existence and multiplicity of solutions for anisotropic elliptic problems ...

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 188, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR ANISOTROPIC ELLIPTIC PROBLEMS WITH VARIABLE EXPONENT AND NONLINEAR ROBIN BOUNDARY CONDITIONS BRAHIM ELLAHYANI, ABDERRAHMANE EL HACHIMI Communicated by Jesus Ildefonso Diaz

Abstract. This article presents sufficient conditions for the existence of solutions of the anisotropic quasilinear elliptic equation with variable exponent and nonlinear Robin boundary conditions, −

N N X ∂ “˛˛ ∂u ˛˛pi (x)−2 ∂u ” X pi (x)−2 + |u| u + λ|u|m(x)−2 u = γg(x, u) ∂x ∂x ∂x i i i i=1 i=1

in Ω, N X

˛ ∂u ˛pi (x)−2 ∂u ˛ ˛ υi = µ|u|q(x)−2 u ∂xi ∂xi i=1

on ∂Ω.

Under appropriate assumptions on the data, we prove some existence and multiplicity results. The methods are based on Mountain Pass and Fountain theorems.

1. Introduction Many problems in physics and mechanics can be modeled with sufficient accuracy using classical Lebesgue and Sobolev spaces, Lp (Ω) and W 1,p (Ω), where p is a fixed constant and Ω is an appropriate domain. But for the electrorheological fluids (Smart fluids), this is not adequate but rather, the exponent should be able to vary. This leads to study the problem in the frame-work of variable exponent Lebesgue and Sobolev spaces, Lp(·) (Ω) and W 1,p(·) (Ω), where p(·) is a real-valued function; see, e.g. [12, 13]. On the other hand, it has been experimentally shown that the above-mentioned fluids may have their viscosity undergoing a significant change; see, e.g. [3]. Consequently, the mathematical modelling of such fluids requires the introduction of the so-called anisotropic variable spaces.Indeed, there is by now a large number of papers and increasing interest about anisotropic problems. With no hope of being complete, let us mention some pioneering works on anisotropic Sobolev spaces 2010 Mathematics Subject Classification. 35J20, 35J25, 35J62. Key words and phrases. Anisotropic elliptic problems; Robin boundary conditions existence and multiplicity. c

2017 Texas State University. Submitted May 24, 2016. Published July 24, 2017. 1

2

B. ELLAHYANI, A. EL HACHIMI

EJDE-2017/188

[20, 24] and some more recent regularity results for minimizers of anisotropic functionals [1, 6, 21]. Therefore, in the recent years, the study of various mathematical problems modeled by quasilinear elliptic and parabolic equations with both anisotropic and variable exponent has received considerable attention. Let us mention many works in that direction by Antontsev and Shmarev; see, e.g. [2] and the references therein. Our paper is mainly devoted to the existence and multiplicity of solutions of quasilinear elliptic equations under nonlinear Robin boundary condition such as −

N N X ∂ ∂u pi (x)−2 ∂u X pi (x)−2 + |u| u + λ|u|m(x)−2 u ∂x ∂x ∂x i i i i=1 i=1

= γg(x, u)

in Ω,

(1.1)

N X

∂u pi (x)−2 ∂u υi = µ|u|q(x)−2 u on ∂Ω, ∂xi ∂xi i=1

where Ω ⊂ Rn is a bounded domain with n ≥ 2, with smooth boundary ∂Ω and υi are the components of the outer normal unit vector and for i ∈ {1, . . . , N }, ¯ q ∈ C(∂Ω). The functions pi and g are supposed to satisfy some pi , m ∈ C(Ω), conditions to be specified below, while λ, γ, and µ are real parameters, with γ, µ > 0. We shall give conditions under which problem (1.1) has infinitely many solutions. According to the behaviour of g and to the kind of results we want to prove, variational methods turn out to be more appropriate. When lims→0 g(x, s)/|s|σ = 0, σ to be made precise later, Mountain Pass theorem provides the existence of at least a solution of (1.1) and, on the other hand, when g is an odd function, Fountain’s theorem yields the existence of infinitely many solutions. A host of publications exist for this type of problems when the boundary condition is replaced by ∂u ∂ν = 0 on ∂Ω and γ = 0 ; see, e.g. [16] and the references therein, where the authors obtained existence results by means of standard variational tools. The associated problem with Dirichlet boundary conditions has also been treated by many authors; see, e.g. [3, 15]. Furthermore, existence of positive solutions for nonlinear Robin problem involving the p(x)-Laplacian have been studied by S. G. Deng; in [9], by using the sub-super solutions and variational methods. We consider here the case where µ is positive and g satisfies more hypotheses than in [17], to use the Mountain Passe and Fountain theorems. It turns out that the condition q − > P++ plays an important role in the proofs of our main results. This article is divided into four sections. In the second section, we introduce some basic properties of the generalized Lebesgue-Sobolev space W 1,p(x) (Ω) and → − anisotropic Sobolev spaces W 1, p (x) (Ω) , and state the existence and multiplicity results concerning the problem (1.1). The third section is devoted to the proofs of the main results and finally, in the fourth section we deal with a generalized equation related to our problem (1.1). 2. Preliminaries and main results To study problem (1.1), we need to introduce the notions of Sobolev space → − W 1,p(x) (Ω) and anisotropic Sobolev spaces W 1, p (x) (Ω), with variable exponent. For convenience, we only recall some basic facts which will be used later. Let

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EXISTENCE AND MULTIPLICITY OF SOLUTIONS

3

Ω ⊂ RN be a measurable subset with meas(Ω) > 0. We write ¯ = {u : u is a continuous function in Ω}, ¯ C(Ω) ¯ = {u ∈ C(Ω) ¯ : ess inf Ω u ≥ 1}. C+ (Ω) Suppose that Ω is a bounded domain of RN with a smooth boundary ∂Ω, and ¯ R) with p(x) > 1, for any x ∈ Ω. p ∈ C(Ω, Denote p− = inf x∈Ω p(x) and p+ = supx∈Ω p(x); then we have, p− > 1 and + p < ∞. Denote by M be either Ω or ∂Ω. Define the variable exponent Lebesgue space Z Lp(x) (M) = u | u : M → R is measurable and |u(x)|p(x) dx < ∞ , M

endowed with the Luxembourg norm Z o n u(x) p(x) dx ≤ 1 . |u|p(x) = |u|Lp(x) (M) = inf τ > 0; τ M R u(x) p(x) Proposition 2.1 ([8]). Let ρ(u) = M τ dx. For u, uk ∈ Lp(x) (M)(k = 1, 2, . . . ), we have: +

−

−

+

(1) |u|Lp(x) (M) ≤ 1 ⇒ |u|pLp(x) (M) ≤ ρ(u) ≤ |u|pLp(x) (M) . (2) |u|Lp(x) (M) > 1 ⇒ |u|pLp(x) (M) ≤ ρ(u) ≤ |u|pLp(x) (M) . (3) |uk |Lp(x) (M) → 0 ⇔ ρ(uk ) → 0. (4) |uk |Lp(x) (M) → ∞ ⇔ ρ(uk ) → ∞. We define the variable exponent Sobolev space W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω)}, endowed with the norm Z o n ∇u(x) p(x) u(x) p(x) + dx ≤ 1 . kuk = inf τ > 0; τ τ Ω Proposition 2.2 (See [12]). Both (Lp(x) (M)), | · |p(x) ) and (W 1,p(x) (Ω), k · k) are separable, reflexive and uniformly convex Banach spaces. Proposition 2.3 (See [12]). The H¨ older inequality holds, namely Z |uv| dx ≤ 2|u|p(x) |v|q(x) ; ∀u ∈ Lp(x) (M), ∀v ∈ Lq(x) (M), M

where

1 p(x)

+

1 q(x)

= 1.

Proposition 2.4 (See [8]). Let ρ(u) = W 1,p(x) (Ω)(k = 1, 2, . . . ), we have +

R Ω

(|∇u(x)|p(x) + |u(x)|p(x) ) dx. For u, uk ∈

−

kuk ≤ 1 ⇒ kukp ≤ ρ(u) ≤ kukp . − + kuk > 1 ⇒ kukp ≤ ρ(u) ≤ kukp . kuk k → 0 ⇔ ρ(uk ) → 0. kuk k → ∞ ⇔ ρ(uk ) → ∞. − − Let → p (·) : Ω → RN be a vectorial function, → p (·) = (p1 (·), p2 (·), . . . , pN (·)) such that 2 ≤ pi ≤ N and put (1) (2) (3) (4)

pM (x) = max{p1 (x), . . . , pN (x)}.

4


EJDE-2017/188

The anisotropic Sobolev space with variable exponent is defined by → −

W 1, p (x) (Ω) = {u ∈ LpM (x) (Ω) :

∂u ∈ Lpi (·) (Ω), ∀i ∈ {1, . . . , N }}, ∂xi

endowed with the norm − kuk→ p (·) =

N N X X ∂u + |u|pi (·) . ∂xi pi (·) i=1 i=1

For convenience, we denote: − − P−− = inf{p− 1 , p2 , . . . , pN },

+ + P++ = sup{p+ 1 , p2 , . . . , pN }

→ −

and write X = W 1, p (x) (Ω). We know that X is reflexive if P−− > 1, (see e.g [22]). We define Z X N N 1 ∂u pi (x) X 1 |u|pi (x) dx, + J(u) = p (x) ∂xi p (x) Ω i=1 i i=1 i Z u G(x, u) = g(x, s)ds. 0

We have Z X N N X ∂u pi (x)−2 ∂u ∂v (J u, v) = + |u|pi (x)−2 uv dx, ∂xi ∂xi ∂xi i=1 Ω i=1 0

for all v ∈ X. In all this paper C, Ci (i = 0, 1, 2, . . . ) represents different positive real constants. We make the following assumptions on the functions q and g. (H0) q ∈ C(∂Ω) satisfies : 1 ≤ q(x) ≤

− (N −1)P− − N −P−

for all x ∈ ∂Ω and q − < P++ .

(H1) g : Ω × R → R is a Caratheodory type function and there exist a constant − ¯ such that: 1 < α(x) < N P−− , for all x ∈ Ω ¯ C > 0 and a function α ∈ C(Ω) N −P−

and |g(x, s)| ≤ C(1 + |s|α(x)−1 )

for all (x, s) ∈ Ω × R.

(H2) There exists M > 0 and θλ ≥ m+ (resp θλ ≤ m− ) if λ ≥ 0 (resp λ < 0 ). such that for all s with |s| ≥ M and x ∈ Ω, we have 0 < θλ G(x, s) ≤ sg(x, s). +

(H3) g(x, s) = ◦(|s|P+ ) as s → 0 and uniformly for x ∈ Ω, (H4) g(x, −s) = −g(x, s), x ∈ Ω, s ∈ R. We say that u ∈ X is a weak solution of (1.1) if Z X Z N N X ∂u pi (x)−2 ∂u ∂v pi (x)−2 + |u| uv dx + λ |u|m(x)−2 uv dx ∂x ∂x ∂x i i i Ω Ω i=1 Z i=1 Z = γg(x, u)v dx + µ |u|q(x)−2 uv dx, Ω

for all v ∈ X.

∂Ω

EJDE-2017/188


5

The energy functional associated with problem (1.1) is Z X N N 1 ∂u X 1 Φ(u) = + |u|pi (x) dx pi (x) ∂xi p (x) Ω i=1 i Zi=1 Z Z 1 1 +λ |u|m(x) dx − γ G(x, u) dx − µ |u|q(x) dx. Ω m(x) Ω ∂Ω q(x)

(2.1)

Proposition 2.5 (See [14, 11]). (1) L ≡ J 0 : X → X ∗ is a continuous, bounded and strictly monotone operator; (2) L is a mapping of type (S+ ), i.e. if un * u in X, and limn→+∞ (L(un ) − L(u), un − u) ≤ 0, then un → u in X; (3) L : X → X ∗ is a homeomorphism. The following are embedding results on anisotropic generalized Sobolev spaces and will be used later. Proposition 2.6 (See [21]). Suppose Ω ⊂ RN is a bounded domain with smooth boundary. For any q ∈ C+ (Ω) satisfying q(x)
0, we have +

|G(x, s)| ≤ |s|P+ + C()|s|α(x) ,

for all (x, s) ∈ Ω × R.

For kuk small enough, we thus obtain Z N Z ∂u pi (x) 1 X 1 pi (x) Φ(u) ≥ + + |u| dx + λ |u|m(x) dx ∂x m(x) P+ i=1 Ω i Ω Z Z + µ − (|u|P+ + C()|u|α(x) ) dx − − |u|q(x) dx q Ω ∂Ω Z Z + + 1 |λ| P+ m(x) ≥ kuk − |u| dx − |u|P+ dx → − + + p (·) − + P+ −1 P+ −1 m Ω Ω P+ 2 N Z Z µ |u|q(x) dx. − C()|u|α(x) dx − − q ∂Ω Ω

(3.1)

EJDE-2017/188


Since P++ < α− ≤ α(x)
0 such that |u|

L

+ P + (Ω)

≤ C20 kuk,

|u|Lq(x) (Ω) ≤ C10 kuk, for all u ∈ X.

(3.2)

By using (3.2) for kuk small enough, we obtain from (3.1) that Φ(u) ≥

1

+ |λ| m− max{|u|m Lm(x) (Ω) , |u|Lm(x) (Ω) } m− − − µ − C()C30 kukα − − C10 kukq . q +

+ + P++ 2P+ −1 N P+ −1 +

− C20 kukP+ → −

kukP+ −

+

Since W 1, p (Ω) ,→ Lm (Ω), we have Φ(u) ≥

1

+

+ + P++ 2P+ −1 N P+ −1

− C20

+ P+

kukP+ −

+

+ − |λ|C max{kukm , kukm } m− −

kukP+ − C()C30 kukα −

− µ 0 C1 kukq . − q

Now, let > 0 be small enough so that: 0 < C20

+ P+

≤

1 + + 2P++ 2P+ −1 N P+ −1

=: c0 .

We have − |λ|C µC10 m+ m− α− max{kuk kukq , kuk } − C()kuk − m− q− + + + − + |λ|C ≥ kukP+ c0 − − max{kukm −P+ , kukm −P+ } m + + + − − µC 0 − kukP+ C()kukα −P+ + −1 kukq −P+ . q +

Φ(u) ≥ c0 kukP+ −

Since P++ < min (α− , m− , q − ), then there exist r1 > 0 and C 0 > 0 such that Φ(u) ≥ C 0 > 0,

for any u ∈ X.

Hence, the proof is complete.

Proof of Theorem 2.8. To apply the Mountain Pass theorem ([25]), we have to prove that Φ(tu) → −∞ as t → +∞, for some u ∈ X. From (H2), it follows that ¯ ∀|s| ≥ M. G(x, s) ≥ C|s|θλ , ∀x ∈ Ω, For u ∈ X and t > 1, we have Z N Z ∂(tu) pi (x) 1 X 1 Φ(tu) ≤ − + |tu|pi (x) dx + λ |tu|m(x) dx ∂xi P− i=1 Ω Ω m(x) Z Z µ − G(x, tu) dx − + |tu|q(x) dx q Ω ∂Ω

8


≤

EJDE-2017/188

+ Z N Z ∂(tu) pi (x) 1 tP+ X m e pi (x) dx + λt + |u| |u|m(x) dx − ∂xi P− i=1 Ω Ω m(x) Z − Z µtq |u|q(x) dx, − Ctθλ |u|θλ dx − + q ∂Ω Ω

where again m e = m+ if λ > 0 and m e = m− if λ ≤ 0. By (H0) and (H2), it follows that, for any λ ∈ R, Φ(tu) → −∞ as (t → +∞). Since Φ(0) = 0, it follows that Φ satisfies the condition of the Mountain Pass lemma, and so Φ admits at least one nontrivial critical point u0 ∈ X; which is characterized by τ = inf sup Φ(h(t)), h∈Γ t∈[0,1]

where Γ = {h ∈ C([0, 1], X); h(0) = 0 and h(1) = e}. Proof of Theorem 2.9. Let X be a reflexive and separable Banach space. It is ∗ ∞ ∗ ∗ well know (see, e.g. [1]) that there are {ej }∞ j=1 ⊂ X and {ej }j=1 ⊂ X (where X is the topological dual of X) such that X = span{ej : 1, 2, . . . },

X ∗ = span{e∗j : 1, 2, . . . },

and he∗j , ei i

( 1 = 0

if i = j, if i = 6 j.

(3.3)

For convenience, we write Xj = span{ej }, Yk = ⊕kj=1 Xj and Zk = ⊕∞ j=k Xj . Denote ( ∗

p (x) =

N p(x)/(N − p(x)) if p(x) < N, +∞ if p(x) ≥ N.

¯ with β(x) < p∗ (x) for x ∈ Ω ¯ and Lemma 3.3 (See [8, 10]). Let β(x) ∈ C+ (Ω), αk := sup{|u|Lβ(x) (Ω) ; kuk = 1, u ∈ Zk }. Then, we have limk→∞ αk = 0. To prove Theorem 2.9, we shall use the Fountain theorem (see [25, Theorem 3.6] ). Obviously, Φ ∈ C 1 (X, R) is an even functional. By using (H0) and (H1) we first prove that if k is large enough, then there exist ρk > νk > 0 such that bk := inf{Φ(u)/u ∈ Zk , kuk = νk } → +∞ as k → +∞; ak := max{Φ(u)/u ∈ Yk , kuk = ρk } → 0

as k → +∞.

(3.4) (3.5)

EJDE-2017/188


Proof of (3.4): For any u ∈ Zk , |u| = rk > 1, we have Z N Z ∂u pi (x) 1 X |λ| pi (x) Φ(u) ≥ + dx − − + |u| |u|m(x) dx m P+ i=1 Ω ∂xi Ω Z Z µ |u|q(x) dx − C (1 + |u|α(x) ) dx − − q ∂Ω Ω Z N Z X − 1 ∂u P− dx − |λ| |u|m(x) dx ≥ + m− Ω P+ i=1 Ω ∂xi Z Z µ α(x) |u|q(x) dx − C (1 + |u| ) dx − − q ∂Ω Ω − 1 C|λ| m(ξ) ≥ kukp− − − |u|Lm(x) − − + P− −1 P− −1 m P+ 2 N µ q(ξ) α(ξ) − C1 |u|α(x) − − |u|Lq(x) (∂Ω) − C2 , for some ξ ∈ Ω. q

9

(3.6)

So, for the study of the previous inequality, we only need to consider either the case where m(x) ≥ α(x) or the case where m(x) < α(x) for all x ∈ Ω. Let us assume that m(x) ≤ α(x) for all x ∈ Ω. Then, we have Lα(x) (Ω) ⊂ m(x) L (Ω). Thus, there is a positive constant C3 > 0 such that |u|Lm(x) (Ω) ≤ C3 |u|Lα(x) (Ω)

for all u ∈ X.

So, for any ξ ∈ Ω, we have m(ξ)

m(ξ)

|u|Lm(x) (Ω) ≤ C m(ξ) |u|Lα(x) (Ω) . −

−

Let us denote e := 1/(2P− −1 N P− −1 ). Then, for any ξ ∈ Ω, we have Φ(u) − e µ m(ξ) α(ξ) q(ξ) ≥ + kukP− − C 0 |u|Lα(x) − C1 |u|Lα(x) (Ω) − − |u|Lq(x) (∂Ω) − C2 q P+ − e µ q(ξ) α(ξ) m(ξ) ≥ + kukP− − C max{|u|Lα(x) (Ω) , |u|Lα(x) (Ω) } − − |u|Lq(x) (∂Ω) − C2 . q P+ Denote E = Lα(x) (Ω) ∩ Lq(x) (∂Ω), A = {u ∈ E : |u|Lα(x) (Ω) ≤ 1, |u|Lq(x) (∂Ω) ≤ 1}, B = {u ∈ E : |u|Lα(x) (Ω) > 1, |u|Lq(x) (∂Ω) ≤ 1}, C = {u ∈ E; |u|Lα(x) (Ω) ≤ 1, |u|Lq(x) (∂Ω) > 1}, D = {u ∈ E; |u|Lα(x) (Ω) ≥ 1, |u|Lq(x) (∂Ω) > 1}. From (3.7), we have  P− e   P++ kuk −     e kukP−−  + P+ Φ(u) ≥ − P− e   + kuk  P+   −   e+ kukP− P +

− C1

if u ∈ A, +

− C1 (αk |u|)α − C2 −

µ q+ q − (βk |u|) +

if u ∈ B,

− −C1

− C1 (αk |u|)α −

µ q q − (βk |u|)

if u ∈ C, +

− C2 , if u ∈ D,

(3.7)

10


EJDE-2017/188

where βk = sup{|u|Lq(x) (∂Ω) ; kuk = 1, u ∈ Zk }. It is obvious that Φ(u) → +∞ as kuk → +∞ in A. For u ∈ B ∪ C, we have − + e Φ(u) ≥ + kukP− − C2 (f αk |u|)αe − C3 , P+ By taking α fk to be either αk or βk and α e = α+ or q + , we obtain −

P− 1 1 C2 + αe+ P−− −αe + − C3 . Φ(u) ≥ e + − + α e α fk α e e P+ −1 α e+ P− −α e+ e+ α fk Since α fk → 0 as k → ∞ and α e+ > P++ , then Ce2 α → ∞ as k → ∞. Consequently, we have Φ(u) → +∞ as kuk → +∞, u ∈ Zk . If u ∈ D, then Φ(u) ≥

+ − + + + e µ P− − C(αkα |u|α ) − − (βkq |u|q ) − C1 . + kuk q P+

By assuming α+ ≤ q + , we obtain + − + + + µ e Φ(u) ≥ + kukP− − C(αkα |u|q ) − − (βkq |u|q ) − C1 q P+ − + + µ + e ≥ + kukP− − (Cαkα + − βkq )|u|q − C1 q P+ + − + + e ≥ + kukP− − C3 (αkα + βkq )|u|q − C1 P+ −

i −P− 1 1 h + q+ + α+ ≥ e + − + C3 q (αk + βk ) P− −q − C1 . q P+ +

+

1 −

Since q + > P++ , we then have [C3 q + (αkα + βkq )] P− −q → ∞, as k → ∞. Consequently, we obtain Φ(u) → +∞ as kuk → +∞, u ∈ Zk . Now, from condition (H2), we have G(x, s) ≥ C1 |s|θλ − C2 , for any (x, s) ∈ Ω × R. +

Then there exist constants C10 , C30 > 0 such that Z N Z ∂u pi (x) C10 X λ Φ(u) ≤ − dx + |u|m(x) dx − C20 kukθλ − C30 , m b Ω P− i=1 Ω ∂xi where m b = m− if λ > 0 and m b = m+ if λ ≤ 0. Hence, we obtain the inequality N N P++ X X ∂u ∂u P++ p (x) p (x) ≤ C , ∂xi L i (Ω) ∂xi L i (Ω) i=1 i=1

Where C is a positive constant. In the case λ > 0, we obtain Φ(u) ≤

+ + C0 C4 λ P+ + kukm − C20 kukθλ − C30 . − kuk m b P−

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EXISTENCE AND MULTIPLICITY OF SOLUTIONS → −

11

→ −

But we have W 1, p (x) (Ω) ,→ Lm(x) (Ω), and W 1, p (x) (Ω) ,→ Lq(x) (∂Ω). Then, as θλ > max(P++ , m+ ) and dim Yk = k, it is easy to see that Φ(u) → −∞ as kuk → +∞ for u ∈ Yk . For the case λ ≤ 0, we have Φ(u) ≤

+ C0 kukP − C20 kukθλ − C30 . P−−

Now, as we have θλ > P++ and dim Yk = k, it is also easy to see that Φ(u) → −∞ as kuk → +∞ for u ∈ Yk . 4. A generalized equation We shall now consider the generalized equation n n X ∂u ∂u pi (x)−2 ∂u X pi (x)−2 |u| u + − ∂xi ∂xi ∂xi i=1 i=1 = λg1 (x, u) + νg2 (x, u)

in Ω,

(4.1)

n X

∂u pi (x)−2 ∂u υi = µf (x, u) ∂xi ∂xi i=1

on ∂Ω,

¯ with 2 ≤ pi (x) ≤ N for all where λ, ν, µ > 0 are real numbers, pi (x) ∈ C(Ω) i ∈ {1, 2, . . . , N }, and g1 , g2 : Ω × R → R are two functions of class C 1 with respect to the Ω-variable, and f : ∂Ω×R → R is of class C 1 with respect to the ∂Ω-variable. We make the following assumptions on the functions q, g1 , g2 and f . ¯ satisfies 1 < qi (x) < (H5) For i = 1, 2, qi ∈ C(Ω) (H6)

− N P− − N −P−

for all x ∈ Ω.

(i) For i = 1, 2, gi : Ω × R → R satisfies the Caratheodory condition, and there exist some positive constant Ci such that |gi (x, s)| ≤ C1 + C2 |s|qi (x)−1

for (x, s) ∈ Ω × R.

(ii) There exists M > 0, σ > P++ such that for all |s| ≥ M and x ∈ Ω, 0 < σG2 (x, s) ≤ g2 (x, s)s. ¯ such that (H7) There exist δ1 > 0, C3 > 0 and q3 ∈ C(Ω) G1 (x, s) ≥ C3 |s|q3 (x) , ∀(x, s) ∈ Ω × (0, δ1 ], where max(q3− , q1+ ) < P−− < P++ < q2− . +

(H8) g2 (x, s) = ◦(|s|P+ −1 ) as s → 0 uniformly for x ∈ Ω. (H9) (i) f : ∂Ω × R → R satisfies the Caratheodory condition and there exists a constant C > 0 such that |f (x, s)| ≤ C(1 + |s|β(x)−1 ),

∀(x, s) ∈ ∂Ω × R;

where β(x) ∈ C(∂Ω) with 1 < β − ≤ β + < P−− and β(x) < for all x ∈ ∂Ω. (ii) There exist R > 0, such that for all |s| ≥ R and x ∈ ∂Ω 0 < σF (x, s) ≤ f (x, s)s.

− (N −1)P− − N −P−

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(H10) There exist δ2 > 0, C4 > 0 and q4 (x) ∈ C(∂Ω) such that F (x, s) ≥ C4 |s|q4 (x) , where 1 < q4
0 be given. Set βk = βk (ν) =

|ψ(u)|.

sup u∈Zk ,kuk≤ν

Then βk → 0 as k → ∞. Theorem 4.4. Assume that (H5), (H6) and (H9) hold. (1) If in addition, (H10) holds, then for every γ, µ > 0, there exists r0 (γ) > 0 such that when 0 ≤ λ, µ ≤ r0 (γ), problem (4.1) has a nontrivial solution u1 such that Φ(u1 ) > 0. (2) If in addition, (H7) and (H10) hold, then for every γ, µ > 0, there exists r0 (γ) > 0 such that when 0 ≤ λ, µ ≤ r0 (γ), problem (4.1) has two nontrivial solutions u1 , v1 such that Φ(u1 ) > 0 and Φ(v1 ) < 0. (3) If in addition, (H7), (H10) and (H11) hold, then for every λ, γ, µ > 0, problem (4.1) has a sequence of solutions {±uk } such that Φ(±uk ) → +∞ as k → +∞. Proof. (1) We denote Z

Z

ψ1 (u) = λ

G1 (x, u(x)) dx,

ψ2 (u) = γ

Ω

G2 (x, u(x)) dx. Ω

When the assumptions in (1) hold, then for sufficiently small kuk, we get +

G2 (x, u) ≤ |u|P+ + C()|u|q2 , Then

Z ψ2 (u) ≤ γ

|u|

+ P+

∀(x, s) ∈ Ω × R, Z

dx + γC() Ω

Ω

Since 1 < q2
0 such that + − ψ2 (u) ≤ γC1 kukP+ + γC()C2 kukq2 . Choose > 0 small enough so that 0 < γC2 < P + −11 P + −1 . Then, we have 2

N Z X i=1

≥ Since

q2−

> N Z X i=1

P++ ,

Ω

Ω

N

+

1 ∂u pi (x) + |u|pi (x) dx − ψ2 (u) pi (x) ∂xi 1

+ P+

+

+ P+ −1

+

−

kukP+ − γC()C2 kukq2 .

2 N there exist r1 > 0 and α > 0 such that

1 ∂u pi (x) + |u|pi (x) dx − ψ2 (u) ≥ α > 0, pi (x) ∂xi

for kuk = r1 .

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We can find r0 (γ) > 0 such that when µ, λ ≤ r0 (γ), we obtain α ψ1 (u) ≤ , ∀u ∈ Sr1 = {u ∈ X; kuk = r1 }. 2 Therefore, λ, µ ≤ r0 (γ). So, we obtain α Φ(u) ≥ > 0, ∀u ∈ Sr1 . 2 Let u ∈ X and t > 1, we have Z N Z X tpi (x) ∂u pi (x) pi (x) dx − λ G1 (x, tu) dx + |u| Φ(tu) = p (x) ∂xi Ω i=1 Ω i Z Z −γ G2 (x, tu) dx − µ F (x, tu) dx. Ω

(4.3)

∂Ω

From Remark 4.2, we obtain Z N Z + X 1 ∂u pi (x) + |u|pi (x) dx − λ G1 (x, tu) dx Φ(tu) ≤ tP+ p (x) ∂xi Ω i=1 Ω i Z Z 1/σ 1/σ − γC5 t |u| dx − µ F (x, tu) dx. Ω

(4.4)

∂Ω

Now, since +

G1 (x, tu) = o((t|u|)q1 ),

+

F (x, tu) = o((t|u|)β )

(because P++ ≤ q1+ and β + < P++
r1 and Φ(u0 ) < 0. Therefore, By the Mountain Pass theorem, problem (4.1) has a nontrivial solution u1 such that Φ(u1 ) > 0. (2) Under the assumpti9ns in (2) hold, (1), we know that there exist r0 (γ) > 0 such that when 0 ≤ λ, µ ≤ r0 (γ), problem has a nontrivial solution u1 such that Φ(u1 ) > 0. For t ∈ (0, 1) small enough, and v0 ∈ C0∞ (Ω) such that 0 ≤ v0 (x) ≤ min{δ1 , δ2 }, we have Φ(tv0 ) =

N Z X

1 ∂(tv0 ) pi (x) + |tv0 |pi (x) dx − λ p (x) ∂xi i=1 Ω i Z Z −γ G2 (x, tv0 ) dx − µ F (x, tv0 ) dx Ω N Z X

Z G1 (x, tv0 ) dx Ω

1 ∂v0 pi (x) + |v0 |pi (x) dx − λC3 p (x) ∂xi i=1 Ω i Z Z −γ G2 (x, tv0 ) dx − µC4 |tv0 |q4 (x) dx. −

≤ tP−

Ω

(4.5)

∂Ω

Z

∂Ω

For t ∈ (0, 1) small enough, we obtain +

G2 (x, tv0 ) = o(|tv0 |P+ ),

as t → ∞

Ω

|tv0 |q3 (x) dx

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15

So, we have Φ(tv0 ) N Z X

+ 1 ∂tv0 pi (x) + |tv0 |pi (x) dx − λC3 tq3 p (x) ∂xi i=1 Ω i Z Z + + − γM tP+ v0 dx − µC4 tq4 |v0 |q4 (x) dx. −

≤ tP−

Ω

Z

|v0 |q3 (x) dx

Ω

(4.6)

∂Ω

Since min(q3+ , q4+ ) < P−− , by factoring + by tq4 if q3+ > q4+ , we obtain

+

the right side of (4.6) by tq3 if q4+ > q3+ , and

lim Φ(tv0 ) < 0.

t→0

Then there exist w ∈ X such that kwk ≤ r1 , and Φ(w) < 0. (3) Φ is an even functional. We denote Z Z Z ψ(u) = λ G1 (x, u) dx + γ G2 (x, u) dx + µ F (x, u) dx Ω

Ω

∂Ω

As βk (ν) is defined in Proposition 2.6, for each positive integer, there exist a positive integer k0 such that βk (n) ≤ 1 for all k ≥ k0 (n). We can assume k0 (n) < k0 (n + 1) for each n. We define {νk : k = 1, 2, . . . } by ( n if k0 ≤ k < k0 (n + 1), νk = (4.7) 1 if 1 ≤ k < k0 . We see that νk → ∞ when k → ∞, then for u ∈ Zk with kuk = νk , we obtain Φ(u) =

N Z X Ω

i=1

≥

1 ∂u pi (x) + |u|pi (x) dx − ψ(u) pi (x) ∂xi 1

− −1 P−

P++ 2

−

N

− P− −1

(νk )P− − 1.

Consequently, inf

Φ(u) → ∞ as k → ∞.

u∈Zk ,kuk=νk

So the hypotheses (3.5) of Fountain theorem are satisfied. Indeed, by (H6), (H9) and Remark 4.2, for kuk ≥ 1 we obtain Φ(u) ≤

+ + C q+ 1/σ P+ + C1 λ|u|q11 (x) − C5 γ|u| 1 + C6 µ|u|ββ(x) + C7 . − kuk σ P−

As the space Yk has finite dimension i.e all norms are equivalents, we then have Φ(u) ≤

+ + + C P+ + C10 λkukq1 − C50 γkuk1/σ + C 0 µkukβ + C7 . − kuk P−

Since min(q1+ , q2+ ) < P++ < σ1 , we obtain Φ(u) → −∞ as kuk → +∞, u ∈ Yk . Finally, the proof of (3) is complete. Acknowledgements. The authors would like to thank the anonymous referees for their helpful remarks.

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Brahim Ellahyani Department of Mathematics, Faculty of Sciences, Mohammed V University, P.B. 10014, Rabat, Morocco E-mail address: [email protected]

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Abderrahmane El Hachimi Department of Mathematics, Faculty of Sciences, Mohammed V University, P.B. 10014, Rabat, Morocco E-mail address: [email protected]