Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2017: 1.89

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

Existence and asymptotic behavior of ground state solutions of semilinear elliptic system

Habib Mâagli
  • King Abdulaziz University, College of Sciences and Arts, Rabigh Campus, Department of Mathematics, P.O. Box 344, Rabigh 21911, Saudi Arabia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Sonia Ben Othman
  • Corresponding author
  • Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Safa Dridi
Published Online: 2016-01-29 | DOI: https://doi.org/10.1515/anona-2015-0157

Abstract

In this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system:

u = a1(x)uα vr, x n (n 3), -Δv = a2(x)vβ us, x n, u,v ¿ 0 in n, lim|x|+ u(x) = lim|x|+ v(x)=0,

where α,β<1, r,s such that ν:=(1-α)(1-β)-rs>0, and the functions a1, a2 are nonnegative in 𝒞locγ(n) (0¡γ¡1) and satisfy some appropriate assumptions related to Karamata regular variation theory.

Keywords: Semilinear elliptic system; asymptotic behavior; Karamata class; ground state solution; subsolution; supersolution

MSC 2010: 35B07; 35B40; 35J60

1 Introduction

Existence and asymptotic behavior of solutions of the semilinear elliptic system

{-Δu=f(x,u,v),xn (n3),-Δv=g(x,u,v),xn,u,v>0in n(1.1)

have received much attention. Namely, systems of type (1.1) in bounded domains of n(n2) with various boundary conditions have been intensively studied and many existence results have been established. We refer the reader to [1, 5, 9, 10, 11, 12].

Several articles have been devoted to the study of systems of type (1.1) with different boundary conditions. See for example [3, 4, 6, 7, 8, 13, 14, 16, 18, 19, 20, 21]. In these works, various existence results of positive bounded solutions or positive blowing-up ones (called also large solutions) have been established. Also, we note that several methods have been used to treat these nonlinear systems such as sub- and supersolutions methods, and variational and topological methods. Most of these results have to do with existence and non-existence of positive entire classical radial solutions. By an entire classical solution of (1.1), we mean a pair of positive functions (u,v) such that u,v𝒞2,γ(n) (0<γ<1) which satisfy (1.1) at every point of n.

In [14], Lair and Wood considered the following system:

{-Δu=a1(x)vr,xn (n3),-Δv=a2(x)us,xn,u,v>0in n,lim|x|+u(x)=lim|x|+v(x)=0,(1.2)

where 0<sr and a1,a2 are radial continuous functions in n. The authors of [14] showed that entire radial solutions of (1.2) exist if 0<sr<1 or if a1 and a2 satisfy the following integrability conditions:

0ra1(r)𝑑r<and0ra2(r)𝑑r<.(1.3)

Later, in [19], Wang and Wood studied the existence of entire (not necessary radial) solutions of system (1.2) where a1 and a2 are nonnegative continuous functions in n. More precisely, the authors proved in [19] the following result.

Theorem 1.1.

  • (i)

    Suppose that rs>0 and there exists C>0 such that

    a1(x)+a2(x)C|x|-2-δ

    for some positive constant δ . Then, system ( 1.2 ) has entire positive solutions.

  • (ii)

    Suppose that rs>1 and that a1 and a2 satisfy for xn,

    a1(x)C11+|x|2,a2(x)C21+|x|2,

    where C1 and C2 are positive constants. Then, system ( 1.2 ) has no entire positive solutions.

In [13], Kawano and Kusano considered the following system:

{-Δu=a1(x)uαvr,xn (n3),-Δv=a2(x)vβus,xn,u,v>0in n,lim|x|+u(x)=lim|x|+v(x)=0,(1.4)

where α,β<0, r,s>0 and a1,a2 are nonnegative functions in 𝒞locγ(n) (0<γ<1). The authors of [13] assumed that there exist locally Hölder continuous functions φ,ψ:+ such that

a1(x)φ(|x|)anda2(x)ψ(|x|)for xn,

which satisfy the integrability conditions as in (1.3). Then, they proved, by using the method of sub- and supersolutions, that system (1.4) has entire nonradial solutions either if α+r>1 and β+s>1, or if α+r<1 and β+s<1.

In this paper, inspired by a method of sub- and supersolutions as in [13], we study the existence of entire classical solutions of system (1.4), where α,β<1, r,s such that

ν:=(1-α)(1-β)-rs>0,(1.5)

and a1,a2 are nonnegative functions in 𝒞locγ(n), 0<γ<1, satisfying a suitable condition relying to Karamata functional class 𝒦 defined on [1,) by

𝒦:={L(t):=cexp(1tz(s)sds):c>0,z𝒞([1,)) and limt+z(t)=0}.

Also, we give an asymptotic precise behavior on a such solution of (1.4).

Our approach relies on the asymptotic behavior of the unique solution to the following semilinear elliptic problem:

{-Δu=a(x)uσ,xn,u>0in n,lim|x|+u(x)=0,(1.6)

where σ<1 and a verifies the following hypothesis:

  • ($\mathrm{H}_{0}$)

    a is a nonnegative function in 𝒞locγ(n), 0<γ<1, satisfying, for xn,

    a(x)L(1+|x|)(1+|x|)λ,

    where λ2 and L𝒦 such that

    1t1-λL(t)𝑑t<.

Here and throughout, the notation f(x)g(x), for two nonnegative functions f and g defined on a set S, means that there exists c>0 such that 1cf(x)g(x)cf(x) for all xS.

We point that in [2], Chemmam Dhifli and Mâagli obtained the asymptotic behavior of the solution to problem (1.6) which is given in the following result.

Theorem 1.2.

Assume (H0). Then, the solution of problem (1.6) satisfies

u(x)θλ(x),

where xRn and θλ is a function defined on Rn by

θλ(x):={(1+|x|L(t)t𝑑t)11-σif λ=2,(L(1+|x|))11-σ(1+|x|)(λ-2)/(1-σ)if 2<λ<n-(n-2)σ,1(1+|x|)n-2(12+|x|L(t)t𝑑t)11-σif λ=n-(n-2)σ,1(1+|x|)n-2if λ>n-(n-2)σ.

By applying Karamata regular variation theory and under the homogeneity condition (1.5), we generalize the results established in [13, 14, 19].

Let us consider the following assumption which is imposed to the functions a1 and a2.

  • (H1)

    ai (i=1,2) is a nonnegative function in 𝒞locγ(n), 0<γ<1, satisfying, for xn,

    ai(x)Li(1+|x|)(1+|x|)λi,

    where λi2 and Li𝒦 such that

    1t1-λiLi(t)𝑑t<.

Also, we introduce the quantities

σ1:=min{n-2,λ2-21-β},σ2:=min{n-2,(λ2-2)+s(n-2)1-β},σ3:=(λ1-2)(1-β)+r(λ2-2)ν,σ4:=(λ2-2)(1-α)+s(λ1-2)ν.

The above values of σi (i=1,,4) are related to the boundary behavior of solutions to system (1.4), as it will be explained in our main results stated in the following theorems.

Theorem 1.3.

Assume (H1) and that the condition of homogeneity (1.5) is satisfied. Suppose that

λ1+rσ1=2,2<λ2,λ2n-(n-2)β,

and L1, L2 satisfy

1L1(t)L2r1-β(t)t𝑑t<if 2<λ2<n-(n-2)β,𝑎𝑛𝑑1L1(t)t𝑑t<if λ2>n-(n-2)β.

Then, system (1.4) has a classical solution (u,v) satisfying

u(x)L~1(1+|x|)𝑎𝑛𝑑v(x)L~2(1+|x|)(1+|x|)σ1,

where L~1,L~2 belong to K, and defined on [1,+) by

L~1(t):={(tL1(ξ)(L2(ξ))r1-βξ𝑑ξ)1-βνif 2<λ2<n-(n-2)β,(tL1(ξ)ξ𝑑ξ)11-αif λ2>n-(n-2)β,𝑎𝑛𝑑L~2(t):={(L2(t))11-β(tL1(ξ)(L2(ξ))r1-βξ𝑑ξ)sνif 2<λ2<n-(n-2)β,1if λ2>n-(n-2)β.

Theorem 1.4.

Assume (H1) and that the condition of homogeneity (1.5) is satisfied. Suppose that

λ1+rσ2n-(n-2)α,2<λ2+s(n-2),(λ1,λ2)(n-(n-2)(α+r),n-(n-2)(β+s)).

Then, system (1.4) has a classical solution (u,v) satisfying

u(x)L~1(1+|x|)(1+|x|)n-2𝑎𝑛𝑑v(x)L~2(1+|x|)(1+|x|)σ2,

where L~1,L~2 belong to K, and defined on [1,+) by:

  • If λ1+rσ2=n-(n-2)α , we have

    L~1(t):={(11+tL1(ξ)(L2(ξ))r1-βξ𝑑ξ)1-βνif 2<λ2+s(n-2)<n-(n-2)β,(11+tL1(ξ)ξ𝑑ξ)11-αif λ2+s(n-2)>n-(n-2)β,𝑎𝑛𝑑L~2(t):={(L2(t))11-β(11+tL1(ξ)(L2(ξ))r1-βξ𝑑ξ)sνif 2<λ2+s(n-2)<n-(n-2)β,1if λ2+s(n-2)>n-(n-2)β.

  • If λ1+rσ2>n-(n-2)α , we have L~1(t):=1 and

    L~2(t):={(L2(t))11-βif 2<λ2+s(n-2)<n-(n-2)β,(11+tL2(ξ)ξ𝑑ξ)11-βif λ2+s(n-2)=n-(n-2)β,1if λ2+s(n-2)>n-(n-2)β.

Theorem 1.5.

Assume (H1) and that the condition of homogeneity (1.5) is satisfied. Suppose that

2<λ1+rσ3<n-(n-2)α𝑎𝑛𝑑2<λ2+sσ4<n-(n-2)β.

Then, system (1.4) has a classical solution (u,v) satisfying

u(x)L~1(1+|x|)(1+|x|)σ3𝑎𝑛𝑑v(x)L~2(1+|x|)(1+|x|)σ4,

where L~1,L~2 belong to K, and defined on [1,+) by

L~1(t):=(L1(t))1-βν(L2(t))rν𝑎𝑛𝑑L~2(t):=(L2(t))1-αν(L1(t))sν.

Remark 1.6.

Since system (1.4) is invariant under the transform

(u,v,α,β,r,s)(v,u,β,α,s,r),

we have also the existence of a classical solution (u,v) for system (1.4) that behave like (1+|x|)σL~(1+|x|), where σ depends on α,β,r,s,λ1,λ2 and L~𝒦, if one of the following conditions holds:

  • λ2+smin{n-2,λ1-21-α}=2, 2<λ1 and λ1n-(n-2)α,

  • λ2+smin{n-2,(λ1-2)-r(n-2)1-α}>2, 2-r(n-2)<λ1, λ1n-(n-2)(α+r) and λ2n-(n-2)(β+s).

We close this section by giving the following notations. For a nonnegative measurable function f in n, we denote by Vf the potential of f defined on n by

Vf(x)=nG(x,y)f(y)𝑑y,whereG(x,y)=Γ(n2-1)4πn2|x-y|n-2

is the Green function of the Laplacien Δ in n (n3).

We point out that for any nonnegative function f in 𝒞locγ(n) (0<γ<1) such that VfL(n), we have Vf𝒞loc2,γ(n) and

-Δ(Vf)=fin n,

see [16, Theorem 6.3].

As usual, we refer to 𝒞0(n) the collection of all continuous functions in n which vanish at infinity. From here on, the letter c denotes a generic positive constant which may vary from line to line.

The rest of the paper is organized as follows. In Section 2, we give an existence result based on the sub- and supersolution method which plays a crucial role to prove the existence of solutions in Theorems 1.3, 1.4 and 1.5. In Section 3, we state some already known results on functions in 𝒦, useful for our study and we give the proofs of our main results. Section 4 is reserved to some applications.

2 Technical condition to existence result

To prove our existence result, we adopt a sub-supersolution method. We consider the more general system

{-Δu=f1(x,u,v),xn,-Δv=f2(x,u,v),xn,lim|x|+u(x)=lim|x|+v(x)=0.(2.1)

Definition 2.1.

A pair of positive functions (u,v)𝒞2,γ(n)×𝒞2,γ(n) is called a subsolution of system (2.1) if

{-Δuf1(x,u,v),xn,-Δvf2(x,u,v),xn,lim|x|+u(x)=lim|x|+v(x)=0.

If the above inequalities are reversed, then (u,v) is called a supersolution of system (2.1).

Lemma 2.2 (see [13]).

Let f1,f2 be nonnegative functions defined on Rn×(0,)×(0,) such that for i{1,2}, fi(,u,v)Clocγ(Rn) (0<γ<1) for all (u,v)(0,)×(0,) and fi(x,,) is continuously differentiable in (0,)×(0,) for all xRn. Suppose that system (2.1) has a supersolution (u¯,v¯) and a subsolution (u¯,v¯) such that

u¯u¯𝑎𝑛𝑑v¯v¯in n.

Then, system (2.1) has a positive entire solution (u,v) satisfying

u¯uu¯𝑎𝑛𝑑v¯vv¯in n.

Proposition 2.3.

Assume (H1) and that the condition of homogeneity (1.5) is satisfied. Suppose that there exist nonnegative functions θ and φ in C0(Rn) satisfying, for xRn,

V(a1θαφr)(x)θ(x)𝑎𝑛𝑑V(a2θsφβ)(x)φ(x).(2.2)

Then, system (1.4) has a classical solution (u,v) satisfying, for xRn,

u(x)θ(x)𝑎𝑛𝑑v(x)φ(x).(2.3)

Proof.

Let c>1 and θ,φ be nonnegative functions in 𝒞0(n) satisfying, for each xn,

1cθ(x)V(a1θαφr)(x)cθ(x)and1cφ(x)V(a2θsφβ)(x)cφ(x).

We point that in view of condition (1.5), there exist m1,m2>0 such that

m1(1-α)-rm2|α|+|r|and-m1s+(1-β)m2|β|+|s|.

Hence, by putting c1=cm1 and c2=cm2, we get

c1,c2>1,c|α|+|r|c11-αc2-randc|s|+|β|c1-sc21-β.

Put

u¯=c1V(a1θαφr),u¯=1c1V(a1θαφr),v¯=c2V(a2θsφβ)andv¯=1c2V(a2θsφβ).

Since θ,φ are nonnegative functions in 𝒞0(n) and a1, a2𝒞locγ(n), then (u¯,v¯) and (u¯,v¯) are in 𝒞2,γ(n) and satisfy u¯u¯ and v¯v¯ in n. Moreover, we have

-Δu¯=c1a1θαφrc-|α|-|r|c1a1(V(a1θαφr))α(V(a2θsφβ))r=c-|α|-|r|c11-αc2-ra1u¯αv¯ra1u¯αv¯rand-Δu¯=c1-1a1θαφrc|α|+|r|c1-1a1(V(a1θαφr))α(V(a2θsφβ))r=c|α|+|r|c1α-1c2ra1u¯αv¯ra1u¯αv¯r.

Similarly, we have

-Δv¯=c2a2θsφβc-|s|-|β|c2a2(V(a1θαφr))s(V(a2θsφβ))β=c-|s|-|β|c21-βc1-sa2u¯sv¯βa2u¯αv¯r

and

-Δv¯=c2-1a2θsφβc|s|+|β|c2-1a2(V(a1θαφr))s(V(a2θsφβ))β=c|s|+|β|c2β-1c1sa2u¯sv¯βa2u¯sv¯β.

Hence, (u¯,v¯) and (u¯,v¯) are, respectively, a subsolution and a supersolution to system (1.4). Then, the result follows by using Lemma 2.2. ∎

3 Existence and asymptotic behavior of solutions

3.1 The Karamata class 𝒦

In what follows, we are quoting, without proof, some fundamental properties of functions belonging to the class 𝒦, collected from [2] and [17].

Lemma 3.1 (see [2, 17]).

  • (i)

    A function L is in 𝒦 if and only if L is a positive function in 𝒞1([1,)) such that

    limt+tL(t)L(t)=0.

  • (ii)

    Let L1,L2𝒦 and p . Then, we have

    L1L2𝒦,L1p𝒦.

  • (iii)

    Let L𝒦 and ε>0 . Then, we have

    limt+t-εL(t)=0.

Lemma 3.2 (see [17]).

Let γR and L be a function in K defined on [1,). We have:

  • (i)

    If γ>-1 , then 1sγL(s)𝑑s diverges and

    tsγL(s)𝑑st+t1+γL(t)1+γ.

  • (ii)

    If γ<-1 , then 1sγL(s)𝑑s converges and

    tsγL(s)𝑑st+-t1+γL(t)1+γ.

Lemma 3.3.

Let LK. Then, we have

t11+tL(s)s𝑑s𝒦.

If further 1L(s)s𝑑s converges, we have

ttL(s)s𝑑s𝒦.

As a typical example of functions in 𝒦, we quote the following.

Example 3.4.

Let m, (μ1,μ2,,μm)m and ω be a positive real number sufficiently large such that the function

L(t)=k=1m(logk(ωt))μk,

is defined and positive on [1,), where logkx=logloglogx (k times). Then, we have that L𝒦.

3.2 Proof of main results

The main idea in order to prove Theorem 1.3, 1.4 and 1.5 is to find functions θ and φ satisfying (2.2), which will be useful to construct a subsolution and a supersolution to system (1.4) of the form

(cV(a1θαφr),cV(a2θsφβ))

and to deduce, by Proposition 2.3, the existence of a solution (u,v) to system (1.4) satisfying (2.3).

We consider the following system:

{-Δw1=p(x)w1αin n,-Δw2=q(x)w2βin n,w1,w2>0in n,lim|x|+w1(x)=lim|x|+w2(x)=0,(3.1)

where p(x)=a1(x)φr(x) and q(x)=a2(x)θs(x). The choice of θ and φ depends closely on the boundary behavior of the solution to the semilinear elliptic problem (1.6) as described in Theorem 1.2. Hence, a solution (w1,w2) of system (3.1) is given by Theorem 1.2 such that w1θ and w2φ. It follows that the functions θ and φ will satisfy

θw1=V(pw1α)V(a1θαφr),φw2=V(qw2β)V(a2θsφβ),

which is our principal purpose.

The rest of this section is devoted to proving our main results.

Proof of Theorem 1.3.

Suppose that

λ1+rσ1=2,2<λ2,λ2n-(n-2)β.

We split the proof into two cases. Case 1. If 2<λ2<n-(n-2)β and

1L1(t)(L2(t))r1-βt𝑑t<,

then we have σ1=λ2-21-β.

Let

θ(x):=(1+|x|L1(t)(L2(t))r1-βt𝑑t)1-βν

and

φ(x):=(L2(1+|x|))11-β(1+|x|)λ2-21-β(1+|x|L1(t)(L2(t))r1-βt𝑑t)sν.

So, we consider system (3.1). Using (H1), we have

p(x)L~1(1+|x|)(1+|x|)λ1+rλ2-21-β=L~1(1+|x|)(1+|x|)2

and

q(x)L~2(1+|x|)(1+|x|)λ2,

where

L~1(t)=L1(t)(L2(t))r1-β(tL1(ξ)(L2(ξ))r1-βξ𝑑ξ)rsν

and

L~2(t)=L2(t)(tL1(ξ)(L2(ξ))r1-βξ𝑑ξ)s(1-β)ν.

Applying Lemma 3.1 and Lemma 3.3, the functions L~1 and L~2 belong to 𝒦 and we have

1L~1(t)t𝑑t=ν(1-α)(1-β)(1L1(t)(L2(t))r1-βt𝑑t)(1-α)(1-β)ν<.

It follows by Theorem 1.2 that system (3.1) has a solution (w1,w2) satisfying for each xn,

w1(x)(1+|x|L~1(t)t𝑑t)11-α=θ(x)andw2(x)(L~2(1+|x|))11-β(1+|x|)λ2-21-β=φ(x).

That is, (2.2) is satisfied and so the result holds by using Proposition 2.3. Case 2. If λ2>n-(n-2)β and

1L1(t)t𝑑t<,

then we have σ1=n-2.

Let

θ(x):=(1+|x|L1(t)t𝑑t)11-α

and

φ(x):=1(1+|x|)n-2.

Consider system (3.1). Using (H1), we have

p(x)L1(1+|x|)(1+|x|)λ1+r(n-2)=L1(1+|x|)(1+|x|)2

and

q(x)L~2(1+|x|)(1+|x|)λ2,

where

L~2(t)=L2(t)(1+|x|L1(ξ)ξ𝑑ξ)s1-α.

Applying Lemma 3.1 and Lemma 3.3, the function L~2 belongs to 𝒦. So, we conclude, by Theorem 1.2, that system (3.1) has a solution (w1,w2) satisfying, for each xn,

w1(x)(1+|x|L1(t)t𝑑t)11-α=θ(x)andw2(x)1(1+|x|)n-2=φ(x).

Hence, (2.2) is satisfied and the result holds by using Proposition 2.3. This completes the proof. ∎

Proof of Theorem 4.

Suppose that

λ1+rσ2n-(n-2)α,2<λ2+s(n-2),(λ1,λ2)(n-(n-2)(α+r),n-(n-2)(β+s)).

We split the proof into two cases. Case 1. Assume that λ1+rσ2=n-(n-2)α.

(i) If 2<λ2+s(n-2)<n-(n-2)β, we put

θ(x):=1(1+|x|)n-2(12+|x|L1(t)(L2(t))r1-βt𝑑t)1-βν

and

φ(x):=(L2(1+|x|))11-β(1+|x|)(λ2-2)+s(n-2)1-β(12+|x|L1(t)(L2(t))r1-βt𝑑t)sν.

So, we consider system (3.1). Using (H1), we have

p(x)L~1(1+|x|)(1+|x|)λ1+r(λ2-2)+s(n-2)1-β=L~1(1+|x|)(1+|x|)n-(n-2)α

and

q(x)L~2(1+|x|)(1+|x|)λ2+s(n-2),

where

L~1(t)=L1(t)(L2(t))r1-β(1t+1L1(ξ)(L2(ξ))r1-βξ𝑑ξ)rsν

and

L~2(t)=L2(t)(1t+1L1(ξ)(L2(ξ))r1-βξ𝑑ξ)s(1-β)ν.

Applying Lemma 3.1 and Lemma 3.3, the functions L~1 and L~2 belong to 𝒦. So, we conclude, by Theorem 1.2, that system (3.1) has a solution (w1,w2) satisfying, for each xn,

w1(x)1(1+|x|)n-2(12+|x|L~1(t)t𝑑t)11-α=θ(x)andw2(x)(L~2(1+|x|))11-β(1+|x|)(λ2-2)+s(n-2)1-β=φ(x).

That is, (2.2) is satisfied and so the result holds by using Proposition 2.3.

(ii) If λ2+s(n-2)>n-(n-2)β, we put

θ(x):=1(1+|x|)n-2(12+|x|L1(t)t𝑑t)11-α

and

φ(x):=1(1+|x|)n-2.

Consider system (3.1). Using (H1), we have

p(x)L1(1+|x|)(1+|x|)λ1+r(n-2)=L1(1+|x|)(1+|x|)n-(n-2)α

and

q(x)L~2(1+|x|)(1+|x|)λ2+s(n-2),

where

L~2(t)=L2(t)(12+|x|L1(ξ)ξ𝑑ξ)s1-α.

Applying Lemma 3.1 and Lemma 3.3, the functions L~2 belongs to 𝒦. So, we deduce, by Theorem 1.2, that system (3.1) has a solution (w1,w2) satisfying, for each xn,

w1(x)1(1+|x|)n-2(12+|x|L1(t)t𝑑t)11-α=θ(x)andw2(x)1(1+|x|)n-2=φ(x).

That is, (2.2) is satisfied and so the result holds by using Proposition 2.3. Case 2. Assume that λ1+rσ2>n-(n-2)α.

(i) If 2<λ2+s(n-2)<n-(n-2)β, we put

θ(x):=1(1+|x|)n-2

and

φ(x):=(L2(1+|x|))11-β(1+|x|)(λ2-2)+s(n-2)1-β.

Consider system (3.1). Using (H1), we have

p(x)L1(1+|x|)(L2(1+|x|))r1-β(1+|x|)λ1+r(λ2-2)+s(n-2)1-β

and

q(x)L2(1+|x|)(1+|x|)λ2+s(n-2).

In view of Lemma 3.1 and Theorem 1.2, system (3.1) has a solution (w1,w2) satisfying, for each xn,

w1(x)1(1+|x|)n-2=θ(x)andw2(x)(L2(1+|x|))11-β(1+|x|)(λ2-2)+s(n-2)1-β=φ(x).

That is, (2.2) is satisfied and so the result holds by using Proposition 2.3.

(ii) If λ2+s(n-2)=n-(n-2)β. Interchanging the role of u and v, the proof is the same as in Case 1 (ii) above.

(iii) If λ2+s(n-2)>n-(n-2)β , we put

θ(x)=φ(x)=1(1+|x|)n-2.

Consider system (3.1). Using (H1), we have

p(x)L1(1+|x|)(1+|x|)λ1+r(n-2)

and

q(x)L2(1+|x|)(1+|x|)λ2+s(n-2).

It follows by Theorem 1.2 that system (3.1) has a solution (w1,w2) satisfying, for each xn,

w1(x)=w2(x)1(1+|x|)n-2.

That is, (2.2) is satisfied and so the result holds by using Proposition 2.3. This completes the proof. ∎

Proof of Theorem 1.5.

Suppose that

2<λ1+rσ3<n-(n-2)αand2<λ2+sσ4<n-(n-2)β.

Let

θ(x):=(L1(1+|x|))1-βν(L2(1+|x|))rν(1+|x|)σ3

and

φ(x):=(L2(1+|x|))1-αν(L1(1+|x|))sν(1+|x|)σ4.

Consider system (3.1). Using (H1), we have

p(x)L~1(1+|x|)(1+|x|)λ1+rσ4

and

q(x)L~2(1+|x|)(1+|x|)λ2+sσ3,

where

L~1(t)=(L1(t))(1-α)(1-β)ν(L2(t))r(1-α)ν

and

L~2(t)=(L2(t))(1-α)(1-β)ν(L1(t))s(1-β)ν.

Applying Lemma 3.1, the functions L~1 and L~2 belong to 𝒦. Put μ1=λ1+rσ4 and μ2=λ2+sσ3. Since μ1(2,n-(n-2)α) and μ2(2,n-(n-2)β), then by Theorem 1.2, we conclude that system (3.1) has a solution (w1,w2) satisfying, for each xn,

w1(x)(L~1(1+|x|))11-α(1+|x|)μ1-21-αandw2(x)(L~2(1+|x|))11-β(1+|x|)μ2-21-β.

By simple calculus, we have μ1-21-α=σ3 and μ2-21-β=σ4, this implies that w1θ and w2φ. The result holds again by using Proposition 2.3. This ends the proof. ∎

4 Applications

Let α,β<1 and r,s satisfying condition (1.5). Let a1,a2 be two nonnegatives functions in Clocγ(n) such that for i{1,2},

ai(x)log(ω(1+|x|))-μi(1+|x|)λi,

where ω is a positive constant large enough, μi and λi2.

We suppose that a1,a2 satisfy (H1) and we consider system (1.4). We are interested in this section to give explicitly the asymptotic behavior of solutions of system (1.4). This is stated in the following applications.

First, we need the following elementary result.

Lemma 4.1.

If μR, then for xRn, we have

12+|x|1t(log(ωt))-μdt{1if μ>1,log(log(ω(1+|x|)))if μ=1,(log(ω(1+|x|)))1-μif μ<1.(4.1)

Furthermore, if we suppose that μ>1, then for xRn we have

1+|x|1t(log(ωt))-μ𝑑t(log(ω(1+|x|)))1-μ.(4.2)

4.1 First application

Assume that λ1+rσ1=2,λ2>2 and λ2n-(n-2)β. Then, by applying Theorem 1.3 and Lemma 4.1, we obtain that system (1.4) has a classical solution (u,v) satisfying the following:

  • (a)

    If 2<λ2<n-(n-2)β and (1-β)(1-μ1)-rμ2<0, then for xn,

    u(x)(log(ω(1+|x|)))(1-β)(1-μ1)-rμ2ν

    and

    v(x)(log(ω(1+|x|)))s(1-μ1)-μ2(1-α)ν(1+|x|)λ2-21-β.

  • (b)

    If λ2>n-(n-2)β and μ1>1, then for xn,

    u(x)(log(ω(1+|x|)))1-μ11-α

    and

    v(x)1(1+|x|)n-2.

4.2 Second application

Assume that λ1+rσ2n-(n-2)α, λ2+s(n-2)>2 and (λ1,λ2)(n-(n-2)(α+r),n-(n-2)(β+s)). It follows by Theorem 1.4 and Lemma 4.1 that system (1.4) has a classical solution (u,v) satisfying the following:

  • (a)

    If λ1+rσ2=n-(n-2)α and 2<λ2+s(n-2)<n-(n-2)β, then for xn,

    u(x)1(1+|x|)n-2{(log(ω(1+|x|)))(1-β)(1-μ1)-rμ2νif (1-β)(1-μ1)-rμ2>0,(log(log(ω(1+|x|))))1-βνif (1-β)(1-μ1)-rμ2=0,1if (1-β)(1-μ1)-rμ2<0

    and

    v(x)1(1+|x|)(λ2-2)+s(n-2)1-β{(log(ω(1+|x|)))s(1-μ1)-μ2(1-α)νif (1-β)(1-μ1)-rμ2>0,(log(ω(1+|x|)))-μ21-β(log(log(ω(1+|x|))))sνif (1-β)(1-μ1)-rμ2=0,(log(ω(1+|x|)))-μ21-βif (1-β)(1-μ1)-rμ2<0.

  • (b)

    If λ1+rσ2=n-(n-2)α and λ2+s(n-2)>n-(n-2)β, then for each xn,

    u(x)1(1+|x|)n-2{(log(ω(1+|x|)))1-μ11-αif μ1<1,(log(log(ω(1+|x|)))11-αif μ1=1,1if μ1>1

    and

    v(x)1(1+|x|)n-2.

  • (c)

    If λ1+rσ2>n-(n-2)α and 2<λ2+s(n-2)<n-(n-2)β, then for xn,

    u(x)1(1+|x|)n-2

    and

    v(x)(log(ω(1+|x|)))-μ21-β(1+|x|)(λ2-2)+s(n-2)1-β.

  • (d)

    If λ1+rσ2>n-(n-2)α and λ+2s(n-2)=n-(n-2)β, then for xn,

    u(x)1(1+|x|)n-2

    and

    v(x)1(1+|x|)n-2{(log(ω(1+|x|)))1-μ21-βif μ2<1,(log(log(ω(1+|x|))))11-βif μ2=1,1if μ2>1.

    If λ1+rσ2>n-(n-2)α and λ2+s(n-2)>n-(n-2)β, then for each xn,

    u(x)1(1+|x|)n-2

    and

    v(x)1(1+|x|)n-2.

4.3 Third application

Assume that 2<λ1+rσ3<n-(n-2)α and 2<λ2+sσ4<n-(n-2)β. Then, by applying Theorem 1.5 and Lemma 4.1 we obtain that system (1.4) has a classical solution (u,v) satisfying, for xn,

u(x)(log(ω(1+|x|)))-(1-β)μ1-rμ2ν(1+|x|)σ3

and

v(x)(log(ω(1+|x|)))-(1-α)μ2-sμ1ν(1+|x|)σ4.

Acknowledgements

The authors would like to express their sincere thanks to the referee for carefully reading the first version of the paper and providing insightful comments.

References

  • [1]

    M. Chaieb, A. Dhifli and S. Zermani, Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in bounded domain, Opuscula Math., to appear.  Google Scholar

  • [2]

    R. Chemmam, A. Dhifli and H. Mâagli, Asymptotic behavior of ground state solutions for semilinear and singular Dirichlet problem, Electron. J. Differ. Equ. 88 (2011), 1–12.  Google Scholar

  • [3]

    S. Chen and G. Lu, Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system, Nonlinear Anal. Theory Methods Appl. 38 (1999), no. 7, 919–932.  Google Scholar

  • [4]

    F. C. Cîrstea and V. D. Rădulescu, Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl. (9) 81 (2002), 827–846.  Google Scholar

  • [5]

    P. Clément, J. Fleckinger, E. Mitidieri and F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), 455–477.  Google Scholar

  • [6]

    J. Garcìa and Melìan, A remark on uniqueness of large solutions for elliptic systems of competitive type, J. Math. Anal. Appl. 331 (2007), 608–616.  Google Scholar

  • [7]

    A. Ghanmi, H. Mâagli, V. D. Rădulescu and N. Zeddini, Large and bounded solutions for a class of nonlinear Schrödinger stationary systems, Anal. Appl. 7 (2009), no. 4, 391–404.  Google Scholar

  • [8]

    A. Ghanmi, H. Mâagli, S. Turki and N. Zeddini, Existence of positive bounded solutions for some nonlinear elliptic systems, J. Math. Anal. Appl. 352 (2009), 440–448.  Web of ScienceGoogle Scholar

  • [9]

    M. Ghergu, Lane–Emden systems with negative exponents, J. Funct. Anal. 258 (2010), 3295–3318.  Web of ScienceGoogle Scholar

  • [10]

    M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl. 37, Oxford University Press, Oxford, 2008.  Google Scholar

  • [11]

    M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monogr. Math., Springer, Berlin, 2012.  Google Scholar

  • [12]

    J. Giacomoni, J. Hernandez and P. Sauvy, Quasilinear and singular elliptic systems, Adv. Nonlinear Anal. 2 (2014), 1–42.  Google Scholar

  • [13]

    N. Kawano and T. Kusano, On positive entire solutions of a class of second order semilinear elliptic systems, Math. Z. 186 (1984), no. 3, 287–297.  Google Scholar

  • [14]

    A. V. Lair and A. W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Equations 164 (2000), 380–394.  Google Scholar

  • [15]

    Y. Peng and Y. Song, Existence of entire large positive solutions of a semilinear elliptic system, Appl. Math. Comput. 155 (2004), no. 3, 687–698. Google Scholar

  • [16]

    S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Academic Press, New York, 1978.  Google Scholar

  • [17]

    R. Seneta, Regular Varying Functions, Lecture Notes in Math. 508, Springer, Berlin, 1976.  Google Scholar

  • [18]

    J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Comm. Partial Differential Equations 23 (1998), 577–599.  Google Scholar

  • [19]

    X. Wang and A. W. Wood, Existence and nonexistence of entire positive solutions of semilinear elliptic systems, J. Math. Anal. Appl. 267 (2002), 361–368.  Google Scholar

  • [20]

    D. Ye and F. Zhou, Invariant criteria for existence of bounded positive solutions, Discrete Contin. Dyn. Syst. 12 (2005), no. 3, 413–424.  Google Scholar

  • [21]

    Z. Zhang, Existence of entire positive solutions for a class of semilinear elliptic systems, Electron. J. Differ. Equ. 2010 (2010), no. 16, 1–5.  Google Scholar

About the article

Received: 2015-11-10

Revised: 2015-12-15

Accepted: 2015-12-16

Published Online: 2016-01-29

Published in Print: 2017-08-01


The research of Habib Mâagli was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia (award number 13-MAT2137-02).


Citation Information: Advances in Nonlinear Analysis, Volume 6, Issue 3, Pages 301–315, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2015-0157.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
H. Mâagli and Z. Zine El Abidine
Complex Variables and Elliptic Equations, 2017, Volume 62, Number 11, Page 1665

Comments (0)

Please log in or register to comment.
Log in