vanish on some portions of Ωincluding its boundary and also showed the existence of an entire blow-up solution to1.5provided that

_∞

0

rmax|x|r prdr <∞. 1.6

Obviously, condition1.6is weaker than the requirements in6.

(3)

In8, Lair and Wood proved that1.5has entire blow-up radial solutions if and only if

_∞

0

rprdr <∞. 1.7

They also demonstrated that for a bounded domainΩ,1.5has no positive blow-up solution when p is continuous in Ω. In addition, they proved that nonnegative, entire bounded solutions do not exist for1.5if

_∞

0

rmin|x|rprdr ∞. 1.8

Although semilinear elliptic systems are the natural extension of single equations in many areas of applications, the results and methods for the study of single equations are often not applicable to the systems of equations. Recently, Lair and Wood9studied the existence of entire positive solutions of the system

Δup1|x|v^α, x∈R^N,

Δvp₂|x|u^β, x∈R^N. 1.9

In the sublinear case 0 < α ≤ β ≤ 1, the authors proved that provided that the nonnegative functionspandqare continuous,c-positive, and satisfy the fast decay conditions

_∞

0

tpitdt <∞, i1,2, F

then the entire positive solutions are bounded, while if p and q satisfy the slow decay conditions

_∞

0

tp_itdt∞, i1,2, D

then the entire positive solutions blow up. For the superlinear caseα, β > 1, the fast decay conditionsFare required to hold. Later, Cˆırstea and R˘adulescu10improved the results of Lair and Wood9and proved that forp, q∈C^0,α_locR^N0< α <1, the following semilinear elliptic system

Δup1xfv, x∈R^N,

Δvp₂xgu, x∈R^N 1.10

has entire solutions iffandgsatisfy

t→ ∞lim f

cgt

t 0 1.11

(4)

for allc >0 and has solutions that are bounded whenDholds. Further, entire solutions exist and are blow-up whenFholds. An analogous condition was also employed by Ghergu and R˘adulescu11to study the following elliptic system with gradient terms:

Δu|∇u|p₁|x|fv, x∈Ω,

Δv|∇v|p2|x|gu, x∈Ω, 1.12

where Ω is a bounded domain or the whole space. Peng and Song 12 also studied the existence of entire blow-up positive solutions of system1.10when thec-positive functions p_i, i1,2 satisfy the decay conditionsF. Peng and Song12also imposed onfandgthe following Keller-Osserman conditions:

_∞

1

_s

0

ftdt −1/2

ds <∞,

_∞

1

_s

0

gtdt −1/2

ds <∞, 1.13

and the convexity conditions

fλa 1−λb≤λfa 1−λfb, λ∈0,1,

gλa 1−λb≤λga 1−λgb, a, b≥0. 1.14

Both papers6,12considered system1.10where the nonnegative functionspi i1,2∈ C0,∞satisfy F and the functions f, g ∈ C0,∞ are nondecreasing and satisfy the Keller-Osserman condition1.13, and

f0 g0 0, fs>0, gs>0, fors >0. 1.15

Recently, Zhang and Liu13studied the following semilinear elliptic system with the magnitude of the gradient

Δu|∇u|p|x|fu, v, x∈R^N,

Δv|∇v|q|x|gu, v, x∈R^N. 1.16

The results of nonexistence of entire positive solutions have been established if f and g are sublinear andpandqhave fast decay at infinity, while iff andg satisfy some growth conditions at infinity, andp,qare of slow decay or fast decay at infinity, then the system

(5)

has infinitely many entire solutions, which are large or bounded. In14, Covei studied the existence of solution of the following semilinear elliptic system:

Δu1p1xf1u1, u2, . . . , ud, x∈R^N, ...

...

Δudpdxfdu1, u2, . . . , ud, x∈R^N.

1.17

Under some conditions onf_i,p_i, the system1.17 has a bounded positive entire solution based on successive approximation. Furthermore, a nonradially symmetric solution also was obtained by using a lower and upper solution method. For more complicated Schr ¨odinger systems, some nice work had been done by Covei in15–17with single equations or a system withp1, . . . , p_d-Laplacian inR^N. For further results on relevant work on single equations and/or systems as well as methods for the study of blow-up solutions of diﬀerential equations, see8,18–32and the references therein.

The authors in13,14only studied the semilinear elliptic system with the magnitude of the gradient term or without the gradient term. For elliptic systems involving nonlinear quadratic gradient terms, no result has been obtained. Thus, motivated by11–17, we study the more general systems case with indefinite number of equations involving a nonlinear quadratic gradient term. In our results, a simple condition 2.5has been used instead of the Keller-Osserman condition1.13commonly used in previous results. The main results obtained are presented inSection 2by Theorems2.3to2.6, while the proofs of the theorems are given inSection 3.

2. Main Results

For convenience in presenting the results, we here define

Pi∞ lim

r→ ∞Pir, Pir _r

0

t^1−N _t

0

s^N−1pisds dt, r ≥0, i1,2, . . . , d,

F∞ lim

r→ ∞Fr, Fr

_r

a

ds

di1sf_ilns,lns, . . . ,lns, r ≥a >1, i1,2, . . . , d.

2.1

Remark 2.1. For anyi∈ {1,2, . . . , d}, since

Fr 1

di1rfilnr,lnr, . . . ,lnr >0, ∀r > a, 2.2

F admits the inverse functionF⁻¹on0, F∞.

(6)

Lemma 2.2see8,23. The slow decay condition _∞

0

tpitdt∞, i1,2, . . . , d 2.3

holds if and only ifP_i∞ ∞.

The first result we obtained is the condition for nonexistence of entire positive blow- up solution, which asserts that if bothf_i, i1,2, . . . , dare bounded, then problem1.1does not have positive entire blow-up solution as detailed by the following theorem.

Theorem 2.3. Supposefi,i1,2, . . . , dsatisfy

max

⎧⎨

⎩ sup

di1ui≥1

f₁u1, u₂, . . . , u_d, . . . , sup

di1ui≥1

f_du1, u₂, . . . , u_d

⎫⎬

⎭<∞, 2.4

and eachp_i, i1,2, . . . , dsatisfy the decay conditionsF. Then problem1.1does not have positive entire blow-up solution.

The other main results we obtained are the conditions, respectively, for the existence of infinitely many positive entire blow-up solutions and infinitely many positive entire bounded solutions, which are summarized in the following three theorems.

Theorem 2.4. If there exists a constanta >1 such that _∞

a

ds

di1sf_ilns,lns, . . . ,lns ∞, 2.5

then the system1.1 has infinitely many classical positive entire solutionsu1, u2, . . . , ud. If, in addition,pi, i1,2, . . . , dsatisfy the decay conditionsD, then all the positive entire solutions of 1.1are blow-up. Moreover, ifp_i, i1,2, . . . , dsatisfy the decay conditionsF, then all the positive entire solutions of1.1are bounded.

Theorem 2.5. If there exists a constanta >1 such that _∞

a

ds

di1sfilns,lns, . . . ,lns <∞, 2.6 and p_i, i 1,2, . . . , d satisfy the decay conditionsF and, in addition, there exist b_i > a,i 1,2, . . . , dsuch that

d i1

Pi∞< F∞−F _d

i1

bi

, 2.7

then the system1.1has a positive radial bounded solutionu1, u₂, . . . , u_dsatisfying

bibifilnb1,lnb2, . . . ,lnbdPir≤uir≤F⁻¹

F _d

i1

bi

^d

i1

Pir

, r≥0. 2.8

(7)

Theorem 2.6.

iIfpi,i1,2, . . . , dsatisfy the decay conditionsDand

slim→ ∞

d i1

filns,lns, . . . ,lns 0, 2.9

then the system1.1has infinitely many positive entire blow-up solutions.

iiIfpi, i1,2, . . . , dsatisfy the decay conditionsFand

sup

s≥0

d i1

sf_ilns,lns, . . . ,lns<∞, 2.10

then the system1.1has infinitely many positive entire bounded solutions.

3. Proofs of the Theorems

Firstly, via the change of variablesφ_i e^uⁱ,i 1,2, . . . , d, we turn the system 1.1 to the following equivalent system with no gradient terms

Δφ1p1|x|φ1f1

lnφ1,lnφ2, . . . ,lnφd

, x∈R^N, ...

... Δφip_i|x|φif_i

lnφ₁,lnφ₂, . . . ,lnφ_d

, x∈R^N, ...

... Δφdp_d|x|φdf_d

lnφ₁,lnφ₂, . . . ,lnφ_d

, x∈R^N.

3.1

Thus we only need to consider system3.1.

Proof ofTheorem 2.3. We use proof by contradiction to testify. We suppose that the system3.1 has the positive entire blow-up solutionφ1, φ2, . . . , φd. Consider the spherical average ofφi

defined by

φ_ir 1

cNr^N−1

|x|rφixdσx, r ≥0, 3.2

wherecN is the surface area of the unit sphere inR^N. Sinceφi are positive entire blow-up solutions, it follows thatφ_i are positive and lim_r_{→ ∞}φ_ir ∞. By the change of variable xry, we have

φ_ir 1 cN

|y|1φi

ry

dσy, r≥0. 3.3

(8)

Then

φ_ir 1 c_N

|y|1∇φi

ry

·ydσ_y, r ≥0. 3.4

Thus by the divergence theorem and3.4, we have

φ_ir r c_N

|y|<1Δφi

ry

dy 1

cNr^N−1

|x|<rΔφixdx 1

cNr^N−1 _r

0

dρ

|x|ρΔφixdσx, ∀r ≥0.

3.5

From33, it follows from3.5that

φ_ir 1

c_Nr^N−1

|x|rΔφixdσx−N−1 c_Nr^N

_r

0

dρ

|x|ρΔφixdσx

1 c_Nr^N−1

|x|rΔφixdσx−N−1

r φ_ir, ∀r≥0.

3.6

Set

U_ir max

0≤t≤rφ_it, 3.7

Then, obviously,U_iare positive and nondecreasing functions. MoreoverU_i≥φ_iandU_ir →

∞asr → ∞. Note from2.4that there existsM >0 such that max

f₁u1, u₂, . . . , u_d, . . . , fdu1, u₂, . . . , u_d

≤M, u₁u₂· · ·u_d≥0. 3.8

Now3.6and3.8lead to

φ_i N−1

r φ_i≤ 1 c_Nr^N−1

|x|rΔφixdσx

p_ir 1 cNr^N−1

|x|rφ_ixfi

lnφ₁x,lnφ₂x, . . . ,lnφ_dx dσ_x

≤Mp_ir 1 c_Nr^N−1

|x|r

1^d

i1

φ_ix

dσ_x Mp_ir

1^d

i1

φ_ir

≤Mpir1Uir,

3.9

for allr≥0. It follows that

r^N−1φ_i

≤Mr^N−1p_ir1U_ir, r≥0. 3.10

(9)

So, for allr≥r0≥0, we have φ_ir≤φ_ir0 M

_r

r0

t^1−N _t

0

s^N−1pis1Uisds dt

≤φ_ir0 M _r

r0

t^1−N1Ut _t

0

s^N−1psds dt

≤φ_ir0 M1Uir _r

r0

t^1−N _t

0

s^N−1pisds dt

≤φ_ir0 M1Uir _r

r0

t^1−N _t

0

s^N−1pisds dt

≤φ_ir0 M1U_ir N−2

_r

r0

tpitdt− 1 r₀^N−1

_r

r0

t^N−1pitdt

≤φ_ir0 M1U_ir N−2

_r

r0

tp_itdt.

3.11

Note that because ofF, we can chooser0>0 suﬃciently large such that max

_∞

r0

rp1rdr, . . . , _∞

r0

rpdrdr

< N−2

4M . 3.12

Since limr→ ∞φ_it ∞, it follows that we can findr1 ≥r0such that Uir max

r0≤t≤rφ_it, ∀r≥r1. 3.13

Thus3.11and3.13yield

U_ir≤φ_ir0

M

1U_ir N−2

_r

r0

tp_itdt, ∀r≥r₁. 3.14

By3.12, we have

Uir≤φ_ir0 1Uir

4 , ∀r≥r1, 3.15

that is,

Uir≤CiU_ir

4 , ∀r≥r1, 3.16

whereCi 1/4 φ_ir0>0, which implies d

i1

Uir≤ 4 3

d i1

Ci, ∀r≥r1. 3.17

(10)

The inequality3.17means thatUi are bounded and soφ_i are bounded which is a contradiction. It follows that 1.1 has no positive entire blow-up solutions, and the proof is completed.

Proof ofTheorem 2.4. We start by showing that1.1has positive radial solutions. Towards this end we fixbi> a,i1,2, . . . , dand we show that the system

φ_iN−1

r φ_ip_irφirfi

lnφ₁r,lnφ₂r, . . . ,lnφ_dr

, i1,2, . . . , d φ_i≥0, on0,∞, φi0 b_i> a

3.18

has a solutionφ1, φ2, . . . , φd. ThusU1x, U2x, . . . , Udx φ1|x|, φ2|x|, . . . , φd|x|

are positive solutions of3.1. Integrating3.18, for anyr ≥0 andi1,2, . . . , d, we have

φ_ir b_i _r

0

t^1−N _t

0

s^N−1p_isφisfi

lnφ₁s,lnφ₂s, . . . ,lnφ_ds

ds dt. 3.19

Let{φ^k_i }_k≥0 be sequences of positive continuous functions defined on0,∞fori 1,2, . . . , dby

φ⁰_i r bi, φ^k1_i r b_i

_r

0

t^1−N _t

0

s^N−1p_isφ^k_i s

×f_i

lnφ^k₁ s,lnφ₂^ks, . . . ,lnφ^k_d s ds dt.

3.20

Obviously, for allr ≥ 0, we haveφ_i^kr≥b_i,φ₀ ≤φ₁. The monotonicity off_iyieldsφ₁r ≤ φ2r,r≥0. Repeating the argument, we deduce that

φ^k_i r≤φ_i^k1r, r≥0, k≥1, 3.21

which means{φ^k_i }_k≥0are nondecreasing sequences on0,∞. Since

φ^k1_i r r^1−N _r

0

s^N−1p_isφ^k_i sfi

lnφ₁^ks,lnφ^k₂ s, . . . ,lnφ_d^ks ds

≤φ^k_i rfi

lnφ^k₁ r,lnφ₂^kr, . . . ,lnφ^k_d r P_ir,

3.22

(11)

we have d

i1

φ^k1_i r≤^d

i1

φ_i^k1r^d

i1

fi

lnφ₁^kr,lnφ^k₂ r, . . . ,lnφ_d^kr Pir

≤^d

i1

φ_i^k1r^d

i1

fi

ln

d i1

φ^k1_i r,ln d

i1

φ^k1_i r, . . . ,ln d

i1

φ_i^k1r

×^d

i1

Pir.

3.23

Letw^k1r ^d_i1φ_i^k1rwhich implies

w^k1r w^k1r ^d_i1fi

lnw^k1r,lnw^k1r, . . . ,lnw^k1r ≤^d

i1

P_ir. 3.24

So, we have _r

0

w^k1t w^k1t ^d_i1fi

lnw^k1t,lnw^k1t, . . . ,lnw^k1tdt≤^d

i1

P_ir, 3.25

that is

F _d

i1

φ^k1_i r

−F _d

i1

bi

≤^d

i1

Pir, ∀r≥0. 3.26

AsF⁻¹increases on0,∞, from3.26, we have that d

i1

φ_i^k1r≤F⁻¹

F _d

i1

bi

^d

i1

Pir

, ∀r≥0. 3.27

It follows from F∞ ∞ thatF⁻¹∞ ∞. By3.27, the sequences{φ^k_i } are bounded and increasing on 0, c0 for any c0 > 0. Thus, {φ_i^k} have subsequences converging uniformly toφ_ion0, c0. Consequently,φ1, φ₂, , . . . , φ_dis a positive solution of3.18; that is, U1, U2, . . . , Udis a entire positive solution of3.1. By noticingφi0 biand thatbi∈0,∞ was chosen arbitrarily, it follows that1.1has infinitely many positive entire solutions.

iIfP_i∞ ∞, since

φ_ir≥b_ib_if_ilnb₁,lnb₂, . . . ,lnb_dPir, r≥0, 3.28

(12)

we have

rlim→ ∞φir ∞, 3.29

which means thatU1, U2, . . . , Udare positive entire blow-up solutions of1.1.

iiIfPi∞<∞, then d

i1

φir≤F⁻¹

F _d

i1

bi

^d

i1

Pi∞

<∞, 3.30

which implies thatU1, U2, . . . , Udare positive entire bounded solutions of1.1.

Proof of the theorem is now completed.

Proof ofTheorem 2.5. If condition2.7holds, then we have

F _d

i1

φ_i^k1r

≤F _d

i1

b_i

^d

i1

P_ir≤F _d

i1

b_i

^d

i1

P_i∞< F∞<∞. 3.31

SinceF⁻¹is strictly increasing on0,∞, we have d

i1

φ^k1_i r≤F⁻¹

F _d

i1

b_i

^d

i1

P_i∞

<∞. 3.32

The last part of the proof is clear from that ofTheorem 2.4. Thus we omit it.

Proof ofTheorem 2.6.iIt follows from3.20that

φ^k_i r≤φ^k1_i r≤b_iφ^k_i rfi

lnφ^k₁ r,lnφ^k₂ , . . . ,lnφ^k_d P_ir

≤biφ_i^krfi

ln

d i1

φ^k_i r,ln d

i1

φ^k_i r, . . . ,ln d

i1

φ^k_i r

Pir.

3.33

LetR >0 be arbitrary. From3.33we get, fork≥1, d

i1

φ^k_i R≤^d

i1

bi^d

i1

φ^k_i R

×^d

i1

fi

ln

d i1

φ_i^kR,ln d

i1

φ^k_i R, . . . ,ln d i1

φ_i^kR _d

i1

PiR.

3.34

(13)

This implies

1≤ ^dⁱ¹bi di1φ^k_i R

^d

i1

fi

ln

d i1

φ^k_i R,ln d

i1

φ^k_i R, . . . ,ln d

i1

φ^k_i R _d

i1

PiR.

3.35

Taking into account the monotonicity of ^d_i1φ^k_i R_k≥1, there exists

LR: lim

k→ ∞

_d

i1

φ_i^kR

. 3.36

We claim thatLRis finite. Indeed, if not, we letk → ∞in3.35, and the assumption2.9 leads us to a contradiction. ThusLRis finite. Sinceφ^k_i are increasing functions, it follows that the mapL:0,∞ → 0,∞is nondecreasing and

d i1

φ_i^kr≤^d

i1

φ^k_i R≤LR, ∀r∈0, R, ∀k≥1. 3.37

Thus the sequences{φ_i^k_k≥1}are bounded from above on bounded sets. Let φ_ir: lim

k→ ∞φ^k_i r, ∀r≥0. 3.38

Thenφ1, φ₂, . . . , φ_dis a positive solution of3.18.

In order to conclude the proof, it is suﬃcient to show thatφ1, φ2, . . . , φdis a blow-up solution of3.18. Let us remark that3.19implies

φ_ir≥b_ib_if_ilnb₁,lnb₂, . . . ,lnb_dPir, r≥0. 3.39

Sincef_iare positive functions and

Pi∞ ∞, 3.40

we can conclude thatφ1, φ2, . . . , φdis a blow-up solution of3.18, and soU1, U2, . . . , Ud is a positive entire blow-up solution of3.1. Thus any blow-up solution of3.1provides a positive entire blow-up solution of1.1. Sinceb_i ∈ 0,∞was chosen arbitrarily, it follows that1.1has infinitely many positive entire blow-up solutions.

iiIf

sup

s≥0

d i1

sfilns,lns, . . . ,lns<∞ 3.41

(14)

holds, then by3.35, we have

LR: lim

k→ ∞

d i1

φ^k_i R<∞. 3.42

Thus

d i1

φ_i^kr≤^d

i1

φ^k_i R≤LR, ∀r∈0, R, ∀k≥1. 3.43

So the sequences{φ^k_i }_k≥1are bounded from above on bounded sets. Let

φr: lim

k→ ∞φ_i^kr, ∀r ≥0. 3.44

Thenφ1, φ2, . . . , φdis a positive solution of3.18.

It follows from3.33and3.35thatφ1, φ₂, . . . , φ_dis bounded, which implies that 1.1has infinitely many positive entire bounded solutions.

In the end of this work we also remark on a system with diﬀerent gradient exponent Δu1|∇u1|^a¹p₁|x|f1u1, u₂, . . . , u_d, x∈R^N,

... ...

Δud|∇ud|^a^d p_d|x|fdu1, u₂, . . . , u_d, x∈R^N,

3.45

where ai ∈ 0,∞, ai/1,2, fi : 0,∞^d → 0,∞ are nonnegative, continuous, and nondecreasing functions for each variable. For these cases, the problem is far more complex, and no analogous results have been established9,10,13,18,21. We also anticipate that the methods and concepts here can be extended to the systems withqi-Laplacian as considered by Covei14–17.

References

1 L. Bieberbach, “Δue^uund die automorphen Funktionen,” Mathematische Annalen, vol. 77, no. 2, pp.

173–212, 1916.

2 J. B. Keller, “On solutions ofΔufu,” Communications on Pure and Applied Mathematics, vol. 10, pp.

503–510, 1957.

3 R. Osserman, “On the inequalityΔu≥fu,” Pacific Journal of Mathematics, vol. 7, pp. 1641–1647, 1957.

4 C. Bandle and M. Marcus, “Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour,” Journal d’Analyse Math´ematique, vol. 58, pp. 9–24, 1992.

5 A. V. Lair, “A necessary and suﬃcient condition for existence of large solutions to semilinear elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 240, no. 1, pp. 205–218, 1999.

(15)

6 K.-S. Cheng and W.-M. Ni, “On the structure of the conformal scalar curvature equation onRⁿ,”

Indiana University Mathematics Journal, vol. 41, no. 1, pp. 261–278, 1992.

7 A. V. Lair and A. W. Wood, “Large solutions of semilinear elliptic problems,” Nonlinear Analysis A, vol. 37, no. 6, pp. 805–812, 1999.

8 A. V. Lair and A. W. Wood, “Large solutions of sublinear elliptic equations,” Nonlinear Analysis A, vol.

39, no. 6, pp. 745–753, 2000.

9 A. V. Lair and A. W. Wood, “Existence of entire large positive solutions of semilinear elliptic systems,”

Journal of Diﬀerential Equations, vol. 164, no. 2, pp. 380–394, 2000.

10 F.-C. S¸t. Cˆırstea and V. D. R˘adulescu, “Entire solutions blowing up at infinity for semilinear elliptic systems,” Journal de Math´ematiques Pures et Appliqu´ees, vol. 81, no. 9, pp. 827–846, 2002.

11 M. Ghergu and V. R˘adulescu, “Explosive solutions of semilinear elliptic systems with gradient term,”

Revista de la Real Academia de Ciencias Exactas A, vol. 97, no. 3, pp. 437–445, 2003.

12 Y. Peng and Y. Song, “Existence of entire large positive solutions of a semilinear elliptic system,”

Applied Mathematics and Computation, vol. 155, no. 3, pp. 687–698, 2004.

13 X. Zhang and L. Liu, “The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 300–

308, 2010.

14 D.-P. Covei, “Radial and nonradial solutions for a semilinear elliptic system of Schr ¨odinger type,”

Funkcialaj Ekvacioj, vol. 54, no. 3, pp. 439–449, 2011.

15 D.-P. Covei, “Large and entire large solution for a quasilinear problem,” Nonlinear Analysis A, vol. 70, no. 4, pp. 1738–1745, 2009.

16 D.-P. Covei, “Existence of entire radically symmetric solutions for a quasilinear system with d- equations,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 3, pp. 433–439, 2011.

17 D.-P. Covei, “Schr ¨odinger systems with a convection term for thep1, . . . , pd-Laplacian inR^N,”

Electronic Journal of Diﬀerential Equations, vol. 2012, no. 67, pp. 1–13, 2012.

18 R. Dalmasso, “Existence and uniqueness of positive solutions of semilinear elliptic systems,”

Nonlinear Analysis A, vol. 39, no. 5, pp. 559–568, 2000.

19 M. Ghergu and V. D. R˘adulescu, Singular Elliptic Equations: Bifurcation and Asymptotic Analysis, vol. 37 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, UK, 2008.

20 S. Chen and G. Lu, “Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system,” Nonlinear Analysis A, vol. 38, no. 7, pp. 919–932, 1999.

21 M. Ghergu, C. Niculescu, and V. R˘adulescu, “Explosive solutions of elliptic equations with absorption and non-linear gradient term,” Indian Academy of Sciences, vol. 112, no. 3, pp. 441–451, 2002.

22 J. Serrin and H. Zou, “Existence of positive entire solutions of elliptic Hamiltonian systems,”

Communications in Partial Diﬀerential Equations, vol. 23, no. 3-4, pp. 577–599, 1998.

23 A. V. Lair and A. W. Wood, “Entire solution of a singular semilinear elliptic problem,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 498–505, 1996.

24 P. Quittner, “Blow-up for semilinear parabolic equations with a gradient term,” Mathematical Methods in the Applied Sciences, vol. 14, no. 6, pp. 413–417, 1991.

25 C. Bandle and E. Giarrusso, “Boundary blow up for semilinear elliptic equations with nonlinear gradient terms,” Advances in Diﬀerential Equations, vol. 1, no. 1, pp. 133–150, 1996.

26 C. S. Yarur, “Existence of continuous and singular ground states for semilinear elliptic systems,”

Electronic Journal of Diﬀerential Equations, vol. 1, pp. 1–27, 1998.

27 X. Wang and A. W. Wood, “Existence and nonexistence of entire positive solutions of semilinear elliptic systems,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 361–368, 2002.

28 X. Li, J. Zhang, S. Lai, and Y. Wu, “The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system,” Journal of Diﬀerential Equations, vol. 250, no. 4, pp. 2197–2226, 2011.

29 S. Lai and Y. Wu, “Global solutions and blow-up phenomena to a shallow water equation,” Journal of Diﬀerential Equations, vol. 249, no. 3, pp. 693–706, 2010.

30 X. Li, Y. Wu, and S. Lai, “A sharp threshold of blow-up for coupled nonlinear Schr ¨odinger equations,”

Journal of Physics A, vol. 43, no. 16, article 165205, p. 11, 2010.

31 J. Zhang, X. Li, and Y. H. Wu, “Remarks on the blow-up rate for critical nonlinear Schr ¨odinger equation with harmonic potential,” Applied Mathematics and Computation, vol. 208, pp. 389–396, 2009.

32 X. Zhang, “A necessary and suﬃcient condition for the existence of large solutions to “mixed” type elliptic systems,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2359–2364, 2012.

33 W. M. Ni, “On the elliptic equationΔuKxu^n2/n−20 , its generalizations, and applications in geometry,” Indiana University Mathematics Journal, vol. 31, no. 4, pp. 493–529, 1982.

(16)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}

Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms

Research Article