Existence,Uniqueness and Blow-Up Rate of LargeSolutionsofQuasi-LinearEllipticEquationswithHigher Order and Large Perturbation

ZHANG Qihuand ZHAO ChunshanDepartment of Mathematics and Information Science,Zhengzhou University of

Light Industry,Zhengzhou 450002,China.

2Department of Mathematical Sciences,Georgia Southern University,Statesboro,

GA 30460,USA.

Received 10 January 2013;Accepted 8 June 2013

Existence,Uniqueness and Blow-Up Rate of Large
SolutionsofQuasi-LinearEllipticEquationswithHigher Order and Large Perturbation

ZHANG Qihu1,?and ZHAO Chunshan21Department of Mathematics and Information Science,Zhengzhou University of

Light Industry,Zhengzhou 450002,China.

2Department of Mathematical Sciences,Georgia Southern University,Statesboro,

GA 30460,USA.

Received 10 January 2013;Accepted 8 June 2013

.We establish the existence,uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem

AMS Subject Classifications:35J25,35J60

Chinese Library Classifications:O175.25

Blow up rate;large positive solution;quasi-linear elliptic problem;uniqueness.

1 Introduction

elliptic problem with singular boundary value condition as follows:

We say that λ(x)uθ?1is a perturbation term.If θ＞p,we say that λ(x)uθ?1is a higher order perturbation,otherwise,it is called a lower order perturbation.For any solution u(x)of(1.1a)-(1.1c),if

we say that λ(x)uθ?1is a small perturbation,otherwise it is called a large perturbation. The p-Laplacian operator

appears in the study of non-Newtonian flows,chemotaxis,and biological pattern formation etc.When p=θ=2,the problem(1.1a)-(1.1c)becomes as follows:

and it has been studiedextensively.Bieberbach[1]studied the large solutions for the particular case?△u=?exp(u)with conditions(1.2c)in smooth bounded two-dimensional domains.Later on,Rademacher[2]continued the study of the large solutions for the particular case?△u=?exp(u)in smooth bounded domains in R3.Bandle-Essen[3] and Lazer-Mckenna[4]extended Bieberbach’s and Rademacher’s results to general case?△u=?b(x)exp(u)in smooth bounded domains of RN,where the function b(x)is continuous and strictly positive onThey showed that the problem has a unique solution together with an estimate of the form u=logd?2+o(d),see[3]for case b≡1 and[4]for case b(x)≥b0＞0 as d→0.

Recently,theuniquenessofsolutionsfor(1.2a)-(1.2c)withh(u)=uq(q＞1)onbounded domains orthewholespace RNwas discussedinmany papers(see,e.g.,[2–23]).Byusing the localization method of[14],it was shown in[14,19]that(1.2a)-(1.2c)with h(u)=uq, q＞1 has at most one blow-up solution under some conditions.Further improvements of these results can be found in[5,6,15,18,20,22,23]and the references therein.

The radial case ofthe problem(1.2a)-(1.2c)on a ball domain BR(x0)with h(u)=uqwas firstly studied by L′opez-G′omez[15].The author also extended the results to a generaldomain by adopting the localization method[14].Later on,Cano-Casanova and L′opez-G′omez improved the results in[15]for h(u)=uq?1to a general function h(u)which satisfies the Keller-Osserman condition[13,17]and h(u)～Huq?1(H＞0 is a constant and q＞1)for sufficiently large u[5,6].In[18],the authors also considered the problem(1.2a)-(1.2c)with h(u)=uq?1on a ball domain BR(x0),but the decay rate of the weight function b(x)was not assumed to be approximated by a distance function near the boundary??.Uniqueness and blow-up rates of solution of(1.2a)-(1.2c)in general domains was also obtained in[19]by combining the localization method with the results in[18].Also see[22,23]formore resultsin thedirectionofgeneralfunction h(u)in(1.2a)-(1.2c).On the boundaryblow-up elliptic problems withnonlineargradientterms,we refertoreferences [24,25].In[7],the author deal with the uniqueness of boundary blow-up solutions on exterior domain of RNwith large coefficient.

It is often important to know what properties are retained when linear diffusion (p=2)which corresponds to the Laplace operator is replaced by nonlinear diffusion (p6=2)which corresponds to the degenerate p-Laplace operator.It is not always possible to extend the results from Laplace operator to the degenerate p-Laplace operator(as many examples have already demonstrated);and even if such extension is possible,one usually has to overcome many non-trivial technical difficulties since many nice properties inherent to the Laplace operator seem lost or difficult to verify once p6=2.On the large solutions of p-Laplace,we refer to[26–39].In[35,37,38],the authors deal with the existence of boundary blow-up solutions for p-Laplacian equation,with lower order and small perturbation.In[39],the authors deal with the existence of solutions which blow up at origin for p-Laplacian equation in RNwith small perturbation.We refer readers to[27]for the existence of large positive solutions of the problem(1.1a)-(1.1c)when θ=p,λ(x)≡λ(a constant)and b(x)is in Cμ(?)(1＞μ＞0).

On the large solutions of elliptic equations,most of results are for the equations with the lower order and small perturbation.Results for the equations with higher order perturbation or large perturbation are rare.In[8],the authors study the existence of boundary blow-up solutions for p=2,with higher order perturbation,but the perturbation is a small perturbation.In[12],the authors consider the existence of boundary blow-up solutions for Laplacian cooperative elliptic systems(p=2)with large perturbation,but the perturbation is a lower order perturbation.In this paper,we are interested in the existence,uniqueness and the blow-up rate of solutions to the problem(1.1a)-(1.1c)with large perturbation.It should be mentioned that we have no comparison principle when we discuss the existence of solutions for the case of θ＞p,and we do not suppose that h(·)is increasing.

Before stating our main results,we make the following assumptions:

(A1)b∈C(?,(0,∞))satisfies

where β is a constant which has both signs.

(A2)λ∈C(?,(0,∞))satisfies

where α is a constant which has both signs.

(A3)h(u)≥0,satisfies h(u)/uθ?1→0+as u→0+,and for some q＞max{p,θ}＞1,

We will discuss theexistenceof large solutionsof(1.1a)-(1.1c)in the following threecases:

Theorem 1.1.Under the condition of Case(I),we assume(A1)-(A3)are satisfied,then(1.1a)-(1.1c)has a solution u(x)satisfy,

where Θ is a constant defined by

Moreover,if β≤0,θ≤p and h(u)/up?1is increasing on(0,∞),then(1.1a)-(1.1c)possesses a unique positive large solution u(x)in ?.

Theorem 1.2.Under the condition of Case(II),we assume(A1)-(A3)are satisfied,then(1.1a)-(1.1c)has a solution u(x)satisfy,

where Θ is a constant defined by

t0is the unique positive solution of

Moreover,if β≤0,θ≤p and h(u)/up?1is increasing on(0,∞),then problem(1.1a)-(1.1c) possesses a unique positive large solution u(x)in ?.

Theorem 1.3.Under the condition of Case(III),we assume(A1)-(A3)are satisfied,then(1.1a)-(1.1c)has a solution u(x)satisfy

Moreover,if β≤0,θ≤p and h(u)/up?1is increasing on(0,∞),then problem(1.1a)-(1.1c) possesses a unique positive large solution u(x)in ?.

2 Some preliminary results

At first let us present some lemmas which will be used in the proof of Theorems 1.1-1.3. Some of them are extensions of the lemmas in[27].Similar results for p=2 can be found in[9,14,16,27,40].

Consider the problem

where ? is a smooth bounded domain in RNwith N≥2,φ∈C1(??),h satisfies(A3)and b∈C(?,R+).

Lemma 2.1.Let p＞1 and denote(2?p)+=max{2?p,0},c(p)=1/(2p?1-1)when 1＜p＜2 and c(p)=[3p(p?1)]/16 when p≥2.Then for all vectors v1and v2in RN,

Proof.This Lemma is exactly the Lemma 4.2 in[41],we refer readers there for a detailed proof by applying Clarkson’s inequality.

Remark 2.1.This Lemma can be proved similarly as the proof of[27,Proposition 2.2] (see also[42]and[16]),which goes back to Benguria-Brezis-Lieb[43].We also refer readers to[29]for maximum and comparison principles for elliptic equations involving p-Laplacian.For readers’convenience,next we give a proof.The method of adding ? and then letting ?→0 goes back to Lindqvist in[41].

we have

Let ?2＞?1＞0 and denote

Since viis vanishing near??,we see that(2.3)holds when wiis replaced by vi.Denote

We note that the integrands in(2.3)(with wi=vi)vanish outside this set.On ?+(?1,?2), we have

We denote

Then by substituting the above expressions for?v1and?v2in I,we obtain

This expression can be simplified by using

Indeed,we can write

Applying Lemma 2.1,we obtain that

Let the above equality be

Then the inequality(2.3)becomes

Note that the left hand side is non-positive.As ?2＞?1→0,the second term on the righthand side of(2.5)converges to

which is positive unless ?+(0,0)is empty,and the first term on the right-hand side of (2.5)converges to

which is positive unless ?+(0,0)is empty.Proof of Lemma 2.2 is completed.

Next we present the definition of sub-solution and super-solution as follows.

Denote

Consider

The corresponding functional of(P?n)is

3 Proof of main results

In this section,we will discuss the existence of large solutions of(1.1a)-(1.1c)in Case (I)-Case(III)as stated in Section 1 and then prove Theorems 1.1-1.3.

3.1 Case(I)

Since the boundary?? of ? is C2smooth,we may assume that there exists a positive constant δ＞0 such that d(·)is a C2function when d(x)≤3δ.Denote

where

and k is large enough.Obviously,g(·,s1,?)∈C1(?).

Lemma 3.1.Under the conditions of case(I),we assume(A1)-(A3)satisfied,then g(·,s1,?)is a supe?

Proof.Define the function g on ? as

where σ∈(0,δ)is small enough,and 0＜? is a small constant.

We have

By computation

where

It is easy to see that

For any fixed ν∈(0,1),when σ＞0 is small enough,we have

then

The third inequality use the conditions of(A2)and Case(I),the fifth inequality used q?θ＞0 and θ?1≥0,the last inequality use the conditions of(A1),(A3)and Case(I).By computation,when d(x)＞σ,we have

There is a large enough k which is dependent on σ such that

Combining(3.4)and(3.5),we can conclude that g(x)is a super-solution of(1.1a)-(1.1c).

Next,under the condition of β≤0,θ≤p and h(u)/up?1is increasing on(0,∞),for any solution u of problem(1.1a)-(1.1c),we will prove that u(x)≤g(x,s1,?).Note that the comparison principle is valid for(1.1a)-(1.1c)now.

where

where

Denote

then

If 0＜? is a small constant,let C=C?2,similar to the discussion of(3.4),we can see that

provided

Note that s1＞0 and q?θ＞0,then s1(q?θ)＞0;the Case(I)means s1(q?θ)+β?α＞0. When σ is small enough,we have

Then

When d(x)＞0 is small enough,we have

Note that g∞(x,?)=g(x,s1,?)is a positive super-solutionof(1.1a)-(1.1c).According to the comparison principle,we obtain that g(x,s1,?)≥u(x).Thus g(x,s1,?)is an upper control function of all of the positive solutions of(1.1a)-(1.1c).

Denote

where ν＞0 is small enough such that b(x)h(ν)≤λ(x)νθ?1,?d(x)≥σ,and

Lemma 3.2.Under the conditions of case(I),we assume(A1)-(A3)satisfied,then v(·,s1,?)is a sub-solution of problem(1.1a)-(1.1c)when σ is small enough.More over,if β≤0,θ≤p and h(u)/up?1is increasing on(0,∞),then for every weak solution u of problem(1.1a)-(1.1c),we have u(x)≥v(x,s1,?).

Denote

By computation

where

It is easy to see that

For any fixed ?∈(0,1),when σ＞0 is small enough,we have

then

Let

By the definition of v,we have

where n1is the inner-ward unit normal vector of???2,and

Then

Thus,v is a sub-solution of(1.1a)-(1.1c).

Next,under the conditions of β≤0,θ≤p and h(u)/up?1is increasing on(0,∞),for any solution u of problem(1.1a)-(1.1c),we will prove that u(x)≥v(x,s1,?).Note that the comparison principle is valid for(1.1a)-(1.1c)now.

where

where

Similar to the proof of(3.10),we can see that

The second inequality used β≤0.Thus

Proof of Theorem 1.1

By the definition of super-solution g(x,s1,?)and sub-solution v(x,s1,?),we have

Now,let’s consider

where C?is defined in(3.2),and

We can see that

and

where n1is the inner-ward unit normal vector of the boundary of ??.

By the definition of g?,it is easy to see

Thus

Note that

Similar to the above discussion and the proof of Lemma 3.2,we can see that v?is a subsolution of(1.1a)-(1.1c).Moreover,

The proof of uniqueness basically follows the proofs in[9],[11]and[22].Let u be an arbitrary solution of(1.1a)-(1.1c)with assumptions on nonlinear function h(u)and weight function b and λ as in theorem 1.1.By Lemma 3.1 and Lemma 3.2,we have

Consequently,for any pair of solution u,v of(1.1a)-(1.1c)

holds in ?σ?,therefore it is true in ?.Letting ?→0 we arrive at u=v.

3.2 Case(II)

Proof of Theorem 1.2

Similar to the proof of Lemmas 3.1 and 3.2,we will construct a pair of sub-solution and super-solution of(1.1a)-(1.1c).Set

where g is defined in(3.1),and ν＞0 is small enough such that b(x)h(ν)≤λ(x)νθ?1,?d(x)≥σ,

where

and t0is the unique positive solution of

We will prove the theorem in three steps.

Step 1.We will prove that g2is a super-solution of(1.1a)-(1.1c).

Under the conditions of Case(II),we have

Denote

Similar to the proof of Lemma 3.1,by computation,we have

where

It is easy to see that

Since t0is the unique positive solution of the following equation

when t＞t0,it follows from q＞max{p,θ}that

Therefore

Similar to the proof of Lemma 3.1,we can see that g2is a super-solution of(1.1a)-(1.1c).

Step 2.Similar to the proof of Step 1 and Lemma 3.2,we can see that v2is a subsolution of(1.1a)-(1.1c).

Step 3.The existence and the uniqueness of solutions.

Similar to the proof of Theorem 1.1,we get the existence of solution u satisfying

Underthe conditions θ≤p,β≤0 and h(u)/uis increasing in(0,∞),similar to the proof of Theorem 1.1,we can get the uniqueness of solutions.

The proof is completed.

3.3 Case(III)

Proof of Theorem 1.3

have the same blowup rate,and the blowup rate is larger than?△pt?d?s2(x),i.e.,

and

Denote

Define

and

We will prove the theorem in three steps.

Step 1.We will prove that g3is a super-solution of(1.1a)-(1.1c).

It follows from(3.14)and conditions(A1)-(A3)that

Combining the above inequality and(3.15)together,when σ＞0 is small enough,similar to the proof of Theorem 1.2,we have

Note that g3=ψ+k.When σ＞0 is small enough,we have

Similar to the proof of Lemma 3.1,when k is large enough,we have

and

By Summarizing the above discussion,we can see that g3is a super-solution of(1.1a)-(1.1c).

Step 2.We will prove that v3is a sub-solution of(1.1a)-(1.1c).Denote

It follows from(3.13)and the conditions(A1)-(A3)that

Combining the above inequality,(3.14)and(3.15)together,when σ＞0 is small enough, we have

It is easy to see that

Note that k?=(1??)t?σ?s2?ν.Similar to the proof of Theorem 1.2,when σ＞0 is small enough,we have

Note that v3=φ?k?on d(x)＜σ.Thus when σ＞0 is small enough,we have

Similar to the proof of Theorem 1.2,v3(x,s2,?)is a sub-solution of(1.1a)-(1.1c).

Step 3.The existence and the uniqueness of solutions.

Also similar to the proof of Theorem 1.1,we get the existence of solution u satisfying

Underthe conditions θ≤p,β≤0 and h(u)/up?1is increasing in(0,∞),similar to the proof of Theorem 1.1,we can get the uniqueness of solutions.

The proof is completed.

Acknowledgments

Research is partly supported by the National Science Foundation of China(10701066& 10971087).

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?Corresponding author.Email addresses:zhangqh1999@yahoo.com.cn,zhangqihu@yahoo.com(Q.Zhang), czhao@GeorgiaSouthern.edu(C.Zhao)

10.4208/jpde.v26.n3.3 September 2013