J. Math. Anal. Appl. 426 (2015) 484–504

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

A priori estimates and existence of positive solutions

for higher-order elliptic equations

Hatem Hajlaoui ∗, Abdellaziz HarrabiInstitut Supérieur des Mathématiques Appliquées et de l’Informatique de Kairouan, Université de Kairouan, Tunisia

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 October 2013Available online 28 January 2015Submitted by V. Radulescu

Keywords:Higher-order elliptic equationsElliptic systemA priori boundsTopological degree

In this paper, we present a new sufficient condition to get a priori L∞-estimates for positive solutions of higher-order elliptic equations in a smooth bounded convex domain of RN with Navier boundary conditions or for radially symmetric solu-tions in the ball with Dirichlet boundary conditions. A priori L∞-estimates for positive solutions of the second-order elliptic system in a smooth bounded convex domain of RN with Dirichlet boundary conditions are also established. As usual, these a priori bounds allow us to obtain existence results. Also, by truncation tech-nique combined with minimax method, we obtain existence of positive solution for higher-order elliptic equations of the form (1.1) below when we only assume that the nonlinearity is a nondecreasing positive function satisfying: lim infs→+∞

f(s)s

> Λ1, lim sups→0

f(s)s

< Λ1, where Λ1 is the first eigenvalue of (−Δ)m with Navier bound-ary conditions and the weak subcritical growth condition: lims→+∞

f(s)sσ = 0, where

σ = N+2mN−2m .

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we are concerned with the existence of positive solutions for the following polyharmonic equations with Navier boundary conditions:

{ (−Δ)mu = f(u) in Ω,

u = Δu = · · · = (Δ)m−1u = 0 on ∂Ω,(1.1)

where Ω is a smooth bounded convex domain of RN , N ≥ 2, and f is a nonlinearity to be specified later; or under Dirichlet boundary conditions in the ball BR of RN , N ≥ 2, centred at the origin with radius R:

* Corresponding author.E-mail addresses: hajlouihatem@gmail.com (H. Hajlaoui), abdellaziz.harrabi@yahoo.fr (A. Harrabi).

http://dx.doi.org/10.1016/j.jmaa.2015.01.0580022-247X/© 2015 Elsevier Inc. All rights reserved.

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 485

{ (−Δ)mu = f(u) in BR,

u = Du = · · · = (D)m−1u = 0 on ∂BR.(1.2)

In the last decades, problems related to existence and nonexistence of positive solutions for higher-order elliptic equations and systems have received a good deal of attention; see [4,9,10,13,15,16,12]. When vari-ational methods cannot be employed, the question of existence of solutions may be dealt via topological methods. Using known topological fixed-point theorems, the proof of existence is essentially reduced to deriving a priori estimates for all possible solutions which are obtained in general by bootstrap or blow-up arguments [4,10,16,15,14].

When f has an asymptotic behaviour like sp at infinity (with 1 < p < N+2mN−2m ), Soranzo [16] established

existence results for Eqs. (1.1) and (1.2) by combining L∞-estimates with topological degree theory, which gave a direct extension of the results for m = 1 in [5]. The methods used in [16] to obtain L∞ bounds are based on the well-known Pohozaev identity and boot-strap argument (in the Navier problem) and blow-up argument (in the Dirichlet problem in the ball). However, we can remark that under assumptions of Theorem 4 in [16], the energy functional associated with (1.1) satisfies the Palais–Smale condition and has the geometry of Mountain Pass lemma, consequently one can deduce the existence of a solution to (1.1)having the minimax structure. Moreover, since f ′(s) ≥ 0 with f(0) = 0 implies that f(s) ≥ 0, the maximum principle ensures that such a solution is positive.

Let us fix notation: Consider the following Hilbert spaces:

Hmϑ (Ω) :=

{v ∈ Hm(Ω); Δjv = 0 on ∂Ω, for j <

m

2

},

Hm0 (Ω) :=

{v ∈ Hm(Ω); Dju = 0 on ∂Ω, for j = 0, 1, . . . ,m− 1

},

which are endowed with the norm:

‖u‖m ={‖Δku‖L2(Ω) if m = 2k,‖∇(Δku)‖L2(Ω) if m = 2k + 1.

Throughout this paper, we assume that N ≥ 2m + 1. The first eigenvalue of (−Δ)m in Ω with Navier boundary conditions and the one of (−Δ)m in BR with Dirichlet boundary conditions will be denoted indifferently by Λ1 and we denote by λ1 the first eigenvalue of −Δ with Dirichlet boundary conditions.

The main goal of this paper is to improve the results of [16] under some weaker conditions. One of the key ingredients in the case of Eq. (1.1) is the L∞ boundary estimates of gradient derived in [16] (see pages 469–471). For this purpose, Soranzo adapted the proof of the idea in [5], by using Troy’s techniques [17]. More precisely, since Eq. (1.1) may be written as the following system:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

u0 = u,

−Δu0 = u1,...−Δum−1 = f(u0),u0 = · · · = um−1 = 0, on ∂Ω,

(1.3)

adapting the moving plan technique used by Troy in his famous paper [17], we can prove that ui are decreasing with respect to the outward normal direction in a neighbourhood of the boundary. Clearly, this allows us to get estimate of L∞ norm of ∇ui near the boundary by interior estimates on ui. Comparing to the Laplacian operator [5], the system case is more restrictive and one has to assume that f ∈ C1(R+)with f ′ ≥ 0. However, an examination of the proof of Lemma 1.1 in [16] shows that we may relax to the following assumption:

486 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

(f1) f : [0, +∞) → R is a nondecreasing and locally Lipschitz function.

Recall that the following standard assumption is required to obtain L∞-estimates:

(f2) lim infs→+∞f(s)s > Λ1.

In fact, (f2) is useful to obtain the following integral estimate:∫Ω

f(u)φ1dx = Λ1

∫Ω

uφ1dx ≤ C0, (1.4)

where φ1 is a fixed positive eigenfunction associated with Λ1 and C0 is a positive constant independent of u (see [5,4,15,16]). We just recall that from (1.4), we obtain a priori bounds of ‖u‖L1(Ω,δdx) with δ =d(x, ∂Ω). This is sufficient to derive L∞-estimates even in general domain and without assuming that f is a nondecreasing function but needs a restrictive growth condition (see [10,13]).

Now, we state the boundary estimates of gradient (see inequality (23), page 471 in [16]). We omit the proof here.

Lemma 1.1. Suppose that f satisfies (f1) and (f2), then there exists a constant C > 0 such that∣∣∣∣∂(Δiu)

∂n(x)

∣∣∣∣ ≤ C, for all i = 0, 1, . . . ,m− 1, x ∈ ∂Ω, (1.5)

for any positive solution u ∈ C2m(Ω) of (1.1).

Our main purpose here is to relax the asymptotic behaviour at infinity imposed to f in [16], that is: There exists a number α ∈ (0, +∞) such that

lims→∞

f(s)sp

= α,

where 1 < p < N+2mN−2m and N > 2m. We use also the following subcritical condition:

(H) There exist two positive constants C and s0 such that

C∣∣f(s)

∣∣ 2NN+2m ≤ 2N

N − 2mF (s) − sf(s), for all s > s0.

In order to clarify the improvements brought by this assumption and to state our main results, we need the following additional assumption:

(f3) lims→+∞f(s)sσ = 0, where σ = N+2m

N−2m .

In Appendix A, we will prove that if f satisfies (f3) and f(s) ≥ C0sN

N−2m (for C0 > 0 and s large), then (H) is also weaker than the following condition introduced in [5] (for m = 1):

(H ′) ∃θ ∈ (0, 2NN−2m ) such that lim sups→+∞

sf(s)−θF (s)s2|f(s)|2m/N ≤ 0.

Moreover, we show that (H) covers a large class of nonlinearities close to the critical growth. For example:∀q ≥ N+2m , there exists αq large enough such that ∀α ≥ αq

N−2m

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 487

f(s) = sN+2mN−2m

(ln(s + α))q , (1.6)

satisfies (H), (f1)–(f3) but not (H ′).In this paper, we are also concerned with the following nonlinear elliptic system:

{−Δu = f(v) in Ω, u = 0 on ∂Ω,

−Δv = g(u) in Ω, v = 0 on ∂Ω,(1.7)

where Ω is a bounded convex domain of RN , N > 2, with boundary ∂Ω sufficiently smooth.A priori bounds for positive solutions of (1.7) have been established via different methods by several

authors, see [4,6,10,13,18] and the references therein. It is proved in [4] that if there are numbers α, β ∈ ]0, ∞[such that

lims→∞

f(s)sp

= α and lims→∞

g(s)sq

= β,

where 1 ≤ p, q < ∞ satisfy 1p+1+ 1

q+1 > N−2N , N ≥ 3, then system (1.7) has a positive solution. Moreover, it is

well known that system (1.7) has no positive solution, for f(u) = up and g(u) = vq, when 1p+1 + 1

q+1 ≤ N−2N ,

N ≥ 3 (see [11]).Combining a priori L∞-estimates with the degree theory, we shall study the existence of positive solutions

for system (1.7), when f is asymptotic to the linear growth at infinity and g is close to the critical growth at infinity (see assumption (S3) below). Also, we will investigate the case when the growth of f and g are both close to s

N+2N−2 (see assumption (S′

3) below).

1.1. A priori L∞-estimates

In this subsection, first we are interested in the a priori L∞-estimates for Eqs. (1.1) and (1.2).

Proposition 1.1. Assume that Ω is a smooth bounded convex domain (for instance of class C2m+1) and f satisfies (f1), (f2) and (H). Then,

1. there exists a constant C > 0 such that ‖u‖m ≤ C, for any positive solution u ∈ C2m(Ω) of (1.1).2. If, moreover, f satisfies (f3), then there exists a constant C > 0 such that

‖u‖L∞(Ω) ≤ C,

for any positive solution u ∈ C2m(Ω) of (1.1).

For Eq. (1.2), let us first mention that we need to assume, instead of (f1) and (f3), that f satisfies the following conditions:

(f ′1) f is a C1 positive function,

(f ′3) lims→+∞

f ′(s)sσ′ = 0, where σ′ = 4m

N−2m .

Our result reads as follows.

Proposition 1.2. Assume that f satisfies (f ′1), (f2) and (H). Then,

1. there exists a constant C > 0 such that ‖u‖m ≤ C, for any radial positive solution u ∈ C2m(BR) of (1.2).

488 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

2. If, moreover, f satisfies (f ′3), then there exists a constant C > 0 such that

‖u‖L∞(BR) ≤ C,

for any radial positive solution u ∈ C2m(BR) of (1.2).

In another framework, Eq. (1.2) was studied by E. Berchio, F. Gazzola and T. Weth. They proved that, if f is a continuous nondecreasing function with f(0) ≥ 0 then every strong positive solution u ∈Hm

0 (BR) ∩L∞(BR) of (1.2) is radially symmetric and strictly decreasing in the radial variable (see Theorem 1 in [2]).

Remark 1.1.

1. The item (1) of Propositions 1.1 and 1.2 holds if we replace (H) by (H ′) (see Appendix B).2. Let u ∈ C2m(Ω) be a positive solution of (1.1) (resp. a radial positive solution of (1.2), with Ω = BR).

If f satisfies (f1) (resp. (f ′1)) and (f2) then, applying Lemma 1.1 (resp. (2.8), see below), the well-known

Pohozaev identity (see Lemma 2 in [16]) implies

∫Ω

F (u)dx− N − 2m2N

∫Ω

uf(u)dx ≤ C. (1.8)

So, for N < 2m, there exists a positive constant C independent of u such that ∫Ωuf(u)dx ≤ C and by

integration by parts, we can get ‖u‖m ≤ C without assuming (H), while if N = 2m, we see clearly that the condition (H) can be reduced to the following: There exist two positive constants q and s0 such that

qF (s) − sf(s) ≥ 0, for all s > s0.

Following the same proofs of item (2) of Propositions 1.1 and 1.2 (see Section 2), the L∞-estimates can be established for N ≤ 2m if we replace (f3) or (f ′

3) by lims→+∞f(s)sσ = 0, where σ < ∞.

To establish a priori L∞-estimates for positive solutions of the system (1.7), let us assume that

(S1) f, g : [0, ∞[ → R are nondecreasing locally Lipschitz functions,(S2) there are numbers a ∈ ]0, ∞[, b ∈ ]0, ∞] such that

lim infs→∞

f(s)s

= a, lim infs→∞

g(s)s

= b

with ab > λ21,

(S3) there is α > 0 such that lim sups→∞f(s)s = α and lims→∞

g(s)

sN+4N−4

= 0,(S4) there exist two positive constants C and s0 such that

C∣∣g(s)∣∣ 2N

N+4 ≤ 2NN − 4G(s) − sg(s) and 0 ≤ 2F (s) − sf(s)

for s > s0, where F (s) =∫ s

0 f(t)dt and G(s) =∫ s

0 g(t)dt.

Then, we have

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 489

Proposition 1.3. Suppose that f and g satisfy (S1)–(S4) and Ω is a smooth bounded convex domain (for instance of class C3). Then, there exists a positive constant C such that

‖u‖L∞(Ω) ≤ C and ‖v‖L∞(Ω) ≤ C,

for all positive solutions (u, v) of (1.7).

To study the case when the growth of f and g are both close to sN+2N−2 , we need, instead of (S2)–(S4), the

following assumptions:

(S′2) there are numbers a, b ∈ ]0, ∞] such that

lim infs→∞

f(s)s

= a, lim infs→∞

g(s)s

= b

with ab > λ21,

(S′3) lims→∞

f(s)

sN+2N−2

= 0 and lims→∞g(s)

sN+2N−2

= 0,(S′

4) there exist two positive constants C and s0 such that

C∣∣g(s)∣∣ 2N

N+2 ≤ 2NN − 2G(s) − sg(s) and 0 ≤ 2F (s) − sf(s)

for s > s0,or

C∣∣f(s)

∣∣ 2NN+2 ≤ 2N

N − 2F (s) − sf(s) and 0 ≤ 2G(s) − sg(s)

for s > s0, where F (s) =∫ s

0 f(t)dt and G(s) =∫ s

0 g(t)dt.

Then, we have

Proposition 1.4. Suppose that f and g satisfy (S1), (S′2), (S′

3), (S′4) and Ω is a smooth bounded convex

domain (for instance of class C3). Then, there exists a positive constant C such that

‖u‖L∞(Ω) ≤ C and ‖v‖L∞(Ω) ≤ C,

for all positive solutions (u, v) of (1.7).

In order to see better the improvement brought by our assumptions, we consider the following instructive examples: Let q ≥ N+4

N−4 , then there exists αq such that ∀α ≥ αq,

g(s) = sN+4N−4

(ln(s + α))q ,

f(s) = as− 1s + γ

+ 1γ, a > 0, γ > 0

satisfy (S1)–(S4), but not the assumptions of [4,13]. Moreover, if a + 1γ < λ1, then all these examples satisfy

f(0) = g(0) = 0, lim sups→0f(s) < λ1 and lim sups→0

g(s) < λ1. On the other hand, for q ≥ N+2 there

s s N−2

490 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

exists αq large enough such that ∀α ≥ αq,

f(s) = g(s) = sN+2N−2

(ln(s + α))q

satisfy (S1) and (S′2)–(S′

4), but not the assumptions of [4,13].

Remark 1.2. Assume that

1p + 1 + 1

q + 1 = N − 2N

,

where p, q > 1 and suppose that f and g satisfy the following conditions:

• lims→∞f(s)sp = lims→∞

g(s)sq = 0,

• there exist two positive constants C and s0 such that either

C∣∣g(s)∣∣ p+1

p ≤ (p + 1)G(s) − sg(s) and 0 ≤ (q + 1)F (s) − sf(s)

for s > s0,or

C∣∣f(s)

∣∣ p+1p ≤ (p + 1)F (s) − sf(s) and 0 ≤ (q + 1)G(s) − sg(s)

for s > s0, where F (s) =∫ s

0 f(t)dt and G(s) =∫ s

0 g(t)dt.

Then, we can prove that there exists a positive constant C such that ‖∇u‖L2(Ω) ≤ C and ‖∇v‖L2(Ω) ≤ C, for any positive solution (u, v) of system (1.7). However, we have not been able to prove that ‖u‖L∞(Ω) and ‖v‖L∞(Ω) are uniformly bounded.

1.2. Existence theorems

In order to establish existence results for Eqs. (1.1) and (1.2), we observe that if f satisfies (f1), (f2), (f3)(respectively (f ′

3)) and (H), define the family of functions ft(s) := f(s + t), depending on a parameter t ∈ I, where I = [0, t0] (t0 > 0), then the functions ft satisfy (f2), (f3) (respectively (f ′

3)) and (H) uniformly with respect to t ∈ I and

(f1,t) ft : [0, +∞) → R are a nondecreasing locally Lipschitz functions, t ∈ I.

Moreover, we have:

∀s0 > 0, there exists M := M(s0) > 0, such that supt∈I

sup0≤s≤s0

ft(s) ≤ M.

Therefore, we can establish a priori L∞-estimates obtained in Propositions 1.1 and 1.2 uniformly in t, that is, there exists a positive constant C independent of t such that ‖ut‖L∞(Ω) ≤ C, where ut is a positive solution of the following equation:

{ (−Δ)mu = ft(u) in Ω,

m−1

u = Δu = · · · = (Δ) u = 0 on ∂Ω.

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 491

(Similar result can be obtained for (1.2).) These uniform a priori bounds combined with degree theory allow us to obtain the following theorems. For this, we need the standard following assumption:

(f4) lim sups→0f(s)s < Λ1.

Theorem 1.1. Suppose Ω is a smooth bounded convex domain. If f is a function satisfying (f1)–(f4) and (H) with f(0) = 0, then Eq. (1.1) has at least one positive solution u ∈ C2m,γ(Ω), for every γ ∈ (0, 1).

Theorem 1.2. Suppose f is a function satisfying (f ′1), (f2), (f ′

3), (f4) and (H) with f(0) = 0, then Eq. (1.2)has at least one positive solution u ∈ C2m,γ(BR), radially symmetric and decreasing along r = |x|, for every γ ∈ (0, 1).

Now, we will use truncation technique to apply the variational method in order to prove existence result for Eq. (1.1) without assuming (H). However, our proof does not give any information on the structure of the solution. The same techniques can be applied to (1.2) to get existence result, however we cannot use the maximum principle to derive that such a solution is positive. Then, we state

Theorem 1.3. Suppose Ω is a smooth bounded convex domain. If f is a function satisfying (f1)–(f4) with f(0) = 0. Then, Eq. (1.1) has at least one positive solution u ∈ C2m,γ(Ω), for every γ ∈ (0, 1).

For the existence of positive solution of system (1.7), we obtain the following theorem:

Theorem 1.4. Let Ω be a bounded convex domain of class C3. Suppose that f and g satisfy conditions (S1) and either (S2), (S3) and (S4) or (S′

2), (S′3) and (S′

4) with f(0) = g(0) = 0, lim sups→0f(s)s < λ1

and lim sups→0g(s)s < λ1. Then, system (1.7) has at last one positive solution (u, v) ∈ C2,γ(Ω), for every

γ ∈ (0, 1).

As mentioned above, using known topological fixed-point theorems, the proof of Theorem 1.4 is based on uniform a priori L∞-estimates obtained by using Propositions 1.3–1.4 with f(s) and g(s) are replaced by f(s + t) and g(s + t), t ∈ [0, t0] for a given t0 > 0.

Our paper is organised as follows. In Section 2, we prove the statements of a priori bounds and L∞-estimates. In Section 3, we give the proofs of Theorems 1.1–1.4. The proofs of some remarks are given in Appendix A and Appendix B.

2. Proofs of a priori L∞-estimates

In all this paper, C denote generic positive constant independent of u, even their value could be changed from one line to another one. For the use in the rest of this paper, we recall the following basic Lp-elliptic regularity result of Agmon et al. [1]. (See also Corollary 2.21, page 45 in [7].)

Lemma 2.1 (Agmon, Douglis, Nirenberg). Let p > 1 and k ≥ 2m. Assume that ∂Ω ∈ Ck. Then for all h ∈ W k−2m,p(Ω), the equation (−Δ)mu = h in Ω admits a unique solution u ∈ W k,p(Ω) ∩ Wm,p

0 (Ω). Moreover, there exists a constant C = C(Ω, k, m) > 0, such that

‖u‖Wk,p(Ω) ≤ C‖h‖Wk−2m,p(Ω).

Clearly this lemma can be applied in the Dirichlet case. However, for the problem with Navier boundary condition, let us observe that Eq. (1.1) can be rewritten, in an equivalent form, as the system (1.3). Suppose

492 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

that ‖f(u)‖Lp(Ω) ≤ C. Since um−1 = 0 on ∂Ω, Lemma 2.1 (for m = 1 and k = 2) implies that um−1 ∈W 2,p(Ω) and ‖um−1‖W 2,p(Ω) ≤ C. Applying Lemma 2.1 to equation −Δum−2 = um−1 and recalling that um−2 = 0 on ∂Ω, we get um−2 ∈ W 4,p(Ω) and ‖um−2‖W 4,p(Ω) ≤ C. Iterating this process, we derive that u = u0 ∈ W 2m,p(Ω) and ‖u‖W 2m,p(Ω) ≤ C. Later, Lemma 2.1 will be referred to as “Lp-elliptic regularity theory.”

Proof of Proposition 1.1. According to Remark 1.1, we treat only the case N > 2m. Let u ∈ C2m(Ω) be apositive solution of (1.1). We split the proof into two steps.

Step 1 : We prove item (1). Since f satisfies (f1) and (f2), we apply Lemma 1.1 and Pohozaev identity, as in Remark 1.1, to obtain the estimate (1.8). Hence, using H we get

∫Ω|f(u)| 2N

N+2m dx ≤ C. Since (Δ)iu = 0, i = 1, 2, . . . , m −1 on ∂Ω, then by Lp-elliptic regularity theory, we derive that there exists C > 0 independent of u such that ‖u‖

W2m, 2N

N+2m (Ω)≤ C. As W 2m, 2N

N+2m (Ω) ↪→ Wm,2(Ω) = Hm(Ω), the claim follows.Step 2 : We prove item (2). We treat only the case m = 2k, the case m = 2k + 1 is completely similar.

First, we remark that there exists a constant C > 0 independent of u such that (−Δ)iu ≥ −C for any i = 1, 2, . . . , m − 1. Indeed, let v be a solution of

{ (−Δ)mv = f(0) in Ω,

v = Δv = · · · = (Δ)m−1v = 0 on ∂Ω,

hence,

{ (−Δ)m(u− v) = f(u) − f(0) in Ω,

u− v = Δ(u− v) = · · · = (Δ)m−1(u− v) = 0 on ∂Ω.

By (f1), there holds f(u) − f(0) ≥ 0, so the maximum principle implies (−Δ)i(u − v) ≥ 0, for all i =1, 2, . . . , m − 1, and the claim follows with −C ≤ mini∈Im infx∈Ω(−Δ)iv. Hence, multiplying (1.1) by −Δu + C, using (f3) and the fact that f(u) ≥ f(0), we derive that for any ε > 0 there exists Cε such that

(−Δ)mu(−Δu) = f(u)(−Δu + C) − f(u)C

≤ εuN+2mN−2m (−Δu + C) + Cε(−Δu + C) − f(u)C

≤ εuN+2mN−2m (−Δu) + Cεu

N+2mN−2m + Cε(−Δu) + Cε.

Integrating by part the last inequality and using Cauchy–Schwartz inequality, we get

∫Ω

∣∣∇((−Δ)ku

)∣∣2dx ≤ ε

∫Ω

|u| 4mN−2m |∇u|2dx + Cε

∫Ω

|u|N+2mN−2m dx + Cε|Ω| 12

(∫Ω

(Δu)2dx) 1

2

+ Cε. (2.1)

By item (1), we known that there exists a constant C independent of u such that ‖u‖m ≤ C. So, (∫Ω

(Δu)2dx) 12 ≤ C, hence, the Hölder inequality and the Sobolev embedding theorem imply

∫Ω

∣∣∇((−Δ)ku

)∣∣2dx ≤ ε

(∫Ω

|u| 2NN−2m dx

) 2mN(∫

Ω

|∇u| 2NN−2m dx

)N−2mN

+ Cε

∫Ω

|u|N+2mN−2m dx + Cε

≤ Cε(‖u‖

4mN−2mm ‖∇u‖2

L2N

N−2m (Ω)+ ‖u‖

N+2mN−2mm

)+ Cε

≤ Cε‖∇u‖2L

2NN−2m (Ω)

+ Cε. (2.2)

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 493

As ‖∇u‖L2(Ω) is bounded, using again the Sobolev embedding, we obtain for j = 1, 2, . . . , N

∥∥∥∥ ∂u

∂xj

∥∥∥∥2

L2N

N−2m (Ω)≤ C

(∥∥∥∥(−Δ)k(

∂u

∂xj

)∥∥∥∥2

L2(Ω)+ ‖∇u‖2

L2(Ω)

)

≤ C

∥∥∥∥∂(−Δ)ku∂xj

∥∥∥∥2

L2(Ω)+ C.

Hence, we get

‖∇u‖2L

2NN−2m (Ω)

≤ C∥∥∇(

(−Δ)ku)∥∥2

L2(Ω) + C. (2.3)

Combining (2.2) and (2.3) and choosing ε small enough, we deduce that ‖∇((−Δ)ku)‖L2(Ω) remains bounded. Hence, u is bounded in Hm+1(Ω) which derives that ‖u‖Lp(Ω) is bounded ∀p > 1, if N = 2(m +1)and ‖u‖

L2N

N−2(m+1) (Ω)is bounded if N > 2(m + 1). Thus, if N = 2(m + 1), the proof is well completed since

we have f(u) is bounded in Lr(Ω), for r large. If N > 2(m + 1), we are in position to apply the boot-strap argument. For this end, we set g(x) := f(u(x)). So, we have ‖u‖Lp1(Ω) ≤ C, p1 = 2N

N−2(m+1) , from (f3), one has ‖g‖Ls1 (Ω) ≤ C, with p0 = N+2m

N−2m and s1 = p1p0

. Hence, Lp-elliptic regularity theory implies that uis bounded in Lr(Ω) for every r > 1, if s1 ≥ N

2m or u is bounded in Lr(Ω) for every r ≤ p2, p2 = Ns1N−2ms1

, if s1 < N

2m . Iterating this process, then there exists a positive constants ck(Ω) independent of u such that ‖g‖Lsk (Ω) ≤ ck(Ω) with sk = pk

p0and u is bounded in Lr(Ω) for every r > 1, if sk ≥ N

2m or u is bounded

in Lr(Ω) for every r ≤ pk+1, pk+1 = NskN−2msk

, if sk < N2m . Finally, we claim that there exists k0 such that

sk0 ≥ N2m and as a consequence u is bounded in Lr(Ω) for every r > 1. Indeed, if not sk < N

2m ∀k ∈ N∗.

Set h(t) = p0t − 2mN and l = N−2m

2N , we have h(l) = l and h(t) − t < 0 for t < l, then if we define rk := 1pk

, we get rk+1 = h(rk) and since r1 = N−2(m+1)

2N < N−2m2N = l, we obtain rk < l and rk+1 < rk, ∀k ∈ N

∗. On the other hand, rk = 1

skp0> 2m

Np0, ∀k ∈ N

∗, then rk is a decreasing bounded sequence, then converges to lwhich is impossible since rk < l. This completes the proof of Proposition 1.1. �

In order to prove Proposition 1.2, we need the following lemma which is obtained by adapting the proof of Corollary 6 in [14].

Lemma 2.2. Let A := ARR2

be the annulus {x ∈ RN ; R

2 < x < R} and g ∈ Lq(A) for some q ∈ (1, ∞). Then, there exists a constant C > 0 depending only on R, N , q, m such that for every solution u ∈C2m(A) ∩ Cm−1(A) of

{(−Δ)mu=g in A,

u=Du=···=(D)m−1u=0 on ∂BR,we have for any ρ ∈ (1

2 , 1)

‖u‖W 2m,q(Aρ) ≤C

(2ρ− 1)m(‖g‖Lq(A) + ‖u‖Lq(A)

),

where Aρ := ARρR.

Proof. We can assume that R = 1, the general case can be obtained by scaling. Let ψ ∈ C2m0 (A) be a cut-off

function with 0 ≤ ψ ≤ 1, ψ = 1 in Aρ, ψ = 0 in A \ Aρ′ where ρ′ = 1+2ρ4 , satisfying |Dγψ| ≤ ( 4

2ρ−1 )|γ| for |γ| ≤ 2m. Then

(−Δ)m(uψ) = gψ +∑

0≤|β|≤2m−1

( ∑0≤|γ|≤2m−|β|

Cβ,γDβuDγψ

)in A.

Hence, by standard linear Lp-estimates of Agmon et al. [1]

494 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

∥∥∇2m(u)∥∥Lq(Aρ) ≤ C

(‖g‖Lq(A) +

2m−1∑k=0

( 2m−k∑l=0

∥∥∇k(u)∥∥Lq(Aρ′ )

(2ρ− 1)−l

))

≤ C

(‖g‖Lq(A) +

2m−1∑k=0

∥∥∇k(u)∥∥Lq(Aρ′ )

(2ρ− 1)k−2m

). (2.4)

For k ∈ {0, . . . , 2m}, we denote Nk := sup 12<σ<1 ‖∇k(u)‖Lq(Aσ)(2σ − 1)k. Then by (2.4), we get

N2m ≤ C

(‖g‖Lq(A) +

2m−1∑k=0

Nk

).

So, we can proceed similarly as in the proof of Corollary 6 in [14] to obtain the result. �Proof of Proposition 1.2. Let u ∈ C2m(BR) be a radial positive solution of (1.2). As for the Proposition 1.1, we split the proof into two steps.

Step 1 : We prove item (1). The assumption (f ′1) guaranties that f is positive, then (−Δ)mu ≥ 0 in BR.

As u = Du = · · · = (D)m−1u = 0 on ∂BR, we can apply Proposition 1 in [16] to derive that u is radially decreasing along |x| = r ∈ (0, R) (see also [8]). On the other hand, it is known that Λ1 is simple and the corresponding eigenfunction is of one sign. Thus, we consider a positive radially symmetric eigenfunction ϕsatisfying ϕ(R2 ) = 1 and

{ (−Δ)mϕ = Λ1ϕ in BR,

ϕ = Dϕ = · · · = Dm−1ϕ = 0, in ∂BR.(2.5)

Multiplying (2.5) by u and (1.2) by ϕ and integrate over BR, we get

Λ1

∫BR

ϕudx =∫BR

ϕf(u)dx. (2.6)

From (f2), there exist λ > Λ1 and a constant C > 0 such that

f(s) ≥ λs− C, (2.7)

for all s ≥ 0. Using (2.6) and (2.7), we deduce that Λ1∫BR

ϕudx and ∫BR

ϕf(u)dx are bounded. Again, we use Proposition 1 in [16] to see that ϕ is radially decreasing and as ϕ(R2 ) = 1, u(R2 )|BR

2| ≤

∫BR

2

ϕudx.

Hence, u(R2 ) = ‖u‖L∞(A) ≤ C. Therefore, ‖f(u)‖L∞(A) is bounded. Applying Lemma 2.2 for g := f(u) and q > N

2m , we get ‖u‖W 2m,q(Aρ) is bounded, for any 12 < ρ < 1, which implies that ‖u‖C2m−1,α(Aρ) ≤ C for any

12 < ρ < 1. In particular, we deduce that

∥∥∇iu∥∥L∞(∂BR) ≤ C, for i = 0, 1, . . . , 2m− 1. (2.8)

Now, we can use Pohozaev identity (see Lemma 2 in [16]) and (H), exactly as previously to complete the proof of item (1).

Step 2 : We prove item (2). As in the proof of Proposition 1.1, we only treat the case m = 2k. In this step, we have to assume (f ′

3) instead of (f3), because one cannot verify that −Δu is uniformly bounded from below as in the case of (1.1).

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 495

Multiplying (1.2) by −Δu, integrating by part and using (f ′3), we obtain

∫BR

∣∣∇((Δ)ku

)∣∣2dx +m−1∑i=1

((−1)i−1

∫∂BR

∂

∂n

((−Δ)m−iu

)(−Δu)idσ

)

≤ ε

∫BR

|u| 4mN−2m |∇u|2dx + Cε.

From (2.8), it follows that there exists C > 0 independent of u such that

∫BR

∣∣∇((Δ)ku

)∣∣2dx ≤ ε

∫BR

|u| 4mN−2m |∇u|2dx + Cε.

The last inequality is similar to (2.1). From this, we check easily that there is a positive constant C such that ‖∇((−Δ)ku)‖L2(BR) ≤ C. Hence, u is bounded in Hm+1(BR). Then, a boot-strap argument, as in the proof of Proposition 1.1, is used and we are done. �

Next, the a priori L∞-estimates for solution of system (1.7) are established.

Proof of Proposition 1.3. Let (u, v) be a positive solution of (1.7). Using (S1) and (S2), we can follow exactly the steps 1, 2 and 3 of the proof of Theorem 2.1 in [4] to show that there is a constant C > 0 independent of u and v such that

∫Ω

F (v)dx− θN − 2N

∫Ω

vf(v)dx +∫Ω

G(u)dx− (1 − θ)N − 2N

∫Ω

ug(u)dx ≤ C, (2.9)

for every θ ∈ ]0, 1[. Choose θ = N2(N−2) , we get

2NN − 4

(∫Ω

F (v)dx− 12

∫Ω

vf(v)dx)

+ 2NN − 4

∫Ω

G(u)dx−∫Ω

ug(u)dx ≤ C.

Using (S3) and (S4), we deduce that ‖g(u)‖L

2NN+4 (Ω)

≤ C and by Lp-elliptic regularity theory, there holds ‖v‖L2(Ω) ≤ C. So, (S3) implies that ‖f(v)‖L2(Ω) ≤ C. Then, we use again Lp-elliptic regularity theory to show that ‖u‖

L2N

N−4 (Ω)and ‖∇u‖L2(Ω) are bounded.

Multiplying −Δv = g(u) by v and using (S3), it follows that for ε > 0 there exists Cε such that

(−Δv)v ≤ εuN+4N−4 v + Cεv,

integrating by parts this inequality and using (S2), we get

∫Ω

|∇v|2dx ≤ ε

∫Ω

uN+4N−4 vdx + Cε

∫Ω

vdx

≤ Cε

∫Ω

uN+4N−4 f(v)dx + C

∫Ω

uN+4N−4 dx + Cε

∫Ω

vdx,

hence, by the Hölder inequality and an integration by parts, there holds

496 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

∫Ω

|∇v|2dx ≤ Cε

∫Ω

u8

N−4 |∇u|2dx + C

(∫Ω

u2N

N−4 dx

)N+42N

+ Cε

(∫Ω

v2dx

) 12

≤ Cε

(∫Ω

u2N

N−4 dx

) 4N(∫

Ω

|∇u| 2NN−4 dx

)N−4N

+ C

(∫Ω

u2N

N−4 dx

)N+42N

+ Cε

(∫Ω

v2dx

) 12

.

Since ‖u‖L

2NN−4 (Ω)

and ‖v‖L2(Ω) are bounded, we get

∫Ω

|∇v|2dx ≤ Cε

(∫Ω

|∇u| 2NN−4 dx

)N−4N

+ Cε.

Using Lp-elliptic regularity theory, we deduce that

∫Ω

|∇v|2dx ≤ Cε

(∫Ω

|Δu| 2NN−2 dx

)N−2N

+ Cε.

On the other hand, using (S3) and the Sobolev embedding theorem, we derive that

(∫Ω

|Δu| 2NN−2 dx

)N−2N

=(∫

Ω

(f(v)

) 2NN−2 dx

)N−2N

≤(∫

Ω

|v| 2NN−2 dx

)N−2N

+ C

≤ C

(∫Ω

|∇v|2dx + 1),

hence,

(∫Ω

|Δu| 2NN−2 dx

)N−2N

≤ Cε

(∫Ω

|Δu| 2NN−2 dx

)N−2N

+ Cε.

It follows that for ε small enough, we have ∫Ω|Δu| 2N

N−2 dx is bounded, since u = 0 on ∂Ω then, Lp-elliptic reg-ularity theory implies that u is bounded in W 2, 2N

N−2 (Ω) and by the Sobolev embedding, we get ∫Ω|u| 2N

N−6 dx

is bounded if N ≥ 7 and ∫Ω|u|rdx is bounded ∀r > 1, if N ≤ 6. Then, for N ≤ 6 we deduce that ‖u‖L∞(Ω)

is bounded. For N ≥ 7, we can apply the boot-strap argument exactly as in the proof of Proposition 1.1and this complete our proof. �Proof of Proposition 1.4. As previously, using (S1) and (S′

2) we can show that there is a constant C > 0independent of u and v such that (2.9) holds. Choose θ = 1

2 in (2.9), we obtain

2NN − 2

∫ (F (v) + G(u)

)dx−

∫ (vf(v) + ug(u)

)dx ≤ C.

Ω Ω

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 497

Therefore, by (S′4) we deduce that f(v) or g(u) is bounded in L

2NN+2 (Ω) and by Lp-elliptic regularity theory,

v and u are bounded in L2N

N−2 (Ω). On the other hand, from (S1) it follows that −Δ(u + v) ≤ (f + g)(u + v). Let p > 1, as in [3], multiplying the last inequality by (u + v)p and integrating by parts, there holds

4p(1 + p)2

∫Ω

∣∣∇(u + v)p+12∣∣2dx =

∫Ω

(f + g)(u + v)(u + v)pdx.

From (S′3), we derive that ∀ε > 0 there exists Cε such that

∫Ω

∣∣∇(u + v)p+12∣∣2dx ≤ Cε

∫Ω

(u + v)N+2N−2−1(u + v)p+1dx + Cε.

Using Sobolev inequality in the left-hand side and applying the Hölder inequality in the right-hand side, we obtain

(∫Ω

(u + v)qdx) 2

q

≤ Cε

(∫Ω

(u + v)2N

N−2 dx

) 4N(p+1)

(∫Ω

(u + v)qdx) 2

q

+ Cε,

where q = N(p+1)N−2 . Therefore, we deduce that ‖u + v‖Lq(Ω) ≤ C, ∀q > 1, since u and v are positive, one

has ‖u‖Lq(Ω) ≤ C and ‖v‖Lq(Ω) ≤ C. From (S′3), it follows that ‖f(v)‖Lα(Ω) and ‖g(u)‖Lα(Ω) are bounded,

with α > N2 and as u = v = 0 on ∂Ω, again by standard Lp-theory, we conclude that there exists a constant

C > 0 independent of u and v such that ‖u‖L∞(Ω) ≤ C and ‖v‖L∞(Ω) ≤ C. So, the proof of Proposition 1.4is completed. �3. Existence of solutions

In this section, we consider the question of existence of nontrivial solutions for Eqs. (1.1), (1.2) and system (1.7).

Proof of Theorems 1.1 and 1.2. We will prove only Theorem 1.1, since the proof of Theorem 1.2 is completely similar. For completeness and for the reader’s convenience, we give the proof here which can be found in [16]. It is based on the theory of fixed point index for compact mapping.

Let us recall the following result (see Proposition 2.1 and Remark 2.1 of de Figueiredo et al. [5]) which is a classical tool in proving existence results.

Proposition 3.1. Let K be a cone in Banach space X and T : K → K a compact mapping such that T (0) = 0. Assume that there are real numbers 0 < r < R and t0 > 0 such that

1. tTx �= x for 0 ≤ t ≤ 1 and x ∈ K, ‖x‖X = r,2. there exists a compact mapping H : BR × [0, +∞[ → K (where Bρ := {x ∈ K : ‖x‖X < ρ}) such that

(a) H(x, 0) = Tx for all x ∈ K,(b) H(x, t) �= x for ‖x‖X = R and t ≥ 0,(c) H(x, t) �= x for all x ∈ BR and t ≥ t0.

Then, there exists a fixed point x of T (i.e. Tx = x), such that ‖x‖X is between r and R.

We apply Proposition 3.1 in the following context. Consider the Banach space X := C(Ω) endowed with the sup norm. It is well known that the solution operator S : X → X defined by S(ϕ) = u where u is the solution of

498 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

{ (−Δ)mu = ϕ in Ω,

u = Δu = · · · = (Δ)m−1u = 0 on ∂Ω,

is a linear compact operator. Let K denote the cone of all nonnegative functions of X. It follows from the maximum principle that S(K) ⊂ K. Observe that f(s) ≥ 0 for s > 0, then the compact mapping defined by Tu = S[f(u)] takes K into K. It follows also that T is compact and T (0) = 0 since f(0) = 0. Obviously, a non-zero fixed point of T is a positive solution u of Eq. (1.1). Elliptic regularity theory implies that u ∈ C2m,γ for every γ ∈ (0, 1) and the strong maximum principle implies that u is a strictly positive solution of (1.1).

In order to show that (1) of Proposition 3.1 holds, assume by contradiction that for all r > 0 there exist t ∈ [0, 1] and u ∈ X with ‖u‖L∞ = r such that tTu = u. Then, we have

{ (−Δ)mu = tf(u) in Ω,

u = Δu = · · · = (Δu)m−1 = 0 on ∂Ω.(3.1)

Using (f4), we see that there exist α < Λ1 and r0 > 0 such that f(s) ≤ αs for all 0 ≤ s ≤ r0. Multiplying Eq. (3.1) (with r = r0) by a positive eigenfunction φ1 associated with Λ1 normalised by

∫Ωφ1dx = 1 and

integrating by parts, there holds

Λ1

∫Ω

uφ1dx = t

∫Ω

f(u)dx ≤ tα

∫Ω

uφ1dx.

Hence. Λ1 ≤ α, which is a contradiction. In order to verify (2) of Proposition 3.1, we introduce

H(u, t) = S[f(u + t)

].

Clearly (a) holds and we show that (c) follows from (f2). Indeed, if there exists (u, t) ∈ K × [0, +∞[ such that H(u, t) = u then (u, t) is solution of the following equation:

{ (−Δ)mu = f(u + t) in Ω,

u = Δu = · · · = (Δ)m−1 = 0 on ∂Ω.(3.2)

From (f2), there exist λ > Λ1 and a positive constant C such that f(s) ≥ λs − C, ∀s ≥ 0. Hence,

λ

∫Ω

(u + t)φ1dx ≤∫Ω

f(u + t)φ1dx + C

As for inequality (1.3), if we multiply (3.2) by φ1 and integrate by parts, we obtain∫Ω

f(u + t)φ1dx = Λ1

∫Ω

uφ1dx.

It follows that λt ≤ (λ − Λ1) ∫Ωuφ1dx + λt ≤ C. Then t is bounded. Therefore, there exists t0 > 0 such

that H(u, t) �= u for any t > t0 and u ∈ K which proves (c).Finally, the a priori estimates established in Section 2 which are valid uniformly in t ∈ [0, t0] imply the

existence of R > 0 such that (b) holds. �Proof of Theorem 1.3. From (f2), we derive that there exist ε0 > 0 and a sequence sn tending to +∞ such that

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 499

f(sn)sn

> Λ1 + ε0, ∀n ∈ N�. (3.3)

Set 1 < p < N+2mN−2m . For n ∈ N

�, we consider the truncated function:

fn(t) =

⎧⎪⎨⎪⎩

0 if t ≤ 0f(t) if 0 ≤ t ≤ snf(sn)spn

tp if t ≥ sn.

It is easy to verify that fn are a nondecreasing locally Lipschitz functions satisfying (f4) and thanks to (3.3), one has fn satisfying (f2) uniformly with respect to n ∈ N

�. Moreover, if we denote by Fn(s) =∫ s

0 fn(t)dt, then there exist Cn > 0 and s′n > sn such that

(p + 1)Fn(s) ≤ sfn(s), for s ≥ s′n (3.4)

and∣∣fn(s)

∣∣ ≤ Cn|s|p, for s ≥ s′n. (3.5)

Now, consider the truncated problem{ (−Δ)mu = fn(u) in Ω,

u = Δu = · · · = (Δ)m−1u = 0 on ∂Ω.(Pn)

Then, the associated energy functional

In(u) = 12‖u‖

2m −

∫Ω

Fn(u)dx

belongs to C1(Hmϑ (Ω)) and In(0) = 0. Fix n ∈ N

�, in view of (3.4) and (3.5), we derive that In satisfies the (PS) condition (see [9] and [4, p. 933]). Moreover, the assumptions f(0) = 0 and lims→0

f(s)s < Λ1 combined

with (3.5) imply that there exists αn > 0 such that In(u) ≥ αn provided ‖u‖m = ρn for some ρn > 0. In addition, if we denote by φ1 the eigenfunction corresponds to Λ1, we may choose R0 > 0, large enough (independent of n) such that In(R0φ1) ≤ 0, ∀n ∈ N

�. Indeed, since fn verify (f2) uniformly with respect to n ∈ N

�, then there exist ε0 > 0 small enough and C0 > 0 such that

Fn(s) ≥ (Λ1 + ε0)s2

2 − C0, for all (s, n) ∈ R× N�. (3.6)

So, we get

In(R0φ1) ≤R2

02 ‖φ1‖2

m − R20

2 Λ1

∫Ω

φ21dx− ε0

R20

2

∫Ω

φ21dx + C0|Ω|

= −ε0R2

02

∫Ω

φ21dx + C0|Ω|.

Consequently, since fn(s) ≥ 0, ∀s ∈ R, we deduce that problem (Pn) has a positive classical solution un ∈C2m(Ω). Moreover, un is characterised by

In(un) = infγ∈Γ

sup In(γ(t)

),

t∈[0,1]

500 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

where Γ = {γ ∈ C([0, 1], Hmϑ (Ω)); γ(0) = 0, γ(1) = R0φ1}. Choose γ0 ∈ Γ such that γ0(t) := tR0φ1,

t ∈ [0, 1] and using (3.6), there holds

In(un) ≤ supt∈[0,1]

In(γ0(t)

)

= supt∈[0,1]

(t2‖R0φ1‖2

m −∫Ω

Fn(tR0φ1)dx)

≤ R20‖φ1‖2

m + C0|Ω|.

As R0 does not depend on n, we conclude that there exists C > 0 such that

0 < In(un) = 12‖un‖m −

∫Ω

Fn(un)dx ≤ C, ∀n ∈ N�. (3.7)

Now, since fn are nondecreasing functions and verify (f2) uniformly with respect to n ∈ N�, we see that fn

satisfy (1.5) uniformly in n. So, the Pohozaev identity (see Lemma 2 in [16]) implies∫Ω

Fn(un)dx− N − 2m2N

∫Ω

unfn(un)dx ≤ C,

combining the last inequality with (3.7), we derive that there is a constant C > 0 independent of n such that

‖un‖m ≤ C. (3.8)

Now, we can easily check that (fn) satisfies

∀ε > 0, ∃Cε > 0, Nε ∈ N such that fn(s) ≤ ε|s|N+2mN−2m + Cε ∀s ∈ R, n > Nε. (3.9)

Indeed, since sn → +∞ and from (f3) we deduce that for any ε > 0, there exists Nε ∈ N� such that sn > Nε

and f(t) ≤ εtN+2mN−2m for t > Nε. Hence, for any n > Nε and t ∈ R+,

• either t > sn, then fn(t)t−N+2mN−2m = f(sn)s−p

n tpt−N+2mN−2m ≤ f(sn)s−

N+2mN−2m

n ≤ ε,• or Nε < t ≤ sn, there holds fn(t) = f(t) ≤ ε|t|

N+2mN−2m ,

• or 0 < t ≤ Nε, then f(t) ≤ max[0,Nε] f(s) = Cε.

Hence, the claim follows.Note that inequality (3.9) is essentially to establish uniform boundedness of ‖un‖L∞(Ω). Let us explain

briefly how we can do. Our approach is very close to the one of Proposition 1.1. First, observe that fn(0) = 0and fn(s) ≥ 0 ∀s ≥ 0. So, the maximum principle implies that (−Δ)un ≥ 0. Multiplying (Pn) by (−Δ)un

and using (3.9), we derive that for any ε > 0 there exists Cε and Nε such that for n > Nε,

(−Δ)mun(−Δun) ≤ εuN+2mN−2mn (−Δun) + Cε.

Integrating by parts and using the Cauchy–Schwartz inequality (recall that m = 2k), we obtain

∫ ∣∣∇((−Δ)kun

)∣∣2dx ≤ ε

∫|un|

4mN−2m |∇un|2dx + Cε|Ω| 12

(∫(Δun)2dx

) 12

.

Ω Ω Ω

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 501

From (3.8), we see that (∫Ω

(Δun)2dx) 12 ≤ C, hence, the Hölder inequality and the Sobolev embedding

theorem, exactly as in (2.2) and (2.3) (with ε = ε0 small enough), imply that there exists a positive constant C independent of n such that ‖∇((−Δ)kun)‖L2(Ω) ≤ C. Hence, un is uniformly bounded in Hm+1(Ω) which implies that ‖un‖Lr(Ω) is uniformly bounded ∀r > 1, if N = 2(m + 1) and ‖un‖

L2N

N−2(m+1) (Ω)is uniformly

bounded if N > 2(m + 1). Set gn(x) := fn(un(x)), one has |gn(x)| ≤ C0(|un(x)|N+2mN−2m + 1), with C0 > 0 is

independent of n.If N = 2(m + 1), then gn is uniformly bounded in Ls0(Ω) for some s0 > N

2m . Therefore, the Lp-elliptic regularity implies that un is uniformly bounded in C2m−1,α(Ω) which derives that ‖un‖L∞(Ω) is uniformly bounded.

If N > 2(m + 1), iterating the Lp elliptic regularity, as in step 2 of the proof of Proposition 1.2, we may conclude that there is a positive constant C independent of n such that ‖un‖L∞(Ω) ≤ C.

To complete the proof of Theorem 1.3, choose n0 sufficiently large such that sn0 > C then, we obtain fn0(un0) = f(un0), that is un0 is a solution of (1.1). �Proof of Theorem 1.4. The proof of Theorem 1.4 is the same of one of Theorem 3.1 in [4] (see also the proof of (ii) of Theorem 1.1 in [13]). It is given here for completeness.

We follow the same idea of the proof of Theorem 1.1 or 1.2. In order to apply Proposition 3.1, we consider the Banach space

X ={U = (u, v) : u, v ∈ C0(Ω), u = v = 0 on ∂Ω

},

endowed with the norm ‖U‖ = ‖u‖L∞(Ω) + ‖v‖L∞(Ω) and we take S : X → X the solution operator defined by S(ϕ, ψ) = (u, v) where u and v are the solutions of

{−Δu = ϕ in Ω, u = 0 on ∂Ω,

−Δv = ψ in Ω, v = 0 on ∂Ω.

S is a linear compact operator and from the maximum principle, we have S(K) ⊂ K where K = {(u, v) ∈X; u(x) ≥ 0, v(x) ≥ 0, x ∈ Ω}. Therefore, the compact mapping defined by T (u, v) = S(f(v), g(u)) takes K into K and satisfies T (0) = 0 since f(0) = g(0) = 0.

To show that (1) of Proposition 3.1 holds, we assume by contradiction that for all r > 0, there exist t ∈ [0, 1] and u, v with ‖(u, v)‖ = r such that

−Δu = tf(v), −Δv = tg(u), in Ω, u = v = 0 on ∂Ω.

Using the assumptions lims→0f(s)s < λ1 and lims→0

g(s)s < λ1, we see that there exists α < λ

2 and r0 > 0such that f(v) ≤ α(u + v) and g(u) ≤ α(u + v), if ‖(u, v)‖ = r0. After multiplying the above two equations by a positive eigenfunction ϕ1 associated with λ1 and integrating by parts, we can show that

λ1

∫Ω

(u + v)ϕ1dx ≤ 2tα∫Ω

(u + v)ϕ1dx

to arrive at a contradiction. In order to verify (2) of Proposition 3.1, we introduce

H((u, v), t

)= S

(f(v + t), g(u + t)

)and we show that (b) and (c) of (2) are fulfilled. Indeed, H((u, v), t) = (u, v) means

−Δu = f(v + t), −Δv = g(u + t), in Ω, u = v = 0 on ∂Ω

502 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

From (S2) or (S′2), there are numbers a′, b′ ∈ ]0, ∞[ and a constant C > 0 such that a′b′ > λ2

1 and

f(s) ≥ a′s− C, g(s) ≥ b′s− C, for all s ≥ 0.

Therefore,

a′∫Ω

(v + t)ϕ1dx ≤∫Ω

f(v + t)ϕ1dx + C

∫Ω

ϕ1dx

and

b′∫Ω

(u + t)ϕ1dx ≤∫Ω

g(u + t)ϕ1dx + C

∫Ω

ϕ1dx.

These expressions lead to a′(b′ + λ1)t ≤ (a′b′ − λ21)

∫Ωuϕ1dx + a′(b′ + λ1)t ≤ C. This implies that t has

to be bounded, proving (c). Finally, we observe that the a priori estimates of Section 2 hold uniformly for t ∈ [0, t0], with f(s) and g(s) are replaced by f(s + t) and g(s + t). So, there exists R > 0 such that (b) holds. �Acknowledgment

We would like to express our deepest gratitude to our Research Laboratory LR11ES53 Algebra, Geometry and Spectral Theory (AGST) Sfax University, for providing us with an excellent atmosphere for doing this work.

Appendix A

In this appendix, we clarify the remarks made on the assumptions (H) and (H ′) in the Introduction. Suppose that there exist two positive constants C0 and s0 such that f(s) ≥ C0s

NN−2m , for s ≥ s0 then

sf(s) ≥ C0s2|f(s)| 2mN . Therefore, if f satisfies (H ′) then for any ε > 0, we have

sf(s) − θF (s) ≤ εs2(f(s)) 2m

N ≤ ε

C0sf(s).

This reads sf(s) ≤ θ1F (s), where θ1 = C0θC0−ε . Since θ < 2N

N−2m , we then get for s ≥ s0,

asf(s) ≤ 2NN − 2mF (s) − sf(s),

where a = 2NN−2m

1θ −1 > 0. On the other hand, from (f3), there exists C > 0 such that |f(s)| 2N

N+2m ≤ Csf(s), for |s| ≥ s0. Hence, (H) is satisfied.

Now, we claim that, for any q ≥ N+2mN−2m , there exists αq such that ∀α ≥ αq, the nonlinearity

f(s) = sN+2mN−2m

(ln(s + α))q , for q ≥ N + 2mN − 2m,

satisfies (H), (f1), (f2) and (f3) but not (H ′). Indeed, obviously f satisfies (f2) and (f3) and a simple computation shows that f ′ is positive which implies that f satisfies (f1). Let s > 0 and F (s) =

∫ s

0 f(t)dt. Integrating by parts, we get

H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504 503

F (s) = N − 2m2N sf(s) + q

N − 2m2N

s∫0

tN+2mN−2m

(ln(t + α))q+1 dt. (A.1)

Hence,

2NN − 2mF (s) − sf(s) = q

s∫0

tN+2mN−2m

(ln(t + α))q+1 dt

≥ qN − 2m

2Ns

2NN−2m

(ln(s + α))q+1 .

But |f(s)| 2NN+2m = |s|

2NN−2m

(ln(t+α))q2N

N+2m. Therefore, if we choose q ≥ N+2m

N−2m , we see that (H) is satisfied. On the

other hand, by a simple verification we show that the function t �→ tN+2mN−2m

(ln(t+α))q+1 is increasing for t ≥ s0

(s0 large). From (A.1), we deduce that there is a positive constant Cs0 such that for any 0 ≤ θ < 2NN−2m

and s ≥ s0, we have

sf(s) − θF (s)s2|f(s)|2m/N

≥ sf(s)s2|f(s)|2m/N

(A− B

ln(αq + s)

)− Cs0 ,

where A = 1 − N−2m2N θ > 0 and B = qθN−2m

2N . Since sf(s)s2|f(s)|2m/N = s

2mN

(ln(α+s))qN−2m

N

tends to +∞ as s → +∞, we conclude that (H ′) is not satisfied.

Appendix B

In this appendix, we show that item (1) of Propositions 1.1 and 1.2 holds if we replace (H) by (H ′). Indeed, by the Hölder inequality and the Sobolev embedding theorem there holds

∫Ω

u2∣∣f(u)∣∣ 2m

N dx ≤(∫

Ω

u2N

N−2m dx

)N−2mN

(∫Ω

∣∣f(u)∣∣dx) 2m

N

≤ C

(∫Ω

u2N

N−2m dx

)N−2mN

≤ C‖u‖2m = C

∫Ω

uf(u)dx.

Using (H ′), we show that for any ε > 0 there exists Cε > 0 such that

∫Ω

uf(u)dx ≤ θ

∫Ω

F (u)dx + ε

∫Ω

u2(f(u)) 2m

N dx + Cε.

From the two above inequalities and (1.8), we obtain

(1 − N − 2m

2N θ − Cε

)‖u‖2

m ≤ Cε.

Since N−2mθ < 1, choose ε small enough, we are done.

2N

504 H. Hajlaoui, A. Harrabi / J. Math. Anal. Appl. 426 (2015) 484–504

References

[1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959) 623–727.

[2] E. Berchio, F. Gazzola, T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math. 620 (2008) 165–183.

[3] H. Brézis, T. Kato, Remarks on the Schrödinger operator with singular compact potentials, J. Math. Pures Appl. 58 (1978) 137–151.

[4] Ph. Clement, D.G. de Figueiredo, E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (56) (1992) 923–940.

[5] D.G. de Figueiredo, P.-L. Lions, R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1) (1982) 41–63.

[6] D.G. de Figueiredo, J. Yang, A priori bounds for positive solutions of a non-variational elliptic system, Comm. Partial Differential Equations 26 (2001) 2305–2321.

[7] F. Gazzola, H.-Ch. Grunau, G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Math., vol. 1991, Springer-Verlag, Berlin, 2010.

[8] Y. Ge, D. Ye, Monotonicity of radially symmetric supersolutions for polyharmonic-type operators, Differential Integral Equations 15 (3) (2002) 357–366.

[9] A. Harrabi, On the Palais–Smale condition, J. Funct. Anal. 267 (2014) 2995–3015.[10] S. Kelemen, P. Quittner, Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet–Neumann

boundary conditions, Commun. Pure Appl. Anal. 9 (3) (2010) 731–740.[11] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1–2) (1993) 125–151.[12] P. Pucci, V. Radulescu, Remarks on a polyharmonic eigenvalue problem, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 161–164.[13] P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces,

Arch. Ration. Mech. Anal. 174 (2004) 49–81.[14] W. Reichel, T. Weth, A priori bounds and a Liouville theorem on a half space for higher-order elliptic Dirichlet problems,

Math. Z. 261 (2009) 805–827.[15] B. Sirakov, Existence results and a priori bounds for higher order elliptic equations and systems, J. Math. Pures Appl. (9)

89 (2) (2008) 114–133.[16] R. Soranzo, A priori estimates and existence of positive solutions of a superlinear polyharmonic equation, Dynam. Systems

Appl. 3 (1994) 465–487.[17] W.S. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 42 (1981) 400–413.[18] H. Zou, A priori estimates for a semilinear elliptic systems without variational structure and their applications, Math.

Ann. 323 (2002) 713–735.