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C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••

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Ordinary differential equations/Partial differential equations

On a p-Kirchhoff problem involving a critical nonlinearity

Anass Ourraoui

University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco

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

Article history:Received 2 September 2013Accepted after revision 31 January 2014Available online xxxx

Presented by Haïm Brézis

This paper deals with a p-Kirchhoff type problem involving the critical Sobolev exponent.Under some suitable assumptions, we show the existence of at least one solution.

© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

r é s u m é

On s’intéresse dans cet article au problème de p-Kirchhoff à exposant critique. On montrel’existence d’au moins une solution sous des hypothèses adéquates.

© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

1. Introduction

In the last years, mathematical models involving p-Kirchhoff-type equations have been extensively studied by many au-thors, who have proposed different methods to analyze the question of the existence of solutions and of related qualitativeproperties (see, for example, [1–6,8,9,12,13] and references therein). For the physical and biological meaning of Kirchhoff-type problems, we refer to [7,14–16].

Inspired by [2,13] and by the above-mentioned papers, we consider the following problem:

h

(∫Ω

|∇u|p dx

)(−�pu) = f (x, u) + |u|p∗−2u, x ∈ Ω, u = 0, x ∈ ∂Ω; (1.1)

where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω , −�pu := −div(|∇u|p−2∇u) is called p-Laplacian,

1 < p < N , p∗ = Np/(N − p) is the critical exponent in the Sobolev embedding. The functions f : Ω × R → R andh : (0,∞) → (0,∞) are assumed to be continuous and satisfy the following conditions:

(H1) h ∈ L1(0, s), s > 0;(H2) lim supt→0+ H(t)

tα < ∞, with α >p∗p , where H(t) = ∫ t

0 h(s)ds;(H3) there exist 0 < β � 1 and a positive constant C such that

H(t)� Ctβ for t > 0;(F1) there exists a positive constant C1 > 0 such that∣∣ f (x, t)

∣∣ � C1(1 + |t|q−1),

E-mail address: anas.our@hotmail.com.

http://dx.doi.org/10.1016/j.crma.2014.01.0151631-073X/© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

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where p < q < p∗;(F2) there exist A > 0 and 0 < δ < αp such that

F (x, t) � A|t|δ, as t → 0,

with F (x, t) = ∫ t0 f (x, s)ds.

So the main result of the paper reads:

Theorem 1.1. If (H1), (H2), (H3) and (F1), (F2) hold true, then the problem (1.1) has a nontrivial solution.

Throughout this paper, by (weak) solutions of (1.1) we understand the critical points of the associated energy functionalφ acting on the Sobolev space W 1,p

0 (Ω),

φ(u) = 1

pH

(‖u‖p) −∫Ω

F (x, u)dx − 1

p∗

∫Ω

|u|p∗dx

where W 1,p0 (Ω) is the Sobolev space endowed with the norm ‖u‖ = (

∫Ω

|∇u|p dx)1p .

We recall that there is a continuous embedding from W 1,p0 (Ω) into Lr(Ω) for all r ∈ [1, p∗], which implies that

φ ∈ C1(W 1,p0 (Ω),R), and then

⟨φ′(u), u

⟩ = h

(∫Ω

|∇u|p dx

)(∫Ω

|∇u|p dx

)−

∫Ω

f (x, u)u dx −∫Ω

|u|p∗dx.

We will denote by λ > 0 the best Sobolev constant of the embedding W 1,p(Ω) ⊂ Lp∗(Ω), that is:

λ = infu∈W 1,p(Ω)\{0}

∫Ω

|∇u|p dx

(∫Ω

|u|p∗ dx)p

p∗.

2. Proof of the main result

The proof of our result relies on the variational principle of Ekeland; it allows us to split the geometry of the problemfrom the compactness aspect. Hereafter, we need some auxiliary lemmas.

Lemma 2.1. There exist γ ,ρ > 0 such that φ � γ for ‖u‖ = ρ .

Proof. By (H3), for ‖u‖ is sufficiently small we have,

φ(u)� C

p‖u‖βp −

∫Ω

F (x, u)dx − 1

p∗

∫Ω

|u|p∗dx

� C

p‖u‖βp − |Ω| p∗−q

p∗(∫

Ω

|u|p∗) q

p∗− 1

p∗ λ−p∗

p ‖u‖p∗ + c

� C

p‖u‖βp − |Ω| p∗−q

p∗ λ−qp ‖u‖q − 1

p∗ λ−p∗

p ‖u‖p∗ + c,

since βp < q < p∗ then there exit γ ,ρ > 0 such that φ(u) > γ for ‖u‖ = ρ . �By (F1) and (H1) we have the following remark.

Remark 2.1.

• The functional H is continuous in [0,∞) and of class C1 on (0,∞) where H(0) = 0.• The functional φ ∈ C1(W 1,p

0 (Ω)) with φ(0) = 0.• Some works (e.g. [1,2,13]) assume that h(t)� h0 > 0 for t � 0, which is not necessary for this work.

Lemma 2.2. The functional φ is bounded from below in Br(0), where Br(0) = {u ∈ W 1,p(Ω): ‖u‖� r}; moreover m̃ = inf φ < 0.

0 Br (0)

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Proof. According to the definition of φ, it is clear that the functional φ is bounded from below in Br(0). Besides, letv ∈ C∞

0 (Ω) \ {0} and t > 0. Using (F2) and (H2), it follows that

φ(tv) = 1

pH

(t p‖v‖p) −

∫Ω

F (x, tv)dx − 1

p∗

∫Ω

|tv|p∗dx

� C1tαp − C2t p∗∫Ω

|v|p∗dx − C3tδ

∫Ω

|v|δ dx < 0,

for t sufficiently small, where Ci > 0, i = 1, . . . ,3. �Now, using the Ekeland variational principle (cf. [10]) to φ on Br(0) endowed with distance τ (u,ω) = ‖u − ω‖, so there

exists a sequence (un)n ⊂ Br such that:

φ(un) → infBr

φ = m̃,

we infer that

φ(un) − φ(ω) � ‖un − ω‖n

,

for all ω = un . As φ is of class C1 then

φ′(un) → 0

and thus we have

φ(un) → m̃ and φ′(un) → 0,

which yields that (un)n is a bounded (P .S)m̃ sequence to φ. Up to a subsequence, still denoted by (un)n , we deduce that(un)n have a weak limit u ∈ W 1,p

0 (Ω). That limit is a solution of problem (1.1). Indeed, φ′(un) = 0, for some subsequencestill denoted by (un)n , for example, in view of El Hamidi and Rakotoson [11], then we have

∇un(x) → ∇u(x), a.e. x ∈ Ω,

un → u in Lr(Ω), 1 < r < p∗,and

|un|p∗−2un → |u|p∗−2u in Lp∗

p∗−q (Ω).

Thus we get⟨φ′(u), v

⟩ = limn→∞

⟨φ′(un), v

⟩,

for any v ∈ W 1,p0 (Ω).

So it remains to prove that u = 0. It is well known thatφ(un) → m̃ then

m̃ + o(1) = φ(un)

� C

p‖un‖βp − C1‖un‖p∗ − C2‖un‖ − C3‖un‖q,

with C1, C2 and C3 > 0. It follows that

C1‖un‖p∗ + C2‖un‖ + C3‖un‖q > −m̃ + o(1),

which implies that

C1‖u‖p∗ + C2‖u‖ + C3‖u‖q > −m̃ > 0,

consequently u = 0.From the previous lemmas and by applying the Ekeland principle, the problem (1.1) has a nontrivial solution.

Remark 2.2. A standard argument shows that φ is coercive whether it is assumed that β > NN−p and we can see that φ is

sequentially weakly lower semi-continuous, so we obtain a global minimum of φ since W 1,p0 (Ω) is a reflexive Banach space.

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Acknowledgement

The author would like to thank the anonymous referee for the valuable comments.

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