Multiplicity of Solutions for a Singular Elliptic System with Concave-convex Nonlinearities and Sign-changing Weight Functions

2020-06-04 06:42WuZijianChenHaibo

数学理论与应用 2020年2期

Wu Zijian Chen Haibo

(School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China)

Abstract In this paper, we consider the multiplicity of solutions for a singular elliptic system with both concave-convex nonlinearities and sign-changing weight functions in bounded domains. Under some appropriate assumptions, we obtain that the system has at least two nontrivial solutions by using the Nehari manifold and fibering map.

Key words Singular elliptic system Concave-convex nonlinearity Sign-changing weight function Nehari manifold

1 Introduction and main result

Let Ω(0∈Ω) be a smooth bounded domain in RNwithN≥3. We are concerned with the following elliptic system:

(1.1)

we easily derive that

The critical points of the functionalIλ,μare in fact weak solutions of (1.1). We say that (u,v)∈Eis a weak solution of (1.1) if the identity

holds for all (ω1,ω2)∈E.

Setα=β=p/2,u=v,λ=μandf=g. Then problem (1.1) reduces to the scalar singular elliptic equations with concave-convex nonlinearities:

(1.2)

Moreover, for the elliptic systems with concave-convex nonlinearities, Wu [3] considered the following problems:

(1.3)

where the weight functionsf,g,hsatisfy the following conditions:

With the help of the Nehari manifold, the author proved this system has at least two nontrivial solutions when the pair of the parameters(λ,μ) belongs to a certain subset ofR2. Furthermore, Hsu [4] considered the following problems:

(1.4)

whereλ,μ>0 andα+β=2*. The author obtained the existence and multiplicity results of positive solutions by variational methods. For more similar problems, we refer to Ahammou [5], Alves et al. [6], Bozhkov and Mitidieri [7], Zhen and Zhang [8], and Squassina [9], and so forth.

However, to our knowledge, there are only a few results in the study of problem (1.1) with concave-convex nonlinearities and sign-changing weight functions. In this paper, we extend the method of [21,22,23] to study the multiplicity of nontrivial solutions of problem (1.1). Let

and

Theorem 1.1Assume that (A),(B) hold. If (λ,μ)∈Θ, then problem (1.1) has at least two solutions (u+,v+) and (u-,v-).

2 Preliminaries

In this section, we introduce some preliminaries. As the energy functionalIλ,μis not bounded below onE, it is helpful to consider the functional on the Nehari manifold

Thus, (u,v)∈Nλ,μif and only if

(2.1)

ObviouslyNλ,μis a much smaller set thanEand, as we shall show,Iλ,μis much better behaved onNλ,μ. Note thatNλ,μcontains all nonzero solutions of (1.1). Moreover, we have the following lemma.

Lemma 2.1The energy functionalIλ,μis coercive and bounded below onNλ,μ.

ProofFor (u,v)∈Nλ,μ, by (2.1), (a+b)s≤2s-1(as+bs),s≥1, the Sobolev and Hölder inequalities, we have

This ends the proof due to 1

The character of the functions of the formKu,v:t→I(tu,tv) (t>0) and the Nehari manifold are closely related. Such functions are known as fibering maps, which were studied by Brown and Wu in [12]. If (u,v)∈E, we have

Lemma 2.2Let (u,v)∈E{0,0} andt>0. Then (tu,tv)∈Nλ,μif and only ifK′u,v(t)=0, that is, the points on the Nehari manifold correspond to the critical points ofKu,v(t). In particular, (u,v)∈Nλ,μif andonly ifK′u,v(1)=0.

ProofNote that

Therefore, we can easily draw the conclusion of the lemma.

Naturally, similar to the method used in Tarantello [24], we splitNinto three parts:

It is easy to see that

Thus, for each (u,v)∈Nλ,μ, we have

(2.2)

ProofUsing the same argument in [21], we omit the details here.

Lemma 2.4One has the following.

ProofThe proof is immediate from (2.2).

ProofWe consider the following two cases.

K″u,v(1)=(2-α-β)‖(u,v)‖2<0.

and so

(2.3)

Similarly, using (2.2), the Sobolev and Hölder inequalities, we have

which implies that

(2.4)

Combining (2.3) and (2.4) we obtain

which is a contradiction. This completes the proof.

In order to understand the Nehari manifold and fibering maps, we consider the functionφu,vdefined by

Obviously, fort>0, (tu,tv)∈Nλ,μif and only if

Note that

Moreover,

(2.5)

and so we can see that if (tu,tv)∈Nλ,μ, then

K″u,v(t)=tq+1φ′u,v(t).

and clearlyφu,vis increasing on (0,tmax) and decreasing on (tmax,+∞) with limn→∞φu,v(t)=-∞. Moreover, it follows from (λ,μ)∈Θ that

That is, for (λ,μ)∈Θ,

(2.6)

Therefore, we have the following lemma.

and

φ′u,v(t+)>0>φu,v(t-).

Iλ,μ(t-u,t-v)≥Iλ,μ(tu,tv)≥Iλ,μ(t+u,t+v)

for eacht∈[t+,t-] and

Iλ,μ(t+u,t+v)≤Iλ,μ(tu,tv)

for eacht∈[0,t+]. This completes the proof.

Next, we remark that it follows from Lemma 5 that

Then we have the following result.

and so

3 Proof of Theorem 1.1

First, we will use the idea of Ni and Takagi [10] to get the following lemmas.

Lemma 3.1If (λ,μ)∈Θ, then for every (u,v)∈Nλ,μ, there exist anε>0 and a differentiable functiong:Bε(0)×Bε(0)⊂E→R+such that

g(0,0)=1,g(ω1,ω2)(u-ω1,v-ω2)∈Nλ,μ, ∀(ω1,ω2)∈Bε(0)×Bε(0).

Moreover, set

then

(3.1)

for all (φ1,φ2)∈E.

ProofWe defineF:R×E→R by

it is easy to seeFis differentiable. SinceF(1,(0,0))=0 andFt(1,(0,0))=K″u,v(1)≠0, we apply the implicit function theorem at point (1,(0,0)) to get the existence ofε>0 and differentiable functiong:Bε(0)×Bε(0)→R+such thatg(0,0)=1 andF(g(ω1,ω2),(ω1,ω2))=0 for ∀(ω1,ω2)∈Bε(0)×Bε(0). Thus,

g(ω1,ω2)(u-ω1,v-ω2)∈Nλ,μ, ∀(ω1,ω2)∈Bε(0)×Bε(0).

Also by the differentiability of the implicit function theorem, for all (φ1,φ2)∈E, we know that

Note that

-=E1-E2-E3

andFt(1,(0,0))=K″u,v(1). So (3.1) holds.

and

(3.2)

for all (φ1,φ2)∈E, whereE1,E2,E3is the same as in Lemma 3.1.

Lemma 3.3If (λ,μ)∈Θ, then

(i) there exists a minimizing sequence {(un,vn)}∈Nλ,μsuch that

Proof(i) By Lemma 2.1 and the Ekeland variational principle [11] onNλ,μ, there exists a minimizing sequence {(un,vn)}⊂Nλ,μsuch that

(3.3)

and

(3.4)

Takingnlarge enough, from Lemma 2.7 we have

from which we deduce that for large enoughn,

which yields

C1≤‖(un,vn)‖≤C2

(3.5)

for suitableC1,C2>0.

Now, we show that

‖Iλ,μ(un,vn)‖→0 asn→∞.

Applying Lemma 3.1 with (un,vn) to obtain the functionsgn(ω1,ω2):Bεn(0)×Bεn(0)→R+for someεn>0, such that

gn(0,0)=1,gn(ω1,ω2)(un-ω1,vn-ω2)∈Nλ,μ, ∀(ω1,ω2)∈Bεn(0)×Bεn(0).

gn(ω1,ρ,ω2,ρ)(un-ω1,ρ,vn-ω2,ρ)∈Nλ,μ,

we deduce from (3.4) that

By the mean value theorem, we have

Thus,

(3.6)

It follows fromgn(ω1,ρ,ω2,ρ)(un-ω1,ρ,vn-ω2,ρ)∈Nλ,μand (3.6) that

Hence we obtain

(3.7)

Note that

‖(gn(ω1,ρ,ω2,ρ)(un-ω1,ρ)-un,gn(ω1,ρ,ω2,ρ)(vn-ω2,ρ)-vn)‖
≤ρ|gn(ω1,ρ,ω2,ρ)|+|gn(ω1,ρ,ω2,ρ)-1|‖(un,vn)‖

and

If we letρ→0+in (3.7), then by (3.5) we can find a constantC>0, independent ofρ, such that

We are done once we show that ‖g′n(0,0)‖ is uniformly bounded inn. By (3.1), (3.5) and the Hölder inequality, we have

We only need to show that there exists aδ>0 such thatK″un,vn(1)≥δ>0 whennis large enough. On the contrary, assume that there exists a subsequence {(un,vn)}⊂Nλ,μsuch thatK″un,vn(1)=on(1). By (2.2) and the Sobolev inequality, we have

and so

(3.8)

Similarly, using (2.2), the Sobolev and Hölder inequalities, we have

which implies that

(3.9)

We obtain from (3.8) and (3.9) that

for sufficiently largen. This is a contradiction. Hence we get

This completes the proof of (i).

(ii) Similarly, by using Lemma 3.2, we can prove (ii). We omit the details here.

(ii) (u+,v+) is a solution of equation (1.1).

ProofFrom Lemma 3.3, let {(un,vn)} be a (PS)αλ,μsequence forIλ,μonNλ,μ, i.e.,

Iλ,μ(un,vn)=αλ,μ+on(1),Iλ,μ(un,vn)=on(1) inE-1(Ω),

(3.10)

then it follows from Lemma 2.1 that {(un,vn)} is bounded inE. Hence, up to a subsequence, there exists (u+,v+)∈Esuch that

(3.11)

By the Hölder inequality and(3.11), we can infer that

(3.12)

and

(3.13)

First, we can claim that (u+,v+) is a nontrivial solution of (1.1). Indeed, by (3.10) and (3.11), it is easy to see that (u+,v+) is a solution of (1.1). Next we show that (u+,v+) is nontrivial. From (un,vn)∈Nλ,μ, we have

(3.14)

Letn→∞ in (3.14), we obtain