Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 228, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
SOLUTIONS TO QUASILINEAR EQUATIONS OF N-BIHARMONIC TYPE WITH DEGENERATE COERCIVITY
SAMI AOUAOUI
Abstract. In this article we show the existence of multiple solutions for quasi- linear equations in divergence form with degenerate coercivity. Our strategy is to combine a variational method and an iterative technique to obtain the solutions.
1. Introduction and statement of main results In this article, we study the quasilinear equation
−div a(x, u)|∇u|N−2∇u
+V(x)|u|N−2u+∆2Nu=f(x, u)+h(x), x∈RN, (1.1) where N ≥2, ∆2Nu= ∆(|∆u|N−2∆u), andh∈LN0(RN), N0 = NN−1, h6= 0 and h≥0. Concerning the functionsV,f anda, we have the following assumptions:
(V1) V :RN →Ris a continuous function such that V(x)≥V0>0, ∀x∈RN, whereV0 is a positive constant.
(V2) For every M > 0, meas({x ∈ RN, V(x) ≤ M}) < +∞, where “meas”
denotes the Lebesgue measure inRN.
(H1) f : RN ×R → Ris a Carath´eodory function. We assume that for every positive real numberk >0, there exist two positive constantsαk> N−1 andCk>0 such that
|f(x, s)| ≤Ck|s|αk, a.e. x∈RN and for alls∈Rwith|s| ≤k.
(H2) There existsν > N such that
νF(x, s)≤f(x, s)s, ∀(x, s)∈RN×R, whereF(x, s) = Z s
0
f(x, t)dt.
(H3) There exist two real numbersA >0 andp > N, such that F(x, s)≥Asp, a.e. x∈RN, ∀s≥0.
2000Mathematics Subject Classification. 35D30, 35J20, 35J61, 58E05.
Key words and phrases. Variational method;N-biharmonic operator; mountain pass theorem;
minimax level; iterative scheme; critical point.
c
2014 Texas State University - San Marcos.
Submitted June 22, 2014. Published October 27, 2014.
1
(H4) There exist two positive constantsβ0andβ1 such that
|f(x, s1)−f(x, s2)| ≤β0zβ1|s1−s2|N−1, a.ex∈RN,∀z∈[0,1] and∀s1, s2∈[−z, z].
(H5) a:RN ×R→Ris a continuous function satisfying the following property:
for everyk >0, there exist 0< ak < a0k <+∞such that ak ≤a(x, s)≤a0k, ∀x∈RN and|s| ≤k.
(H6) There exists a constantL >0 such that
|a(x, s1)−a(x, s2)| ≤L|s1−s2|N−1, a.e. x∈RN, ∀s1, s2∈[−1,1].
Examples. WhenN = 2, forf, we can choose:
(1) f(x, s) =λ|s|α−1s, λ >0,α >1.
(2) f(x, s) =λ|s|α−1s+|s|β−1s(ep0s2−1),λ >0,α >1,β >1 andp0>0.
Fora, we can choose:
(1) a(x, s) = 1 +|s|σ−1s, σ >1.
(2) a(x, s) =1+s12.
Many articles about problems similar to (1.1), having a divergence part of the form −div(A(x, u)|∇u|p−2∇u) with degenerate coercivity, have been published.
Among them, the following model is of special interest:
−div(|∇u|p−2∇u
(1 +|u|)q ) =f in Ω,
where Ω is some open (bounded in the majority of cases) domain ofRN, N ≥2, q >0,p >1 andfis datum satisfying some summability condition. See for example [4, 7, 8, 9, 10, 11] and references therein. We want to mention also the model
−div(A(x, u)|∇u|p(x)−2∇u) +|u|p(x)−2u=Z(x, u,∇u), inRN, N ≥3, wherepis some bounded and Lipschitz continuous function. This model was studied in [6] in the very special framework of the generalized Sobolev space with variable exponents. In the previously cited works, the authors use approximations in order to overcome the lack of coercivity. Then, establish a priori estimates on the sequence of approximative solutions, and then use the passage to the limit to finally obtain a weak solution for the initial equation.
In this article, we develop a new method to deal with such kind of problems. The main idea in this new method is inspired by the work [14]. In [14], de Figueiredo, Girardi and Matzeu considered the semilinear elliptic equation
−∆u=f(x, u,∇u) in Ω,
u= 0 on∂Ω, (1.2)
where Ω is a bounded smooth domain of RN, N ≥ 3. Because the dependence of the nonlinearity on the gradient of the solution, (1.2) is non-variational and a direct attack to it using critical point theory is not possible. The new approach by de Figueiredo, Girardi and Matzeu consists of associating with (1.2) a family of semilinear elliptic problems with no dependence on the gradient. Namely, for each w∈H01(Ω), they considered the problem
−∆u=f(x, u,∇w) in Ω,
u= 0 on∂Ω. (1.3)
Thus, the authors have “frozen” the gradient term. Problem (1.3) is of variational nature and could be treated by variational method. We have to mention that this idea was used in some later works dealing also with nonlinear problems involving nonlinearities with dependence on the gradient. We can cite [15, 17, 18, 26, 27, 29].
In this article, we try to use this idea to discuss a completely different kind of problem. In fact, in the problem (1.1) and in contrast with (1.2) (and similar equations), the nonvariational nature is not due to the dependence of the right- hand term on the gradient of the solution but it is in reality due to the presence of the coefficienta(x, u) in the divergence part. Hence, we will try to “freeze” the terma(x, u). The “associated” problem will be variational and consequently could be treated using the critical point theory. An iterative scheme will be performed in order to obtain weak solutions for the initial problem (1.1). This method allows us to obtain a multiplicity result which, knowing that in the majority of cases the classical nonvariational methods give the existence of one solution, could be seen as an interesting result.
The existence of theN−biharmonic operator, ∆2N, is remarkable. The impor- tance of studying fourth-order equations lies in the fact that they can describe some physical phenomena as the deformations of an elastic beam in equilibrium state (see [24, 36]). Laser and McKenna [23] pointed out that this type of nonlinearity pro- vides a model to study travelling waves in suspension bridges. For this reason, there is a wide literature that deals with existence and multiplicity of solutions for nonlinear fourth-order elliptic problems in bounded and unbounded domains. See for example [19, 20, 25, 28] and references therein. On the other hand, the study of nonlinear equations involving theN-Laplacian operator,N≥2, which is a bor- derline case for the Sobolev embedding, could be considered as one of the most interesting topics of research during last decades. A special interest has been given to equation ofN-Laplacian type containing nonlinear terms which have a subcriti- cal or critical exponential growth. See [1, 2, 3, 12, 16, 21, 22, 30, 31, 32, 33, 34, 35]
and references therein. Here, we highlight the fact that in the present work we deal with a more general type of nonlinearity which includes the case of exponential growth.
The appropriate space in which the problem (1.1) will be studied is the subspace ofW2,N(RN),
E=n
u∈W2,N(RN) : Z
RN
V(x)|u|Ndx <+∞o ,
which is a Banach reflexive space equipped with the norm kuk=Z
RN
(|∇u|N +V(x)|u|N +|∆u|N)dx1/N .
In view of (V1), we clearly have
E ,→W2,N(RN),→Lq(RN), ∀N ≤q≤+∞.
Also there exists a positive constantδ0>0 such that
|u|L∞(RN)≤δ0kuk, ∀u∈E. (1.4) Furthermore, since (V2) holds, we obtain (see [31]) the compactness of the embed- ding
E ,→Lp(RN), for allp≥N.
This compact embedding will be crucial in the proof of our multiplicity result.
Definition 1.1. A function u ∈ E is said to be a weak solution of the problem (1.1) if it satisfies
Z
RN
a(x, u)|∇u|N−2∇u∇v dx+ Z
RN
V(x)|u|N−2uv dx+ Z
RN
|∆u|N−2∆u∆v dx
= Z
RN
f(x, u)v dx+ Z
RN
hv dx, ∀v∈E.
The main result in the present paper is given by the following theorem.
Theorem 1.2. Assume that(V1), (V2), (H1)–(H6)hold. Then, there existA0>0 and d0 > 0 with the following property: if A > A0, and |h|LN0
(RN) < d0, then problem (1.1)admits at least two nontrivial weak solutions.
2. Proof of main resutls
The proof of Theorem 1.2 will be divided into several steps. First, forw ∈E, we introduce the functionalIw defined onE by
Iw(u) = Z
RN
a(x, w)|∇u|N +V(x)|u|N +|∆u|N
N dx−
Z
RN
F(x, u)dx− Z
RN
hu dx.
Lemma 2.1. Assume that(V1), (V2), (H1), (H5)hold. Then, there exist0< ρ <
1
δ0,µ >0, andd >0 independent ofw such that Iw(u)≥µ, forkuk=ρ, provided that kwk ≤ δ1
0 and|h|LN0(RN)< d.
Proof. Forkwk ≤ δ1
0, by (1.4) it yields |w|L∞(RN)≤1 and by (H5) we can assert that there exist 0< a1< a01<+∞such that
a1≤a(x, w(x))≤a01, ∀x∈RN. (2.1) Forkuk ≤1/δ0, then by (1.4) it yields
|u(x)| ≤1, a.e.x∈RN.
By (H1), we get the existence of two constantsα > N−1 andc1>0 such that
|f(x, u(x))| ≤c1|u(x)|α, a.e x∈RN. (2.2) This implies
Z
RN
F(x, u)dx≤c2kukα+1. This inequality and (2.1) give
Iw(u)≥min{1, a1}kukN
N −c2kukα+1− |h|LN0(RN)kuk.
Sinceα+ 1> N, then one can easily find 0< ρ < min{1, δ1
0} small enough such that
min{1, a1}ρN
N −c2ρ1+α≥min{1, a1}ρN 2N. It follows that
Iw(u)≥min{1, a1}ρN
2N − |h|LN0(RN)ρ, forkuk=ρ.
We complete the proof of Lemma 2.1 by taking d = min{1, a1}ρN−14N and µ =
min{1, a1}4NρN.
Lemma 2.2. Assume that (V1), (V2), (H1), (H3), (H5) hold. Then, there exists ϑ ∈ E independent of w such that kϑk > ρ and Iw(ϑ) < 0 for all w ∈ E with kwk ≤ δ1
0.
Proof. Letϕ∈C0∞(RN) be such thatϕ6= 0 andϕ≥0. Fort >0, we have Iw(tϕ)≤max{1, a01}tN
NkϕkN − Z
RN
F(x, tϕ)dx
≤max{1, a01}tN
NkϕkN −Atp|ϕ|pLp(
RN). Sincep > N, we have
max{1, a01}tN
NkϕkN −Atp|ϕ|pLp(
RN)→ −∞, ast→+∞.
Thus, we can choose ϑ =t0ϕwhere t0 is large enough such that kt0ϕk >1 > ρ.
This completes the proof.
Now, by the Mountain Pass Theorem without the Palais-Smale condition (see [5, 37]), there exists a sequence (un,w)⊂Esuch thatIw0 (un,w)→0 andIw(un,w)→ cw= infγ∈Γsup0≤t≤1Iw(γ(t)), where
Γ ={γ∈C([0,1], E), γ(0) = 0, γ(1) =t0ϕ=ϑ}.
Lemma 2.3. Assume that (V1), (V2), (H1)–(H3), (H5) hold. Let w ∈ E with kwk ≤ δ1
0. Then, for every0< η < δ1
0, there exist Aη >0 and dη >0 such that:
if A > Aη, and|h|LN0
(RN)< dη then the functional Iw admits a nontrivial critical point uw ∈ E such that 0 < µ≤ Iw(uw) = cw, where µ is given by Lemma 2.1.
Moreover,kuwk ≤η.
Proof. We have
Iw(un,w)−1
νhIw0 (un,w), un,wi ≤cw+on(1)(1 +kun,wk).
Using (H2) and (2.1), we have min{1, a1}(1
N −1
ν)kun,wkN ≤cw+on(1)(1 +kun,wk) +|h|LN0(RN)kun,wk. (2.3) Then, (un,w) is a bounded sequence inE. Now, by Young’s inequality, there exists c3>0 such that
|h|LN0(RN)kun,wk ≤ min{1, a1}
2 (1
N − 1
ν)kun,wkN +c3|h|NLN00
(RN). Putting this inequality in (2.3), we obtain
min{1, a1}
2 (1
N −1
ν)kun,wkN ≤cw+on(1)(1 +kun,wk) +c3|h|NLN00
(RN). By passing to the upper limit, we obtain
lim sup
n→+∞
kun,wkN ≤ 2cw
min{1, a1}(N1 −ν1)+c4|h|NLN00
(RN). (2.4)
Now, observe that by the even definition ofcw, we have cw≤max
t≥0 Iw(tϕ)≤max
t≥0
max{1, a01}tNkϕkN
N −Atp|ϕ|pLp(RN)
.
It is clear that the function
K(t) =max{1, a01}tNkϕkN
N −Atp|ϕ|pLp(RN)
defined on [0,+∞[ attains its maximum at tmax= (max{1, a01}kϕkN
Ap|ϕ|pLp(
RN)
)p−N1 . Thus,
maxt≥0 K(t) = max{1, a01}kϕkN(1 N −1
p)max{1, a01}kϕkN pA|ϕ|pLp(RN)
p−NN .
Hence,
cw≤max{1, a01}kϕkN(1 N −1
p)max{1, a01}kϕkN pA|ϕ|pLp(RN)
p−NN
. (2.5)
Denote
Σ(A) = max{1, a01}kϕkN(1 N −1
p)(max{1, a01}kϕkN pA|ϕ|pLp(RN)
)p−NN . Fix 0< η < δ1
0. It is clear that there existsAη>0 large enough such that Σ(A)≤min{1, a1}
4 (1
N −1 ν)ηN, provided thatA > Aη. On the other hand, we can choose |h|LN0
(RN)small enough such that
c4|h|NLN00
(RN)≤ ηN 2 . Hence, by (2.4) and (2.5) we deduce that
lim sup
n→+∞
kun,wkN ≤ηN.
It follows, that there existsn0>1 large enough such that kun,wk ≤ 2Σ(A)
min{1, a1}(N1 −ν1)+c4|h|NLN00
(RN)
1/N
≤η < 1
δ0, ∀n≥n0. Up to a subsequence, (un,w) is weakly convergent to some pointuwinE. We claim that, up to a subsequence, (un,w) is strongly convergent touw inE. First, observe that by (2.2) we have
Z
RN
|f(x, un,w)|N0dx≤c5 Z
RN
|un,w|αN0dx.
Thus, we get the boundedness of the sequence (f(·, un,w)) inLN0(RN). This fact together with the compact embeddingE ,→,→LN(RN) imply
Z
RN
|f(x, un,w)(un,w−uw)|dx→0, n→+∞. (2.6)
Using (2.6) and the weak convergence of (un,w) touw inE, we obtain Z
RN
a(x, w)(|∇un,w|N−2∇un,w− |∇uw|N−2∇uw)∇(un,w−uw)dx +
Z
RN
V(x)(|un,w|N−2un,w− |uw|N−2uw)(un,w−uw)dx +
Z
RN
(|∆un,w|N−2∆un,w− |∆uw|N−2∆uw)∆(un,w−uw)dx
→0, as n→+∞.
Recalling the standard inequality
(|x|N−2x− |y|N−2y)(x−y)≥2−N|x−y|N, ∀x, y∈Rr, ∀r≥1, (2.7) we can deduce that, up to a subsequence, (un,w) is strongly convergent touw in E. Consequently,uwis a critical point ofIw andIw(uw) =cw≥µ >0. Moreover, taking into account that
kun,wk ≤η, ∀n≥n0,
and passing to the limit asn→+∞, we obtainkuwk ≤η.
Lemma 2.4. Assume that (V1)–(V2), (H1), (H5) hold. Let w ∈ E be such that kwk ≤ δ1
0. Then, the functionalIw admits a nontrivial weak solutionUw∈E such that Iw(Uw) ≤ −σ < 0 and kUwk ≤ ρ, where ρ is given by Lemma 2.1 and σ is some positive constant independent ofw.
Proof. Letϕbe the function introduced and used in Lemma 2.2. Clearly, we can choose 0≤ϕ(x)≤1 for allx∈RN. For 0< t <1, we have
Iw(tϕ)≤max{1, a01}tN
NkϕkN − Z
RN
F(x, tϕ)dx−t Z
RN
hϕ dx. (2.8) By (H1), we can easily obtain
lim
t→0+
Z
RN
F(x, tϕ)
t dx= 0.
Moreover, since
Z
RN
hϕ dx >0,
by (2.8) one can easily find 0< t1 <inf(1,kϕkρ ) small enough and independent of w, andσ >0 also independent ofw such that
Iw(t1ϕ)≤ −σ <0.
Now, denote
θw= inf{Iw(u), kuk ≤ρ}.
In view of Lemma 2.1 and by the Ekeland’s variational principle (see [13]), there exists a sequence (Un,w)⊂E such that
kUn,wk ≤ρ, Iw(Un,w)→θw, and Iw0 (Un,w)→0.
Up to a subsequence, (Un,w) is weakly convergent to some pointUwinE. Observe thatρ < δ1
0 and arguing as for (2.6), we can prove that Z
RN
f(x, Un,w)(Un,w−Uw)dx→0, asn→+∞.
Proceeding exactly as for the sequence (un,w), we can easily show that, up to a subsequence, (Un,w) is strongly convergent toUwin E. Therefore, the pointUw is a critical point ofIw satisfying
Iw(Uw) =θw≤ −σ <0, and kUwk ≤ρ.
This completes the proof.
Proof of Theorem 1.2 completed. To conclude the proof, an iterative scheme will be performed. Let 0< η <1/δ0and fixu0∈Esuch thatku0k ≤η. By Lemma 2.3, under the conditionA > Aη, and|h|LN0(RN)< dη, the functionalIu0 admits a nontrivial critical pointu1∈Esuch that
Iu0(u1)≥µ >0, ku1k ≤η.
Similarly, the functionalIu1 admits a critical pointu2such that Iu1(u2)≥µ >0, ku2k ≤η.
This way, we construct a sequence (un)⊂E such that kunk ≤η, Iun−1(un)≥µ >0, andun is a critical point of the functionalIun−1. Thus, we have
Z
RN
a(x, un−1)|∇un|N−2∇un∇v dx +
Z
RN
V(x)|un|N−2unv dx+ Z
RN
|∆un|N−2∆un∆v dx
= Z
RN
f(x, un)v dx+ Z
RN
hv dx, ∀v∈E.
(2.9)
Similarly, we have Z
RN
a(x, un)|∇un+1|N−2∇un+1∇v dx +
Z
RN
V(x)|un+1|N−2un+1v dx+ Z
RN
|∆un+1|N−2∆un+1∆v dx
= Z
RN
f(x, un+1)v dx+ Z
RN
hv dx, ∀v∈E.
(2.10)
Taking v = un+1−un as test function in (2.9) and (2.10), and subtracting one equation from the other, we obtain
Z
RN
a(x, un)
|∇un+1|N−2∇un+1− |∇un|N−2∇un
∇(un+1−un)dx
+ Z
RN
(a(x, un)−a(x, un−1))|∇un|N−2∇un∇(un+1−un)dx +
Z
RN
V(x)
|un+1|N−2un+1− |un|N−2un
(un+1−un)dx +
Z
RN
|∆un+1|N−2∆un+1− |∆un|N−2∆un
∆(un+1−un)dx
= Z
RN
(f(x, un+1)−f(x, un))(un+1−un)dx.
(2.11)
Sincekunk ≤η, for alln≥1, it follows by (1.4) that
|un|L∞(RN), |un+1|L∞(RN)≤δ0η <1, ∀n≥0.
By (H4), it yields
|f(x, un+1(x))−f(x, un(x))| ≤β0(δ0η)β1|un+1(x)−un(x)|N−1, a.e. x∈RN, for alln≥0. Consequently
Z
RN
(f(x, un+1)−f(x, un))(un+1−un)dx≤β0(δ0η)β1 Z
RN
|un+1−un|Ndx. (2.12) If we takeη small enough such that
β0(δ0η)β1 ≤V0min{1, a1}2−N
4 ,
then from (2.12) we infer Z
RN
(f(x, un+1)−f(x, un))(un+1−un)dx
≤ min{1, a1}2−N 4
Z
RN
V(x)|un+1−un|Ndx
≤ min{1, a1}2−N
4 kun+1−unkN.
(2.13)
On the other hand, by Young’s inequality we have Z
RN
|a(x, un)−a(x, un−1)| |∇un|N−1|∇(un+1−un)|dx
≤min{1, a1}2−N 4
Z
RN
|∇(un+1−un)|Ndx +c6
Z
RN
|a(x, un)−a(x, un−1)|N0|∇un|Ndx, and by (H6) it follows that
Z
RN
|a(x, un)−a(x, un−1)| |∇un|N−1|∇(un+1−un)|dx
≤min{1, a1}2−N
4 kun+1−unkN+c6LN0|un−un−1|NL∞(RN)kunkN
≤min{1, a1}2−N
4 kun+1−unkN+ (c6LN0δ0NηN)kun−un−1kN.
(2.14)
Using (2.7), (2.11), (2.13) and (2.14), we obtain min{1, a1}
2N+1 kun+1−unkN ≤(c6LN0δ0NηN)kun−un−1kN. (2.15) Set
Γ(η) =c6LN0δ0NηN2N+1 min{1, a1}
1/N
.
By (2.15), it yields
kun+1−unk ≤Γ(η)kun−un−1k, ∀n≥1. (2.16)
Clearly, we can choose η small enough such that Γ(η) <1. Therefore, by (2.16) (un) is a Cauchy sequence and by consequence it is strongly convergent to some pointu∈E. Passing to the limit asn→+∞in (2.9), we conclude thatusatisfies
Z
RN
a(x, u)|∇u|N−2∇u∇v dx+ Z
RN
V(x)|u|N−2uv dx+ Z
RN
|∆u|N−2∆u∆v dx
= Z
RN
f(x, u)v dx+ Z
RN
hv dx, ∀v∈E.
According to Definition 1.1, this means thatuis a weak solution of problem (1.1).
On the other hand, we haveIun−1(un)≥µ >0, for alln≥2. Hence, Z
RN
a(x, un−1)|∇un|N+V(x)|un|N +|∆un|N
N dx
− Z
RN
F(x, un)dx− Z
RN
hundx≥µ >0.
Passing to the limit asn→+∞, it follows Ψ(u) =
Z
RN
a(x, u)|∇u|N+V(x)|u|N +|∆u|N
N dx
− Z
RN
F(x, u)dx− Z
RN
hu dx≥µ >0.
Now, using Lemma 2.4 it is immediate that an iterative scheme could be performed to construct a sequence (Un)⊂E such that, for alln≥1,
kUnk ≤ρ < 1
δ0, IUn−1(Un)≤ −σ <0,
and Un is a critical point of the functional IUn−1. Moreover, using the same ar- guments as for the sequence (un), we can easily prove that the sequence (Un) is strongly convergent to some pointU ∈Ewhich is a weak solution of problem (1.1).
Furthermore, we have Ψ(U) =
Z
RN
a(x, U)|∇U|N +V(x)|U|N +|∆U|N
N dx
− Z
RN
F(x, U)dx− Z
RN
hU dx≤ −σ <0.
Hence,u6=U. This completes the proof of Theorem 1.2.
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