Eigenvalues of some p(x)-biharmonic problem under ... - Project Euclid

Eigenvalues of some p(x)-biharmonic problem under Neumann boundary conditions Mounir Hsini2 , Nawal Irzi2 and Khaled Kefi1,2 Community College of Rafha Northern Border University Kingdom of Saudi Arabia1 Mathematic Department, Faculty of Science of Tunis2 E-mail : [email protected] E-mail : [email protected] E-mail : khaled [email protected]

Abstract. In this paper we study the following p(x)-biharmonic problem in Sobolev spaces with variable exponents    42 u = λ ∂F x ∈ Ω,  ∂u (x, u),   p(x) ∂u x ∈ ∂Ω, ∂n = 0,      ∂ (|4u|p(x)−2 4u) = a(x)|u|p(x)−2 u x ∈ ∂Ω. ∂n By means of the variational approach and Ekeland’s principle, we establish that the above problem admits a nontrivial weak solution under appropriate conditions. 2010 Mathematics Subject Classification: 35J65, 35D05, 35J60, 35D30, 35J58. Key words: p(x)-biharmonic operator, Ekeland’s variational principle, generalized Sobolev spaces, weak solution.

1

Introduction

Stimulated by the development of the study of elastic mechanics (see [29]), electrorheological fluids (see [26]), image processing (See [6]) and mathematical description of the processes filtration of an idea baroscopic gas through a porous medium (see [4]), interest in variational problems and differential equations with variable exponents has grown in recent decades. Meanwhile, elliptic problems involving operators in divergence form can be found in [5, 22]. Some other results dealing with the p(x)-Laplace, the p(x)-biharmonic operators in Sobolev spaces with variable exponents can be found in [12, 15, 16, 17, 18, 20, 21]. The purpose of this paper is to study the existence of an eigenvalue for the following the p(x)-biharmonic

1

problem    42 u = λ ∂F  ∂u (x, u),   p(x) ∂u ∂n = 0,      ∂ (|4u|p(x)−2 4u) = a(x)|u|p(x)−2 u, ∂n

x ∈ Ω, x ∈ ∂Ω,

(1)

x ∈ ∂Ω.

Where Ω is a bounded smooth domain in RN (N ≥ 3) with sufficiently smooth boundary ∂Ω, ∆2p(x) u =
∆ |∆u|p(x)−2 ∆u is the p(x)-biharmonic operator of fourth order, n is a unit outward normal to ∂Ω, a ∈ L∞ (∂Ω) with a− := inf a(x) > 0, λ is a positive real number and the functions p and F satisfy x∈∂Ω

the following assumptions: p ∈ C(Ω) with p− := inf p(x) > 1 and F ∈ C 1 (Ω × R, R). x∈Ω

The p(x)-biharmonic problem under Neumann boundary conditions was studied by many authors in recent years. Let us recall that Ben Haddouch et al in [3] studied the following problem    42 u = λ|u|q(x)−2 u, x ∈ Ω, p(x) (2)   ∂u = ∂ (|4u|p(x)−2 4u) = 0. x ∈ ∂Ω. ∂n ∂n The authors established the existence of a continuous family of eigenvalues by using the mountain pass lemma and Ekeland’s variational principle. Moreover Taarabti et al in [27] studied the following nonhomogeneous eigenvalue problem    42 u = λV (x)|u|q(x)−2 u, x ∈ Ω, p(x) (3)   ∂u = ∂ (|4u|p(x)−2 4u) = 0. x ∈ ∂Ω. ∂n ∂n They used the Ekeland’s variational principle in order to prove the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin. Moreover, Bin ge and Yuhu Wu in [13], proved the existence of a continuous family of eigenvalues by considering different situations concerning the growth rates involved in the above quoted problem. Inspired by the above-mentioned papers, we study problem (1) under the assumptions:

(H1) F : Ω × R −→ R is a C 1 function such that F (x, tu) = tq(x) F (x, u), (t > 0) for all x ∈ Ω, u ∈ R. ∂F (x, t)| ≤ cV (x)|t|q(x)−1 , ∀ t ∈ R, ∀ x ∈ Ω, where c is a positive constant, V ∈ Ls(x) (Ω), ∂t s, q ∈ C(Ω) are such that for all x ∈ Ω, we have 1 < q(x) < p(x) < N2 < s(x).

(H2) |

2

(H3) There exists Ω0 ⊂⊂ Ω with |Ω0 | > 0 such that F (x, t) > 0 in Ω0 . Remark 1. Due to assumption (H1), F leads to the so-called Euler identity, t

∂F (x, t) = q(x)F (x, t), ∀x ∈ Ω, t ∈ R ∂t

(4)

Our main results established, for small perturbation, the existence of a continuous family of eigenvalues in a neighborhood of the origin. In a second hand we showed the existence of a global minimizer of the Euler Lagrange functional associated to problem (1).

2

Terminology and abstract setting

To study p(x)-biharmonic problems, we need some results on the spaces Lp(x) (Ω), W 1,p(x) (Ω) and W k,p(x) (Ω); see [10, 14, 24, 25] for details, complements and proofs. Set C+ (Ω) := {h : h ∈ C(Ω), h(x) > 1, for all x ∈ Ω}. For any p ∈ C+ (Ω), we denote 1 < p− := min p(x) ≤ p+ = max p(x) < ∞ and x∈Ω p(x)

L

x∈Ω

Z (Ω) = {u : Ω → R measurable and

|u(x)|p(x) dx < ∞}.

Ω

The spaces Lp(x) (Ω) have been introduced by Orlicz [23]. The space Lp(x) (Ω) is endowed with the Luxemburg norm, which is defined by Z u(x) p(x) | dx ≤ 1}. |u|p(x) = inf{µ > 0 : | µ Ω p Clearly, when p(x) ≡ p, the space Lp(x) (Ω) reduces Z to the classical Lebesgue space L (Ω) and the norm 1

|u|p dx) p in Lp (Ω).

|u|p(x) reduces to the standard norm kukLp = ( For any positive integer k, let

Ω

W k,p(x) (Ω) = {u ∈ Lp(x) (Ω) : Dα u ∈ Lp(x) (Ω), |α| ≤ k}, where α = (α1 , α2 , ..., αN ) is a multi-index, |α| =

N X

αi and Dα u =

∂ |α| u ∂ α1 x1 .....∂ αN xn .

Then W k,p(x) (Ω) is

i=1

a separable and reflexive Banach space equipped with the norm X kukk,p(x) = |Dα u|p(x) . |α|≤k 0

Let Lp (x) (Ω) be the conjugate space of Lp(x) (Ω) with p1 + p10 = 1. Then the following H¨older-type inequality Z 1 1 0 | uvdx| ≤ ( − + 0 − )|u|p(x) |v|p0 (x) , u ∈ Lp(x) (Ω), v ∈ Lp (x) (Ω). (5) p (p ) Ω 3

holds. Moreover, if h1 , h2 and h3 : Ω → (1, ∞) are Lipschitz continuous functions such that 1/h1 (x) + 1/h2 (x) + 1/h3 (x) = 1, then for any u ∈ Lh1 (x) (Ω), v ∈ Lh2 (x) (Ω) and w ∈ Lh3 (x) (Ω) the following inequality holds (see [9, Proposition 2.5]): Z
uvw dx ≤ 1 + 1 + 1 |u|h (x) |v|h (x) |w|h (x) . (6) 1 2 3 − − − h2 h3 h1 Ω Inequality (5) and its generalized version (6) are due to Orlicz [23]. The modular on the space Lp(x) (Ω) is the map ρp(x) : Lp(x) (Ω) → R defined by Z |u|p(x) dx. ρp(x) (u) := Ω

Proposition 1. (See [19]) For all u, v ∈ Lp(x) (Ω), we have 1. |u|p(x) < 1 (resp. = 1, > 1) ⇔ ρp(x) (u) < 1 (resp. = 1, > 1). −

−

+

+

2. min(|u|pp(x) , |u|pp(x) ) ≤ ρp(x) (u) ≤ max(|u|pp(x) , |u|pp(x) ). 3. ρp(x) (u − v) → 0 ⇔ |u − v|p(x) → 0. Another property interesting the variable exponent Lebesgue space Lp(x) (Ω) is Proposition 2. (See [7]) Let p and q be a measurable functions such that p ∈ L∞ (Ω) and 1 ≤ p(x)q(x) ≤ ∞, for a.e. x ∈ Ω. Let u ∈ Lq(x) (Ω), u 6= 0. Then +

−

−

+

min(|u|pp(x)q(x) , |u|pp(x)q(x) ) ≤ ||u|p(x) |q(x) ≤ max(|u|pp(x)q(x) , |u|pp(x)q(x) ). In order to prove the existence of a weak solution for problem (1), we introduce the space n o ∂u X = u ∈ W 2,p(x) (Ω) : |∂Ω = 0 . ∂n This space was firstly considered by El Amrouss et al in [1] who have proven that X is a nonempty and well defined closed subspace of W 2,p(x) (Ω). Let Z Z ∆u p(x) u kuka := inf{µ > 0 : | | dx + a(x)| |p(x) dσ ≤ 1} for u ∈ X. µ Ω µ ∂Ω Since a ∈ L∞ (Ω) and essinf x∈Ω a > 0, we deduce that kuka is an equivalent norm to kuk2,p(x) in X. In our paper, we will use the norm kuka and the modular is defined as ρap(x) : X → R by ρap(x) (u)

Z

p(x)

|∆u|

=

Z dx +

Ω

a(x)|u|p(x) dσ,

∂Ω

which satisfies the same properties as Proposition 1. Accordingly, we have, similar to Theorem 1.3 in [11], the following propositions. Proposition 3. For all u ∈ Lp(x) (Ω), we have 4

1. kuka < 1 (resp. = 1, > 1) ⇔ ρap(x) (u) < 1 (resp. = 1, > 1). −

−

+

+

2. min(kukpa , kukpa ) ≤ ρap(x) (u) ≤ max(kukpa , kukpa ). 3. kun ka → 0 (respectively, → ∞) ⇔ ρap(x) (un ) → 0 (respectively, → ∞). Arguments similar to those used in the proof of Proposition 4.2 in [2], showed that Z Z 1 1 p(x) Proposition 4. Let Ia (u) = |∆u| dx + a(x)|u|p(x) dσ, then Ω p(x) ∂Ω p(x) 1. Ia : X → R is sequentially weakly lower semi continuous, Ia ∈ C 1 (X, R). 2. The mapping Ia0 : X → X ∗ is a strictly monotone, bounded homeomorphism and is of type (S+ ), that is, if un * u and lim sup Ia0 (un )(un − u) ≤ 0, then un → u. n→+∞

We recall that the critical Sobolev exponent is defined as follows:  N N p(x)   , p(x) < , N − p(x) 2 p∗ (x) =  N  +∞, p(x) ≥ . 2 We point out that if q ∈ C + (Ω) and q(x) < p∗ (x) for all x ∈ Ω, then X is continuously and compactly embedded in Lq(x) (Ω). The Lebesgue and Sobolev spaces with variable exponents coincide with the usual Lebesgue and Sobolev spaces provided that p is constant. According to [[25, pp. 8-9]], these function spaces Lp(x) and W 1,p(x) have some non-usual properties, such as: (i) Assuming that 1 < p− ≤ p+ < ∞ and p : Ω → [1, ∞) is a smooth function, then the following co-area formula Z ∞ Z p |u(x)| dx = p tp−1 |{x ∈ Ω; |u(x)| > t}| dt 0

Ω

has no analogue in the framework of variable exponents. (ii) Spaces Lp(x) do not satisfy the mean continuity property. More exactly, if p is nonconstant and continuous in an open ball B, then there is some u ∈ Lp(x) (B) such that u(x + h) 6∈ Lp(x) (B) for every h ∈ RN with arbitrary small norm. (iii) Function spaces with variable exponents are never invariant with respect to translations. The convolution is also limited. For instance, the classical Young inequality |f ∗ g|p(x) ≤ c |f |p(x) kgkL1 remains true if and only if p is constant.

5

3

Main results and auxiliary properties

Throughout the paper, the letters c, ci , i = 1, 2, ... denote positive constants which may change from line to line. 0 s(x)q(x) In the sequel, denote by s (x) the conjugate exponent of the function s(x) and put α(x) := s(x)−q(x) then we have: Remark 2. Under assumption (H2 ), one has s0 (x)q(x) < p∗ (x), for all x ∈ Ω, α(x) < p∗ (x), for all 0 x ∈ Ω, hence the embedding X ,→ Ls (x)q(x) (Ω) and X ,→ Lα(x) (Ω) are compact and continuous. Proposition 5. (See Theorem 2.4 in [8]) Let Ω ∈ RN be an open bounded domain with Lipschitz boundary. Let m be a positive integer. Suppose that p ∈ C 0 (Ω) with p− > 1 and mp+ < N . If q ∈ S(∂Ω) where S(∂Ω) the set of all measurable real functions defined on Ω, and there exists a positive constant ε such that (N − 1)p(x) for x ∈ ∂Ω, 1 ≤ q(x) < q(x) + ε ≤ N − mp(x) then the boundary trace embedding W m,p(.) (Ω) ,→ Lq(.) (∂Ω) is compact. Remark 3. Since p > 21 , then by proposition 5 one has W 2,p(x) (Ω) ,→ Lp(x) (∂Ω) is compact. Note that an eigenvalue for problem (1) satisfy the following definition. Definition 1. We say that λ ∈ R is an eigenvalue of problem (1), if there exists u ∈ X \ {0} such that Z Z Z ∂F (x, u)vdx, |4u|p(x)−2 4u4vdx + a(x)|u|p(x)−2 uvdσ = λ Ω ∂Ω Ω ∂u for any v ∈ X and we recall that if λ is an eigenvalue of the problem (1), then the corresponding u ∈ X \ {0} is a weak solution of (1). Proposition 6. If u ∈ X is a weak solution of (1) and u ∈ C 4 (Ω) then u is a classical solution of (1). Proof. Let u ∈ C 4 (Ω) be a weak solution of problem (1) then for every v ∈ X, we have Z Z Z ∂F p(x)−2 p(x)−2 (x, u)vdx, |4u| 4u4vdx + a(x)|u| uvdσ = λ Ω ∂Ω Ω ∂u By applying Green formula, we have: Z Z Z p(x)−2 p(x)−2 4(|4u| 4u)vdx = − ∇(|4u| 4u).∇vdx + Ω

and

Ω

Z |4u|

p(x)−2

Z 4u4vdx = −

Ω

As v ∈ X then

∂Ω

p(x)−2

∇(|4u|

Z 4u).∇vdx +

Ω ∂ ∂n (v)

v

(|4u|p(x)−2 4u)

∂Ω

= 0. For v ∈ D(Ω), we have 4(|4u|p(x)−2 4u) = λ 6

∂ (|4u|p(x)−2 4u)dσ, ∂n

∂F (x, u) a.e x ∈ Ω. ∂u

∂ (v)dσ, ∂n

For each v ∈ X, we have Z ∂Ω

∂ (|4u|p(x)−2 4u)vdσ = ∂n

then for all v ∈ D(Ω) we have Z ∂Ω

∂ (|4u|p(x)−2 4u)vdσ = ∂n

Z

a(x)|u|p(x)−2 uvdσ,

∂Ω

Z

a(x)|u|p(x)−2 uvdσ,

∂Ω

which implies that ∂ (|4u|p(x)−2 4u) − a(x)|u|p(x)−2 u = 0 a.e x ∈ Ω. ∂n  The first result in this paper is the following. Theorem 1. Assume hypothesis (H1), (H2) and (H3) are fulfilled. Then, there exists λ∗ > 0, such that any λ ∈ (0, λ∗ ) is an eigenvalue of problem (1). In the second, we establish that the Euler-Lagrange functional associated to problem (1) has a global minimizer. Theorem 2. Assume that hypothesis (H1), (H2) and (H3) holds. Then any λ > 0 is an eigenvalue of problem (1). In order to formulate the variational problem (1), let us introduce the functionals Φ and J : X → R defined by: Z Φ(u) = Ω

1 |4u|p(x) dx + p(x)

Z ∂Ω

a(x) p(x) |u| dσ and J(u) = p(x)

Z F (x, u)dx. Ω

The Euler Lagrange functional corresponding to problem (1) is defined by Ψλ : X → R, where Ψλ (u) := Φ(u) − λJ(u). Standard arguments showed that Ψλ ∈ C 1 (X, R) and Z Z Z ∂F hdΨλ (u), vi = |4u|p(x)−2 4u4vdx + a(x)|u|p(x)−2 uvdσ − λ (x, u)vdx, Ω ∂Ω Ω ∂u for any v ∈ X. Hence a solution of problem (1) is a critical point of Ψλ . We start with the following auxiliary lemmas. Lemma 1. Suppose we are under hypotheses of Theorem 1. Then for all ρ ∈ (0, 1), there exist λ∗ > 0 and b > 0 such that for all u ∈ X with kuka = ρ Ψλ (u) ≥ b > 0

for all 7

λ ∈ (0, λ∗ ).

0

Proof. Since the embedding X ,→ Ls (x)q(x) (Ω) is continuous, then |u|s0 (x)q(x) ≤ c2 kuka , for all u ∈ X.

(7)

Let us assume that kuka < min(1, 1/c2 ), where c2 is the positive constant of inequality (7). Then, we have |u|s0 (x)q(x) < 1, using H¨ older inequality (5), proposition 3, remark 1 and inequality (7), we deduce that for any u ∈ X with kuka = ρ, the following inequalities hold true Z Z Z 1 a(x) p(x) p(x) Ψλ (u) = |4u| dx + |u| dσ − λ F (x, u)dx Ω p(x) ∂Ω p(x) Ω 1 + ≥ kukpa − λc1 |V |s(x) ||u|q(x) |s0 (x) p+ − 1 + ≥ kukpa − λc1 |V |s(x) |u|qs0 (x)q(x) + p − 1 + − kukpa − λc1 |V |s(x) cq2 kukqa ≥ + p − − 1 + − 1 p+ − − ρ − λc1 cq2 |V |s(x) ρq = ρq ( + ρp −q − λc1 cq2 |V |s(x) ). = + p p

By the above inequality, we remark that if we define +

ρp −q λ = 2p+

−

1

∗

− c1 cq2 |V |s(x)

,

(8)

then for any λ ∈ (0, λ∗ ) and u ∈ X with kuka = ρ there exists b > 0 such that Ψλ (u) ≥ b > 0. The proof of Lemma 1 is complete.



The following result asserts the existence of a valley for Ψλ near the origin. Lemma 2. There exists ϕ ∈ X such that ϕ ≥ 0, ϕ 6= 0 and Ψλ (tϕ) < 0, for t > 0 small enough. Proof. Assumption (H2) implies q(x) < p(x) for all x ∈ Ω0 . In the sequel, denote by q0− = − − infΩ0 q(x) and by p− 0 = infΩ0 p(x). Let 0 such that q0 + 0 < p0 . On the other hand, since q ∈ C(Ω0 ), there exists an open set Ω1 ⊂ Ω0 such that |q(x) − q0− | < 0 for all x ∈ Ω1 . It follows that q(x) ≤ q0− + 0 < p− 0 , for all x ∈ Ω1 . ∞ Let ϕ ∈ C0 (Ω) such that supp(ϕ) ⊂ Ω1 ⊂ Ω0 , ϕ = 1 in a subset Ω0 1 ⊂ supp(ϕ), 0 ≤ ϕ ≤ 1 in Ω1 .

8

we obtain Z Ψλ (tϕ) = ≤ ≤ ≤

Z Z 1 a(x) p(x) p(x) |4(tϕ)| dx + |tϕ| dσ − λ F (x, tϕ)dx Ω p(x) ∂Ω p(x) Ω Z Z Z  1  p(x) p(x) p(x) p(x) t |4ϕ| dx + t a(x)|ϕ| dσ − λ tq(x) F (x, ϕ)dx p− Ω0 ∂Ω Ω1 0 − Z p t0 a q0− +0 F (x, ϕ)dx, − ρp(x) (ϕ) − λt p0 Ω1 − Z tp0 p− p+ q0− +0 F (x, ϕ)dx. ) − λt max(kϕk , kϕk a a p− Ω1 0

Therefore Ψλ (tϕ) < 0 for t < δ

− 1/(p− 0 −q0 −0 )

with ( 0 < δ < min 1,

λp− 0

R Ω1

F (x, ϕ)dx +

−

max(kϕkpa , kϕkpa )

) .

Since ϕ = 1 in Ω0 1 , then kϕka > 0, so the proof of Lemma 2 is complete.



Proof of Theorem 1 completed. Let λ∗ > 0 be defined as in (8) and λ ∈ (0, λ∗ ). By Lemma 1 it follows that on the boundary of the ball centered at the origin and of radius ρ in X, denoted by Bρ (0), we have inf Ψλ > 0. (9) ∂Bρ (0)

On the other hand, by Lemma 2, there exists ϕ ∈ X such that Ψλ (tϕ) < 0 for all t > 0 small enough. Moreover, using H¨ older inequality (5), proposition 3 and inequality (7), we deduce that for any u ∈ Bρ (0) we have 1 + q− Ψλ (u) ≥ + kukpa − λc1 cq− 2 |V |s(x) kuka . p It follows that −∞ < c := inf Ψλ < 0. Bρ (0)

Let 0 <  < inf ∂Bρ (0) Ψλ − inf Bρ (0) Ψλ . Using the above information, the functional Ψλ : Bρ (0) −→ R, is lower bounded on Bρ (0) and Ψλ ∈ C 1 (Bρ (0), R). Then by Ekeland’s variational principle, there exists u ∈ Bρ (0) such that    c ≤ Ψλ (u ) ≤ c + ,   0 < Ψλ (u) − Ψλ (u ) + · k u − u ka ,

u 6= u .

Since Ψλ (u ) ≤ inf Ψλ +  ≤ inf Ψλ +  < inf Ψλ , Bρ (0)

Bρ (0)

9

∂Bρ (0)

we deduce that u ∈ Bρ (0). Now, we define Iλ : Bρ (0) −→ R by Iλ (u) = Ψλ (u) + · k u − u ka . It is clear that u is a minimum point of Iλ and thus Iλ (u + t · v) − Iλ (u ) ≥ 0, t for small t > 0 and any v ∈ B1 (0). The above relation yields Ψλ (u + t · v) − Ψλ (u ) + · k v ka ≥ 0. t Letting t → 0 it follows that hdΨλ (u ), vi + · k v ka ≥ 0 and we infer that k dΨλ (u ) ka ≤ . We deduce that there exists a sequence {wn } ⊂ Bρ (0) such that Ψλ (wn ) −→ c < 0

and dΨλ (wn ) −→ 0X ∗ .

(10)

It is clear that {wn } is bounded in X. Thus, there exists w in X such that, up to a subsequence, {wn } converges weakly to w in X. Since α(x) < p∗ (x) for all x ∈ Ω we deduce that there exists a compact embedding E ,→ Lα(x) (Ω) and consequently {wn } converges strongly in Lα(x) (Ω). For the strong convergence of {wn } in X, we need the following proposition: Proposition 7. Z lim

n→∞ Ω

∂F (x, wn )(wn − w)dx = 0. ∂u

Proof. Using H¨ older inequality (5) we have: Z | Ω

∂F (x, wn )(wn −w)|dx ≤ c|V |s(x) ||wn |q(x)−2 wn (wn −w)|s0 (x) ≤ c|V |s(x) |||wn |q(x)−2 wn | q(x) |wn −w|α(x) . ∂u q(x)−1

Now if ||wn |q(x)−2 wn |

q(x) q(x)−1

> 1, by proposition 2, we get ||wn |q(x)−2 wn |

+

q(x) q(x)−1

≤ |wn |qq(x) .

The compact embedding X ,→ Lq(x) (Ω) ends the proof.



Since dΨλ (wn ) → 0 and wn is bounded in X, one has |hdΨλ (wn ), wn − wi| ≤ |hdΨλ (wn ), wn i| + |hdΨλ (wn ), wi| ≤ kdΨλ (wn )ka kwn ka + kdΨλ (wn )ka kwka . Moreover, using Proposition 7, one has lim hdΨλ (wn ), wn − wi = 0.

n→∞

Hence Z lim

n→∞ Ω

p(x)−2

|4wn |

Z 4wn (4wn − 4w)dx +

a(x)|wn |p(x)−2 wn (wn − w)dσ = 0.

∂Ω

Now, proposition 4, ensures that {wn } converges strongly to w in X. Since Ψλ ∈ C 1 (X, R), we conclude dΨλ (wn ) → dΨλ (w), as n → ∞. 10

(11)

Relations (10) and (11) showed that dΨλ (w) = 0 and thus w is a weak solution for problem (1). Moreover, by relation (10), it follows that Ψλ (w) < 0 and thus, w is a nontrivial weak solution for (1). The proof of Theorem 1 is complete. Proof of Theorem 2. Using H¨ older inequality (5) for kuka > 1, one has Z Z Z 1 1 p(x) p(x) |4u| dx + |a(x)|u| dσ − λ F (x, u)dx Ψλ (u) = ∂Ω p(x) Ω Ω p(x) 1 − ≥ kukpa − λc1 |V |s(x) ||u|q(x) |s0 (x) p+ − 1 − kukpa − λc1 |V |s(x) |u|qs0 (x)q(x) ≥ + p − 1 − − kukpa − λc1 |V |s(x) cq2 kukqa → +∞, as kuka → +∞. ≥ + p

As a conclusion, since Ψλ is weakly lower semi-continuous then it has a global minimizer which is solution of problem (1), moreover lemma 2 ensures that this minimizer is nontrivial, which ends the proof.

3.1

Example

Put F (x, t) = V (x)tq(x) , where the function V (·) was as in the assumption (H2) and consider the problem    x ∈ Ω, 42 u = λ ∂F  ∂u (x, u),   p(x) ∂u (12) x ∈ ∂Ω, ∂n = 0,      ∂ (|4u|p(x)−2 4u) = a(x)|u|p(x)−2 u, x ∈ ∂Ω. ∂n Where Ω is a bounded smooth domain in RN (N ≥ 3) with sufficiently smooth boundary ∂Ω, n is a unit outward normal to ∂Ω, a ∈ L∞ (∂Ω) with a− := inf a(x) > 0 and λ is a positive real number. x∈∂Ω

First, observe that the function F satisfy the assumptions (H1), (H2) and (H3). Then, Theorem 1 asserts that there exists λ∗ > 0 under which problem (12) has a nontrivial weak solution. Moreover, due to Theorem 2, one has a solution for any λ > 0.

Acknowledgements The authors thank the anonymous referee for a careful and constructive analysis of this paper and for the remarks and comments, which have considerably improved the initial version of the present work.

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