Nonlinear Analysis: Real World Applications 16 (2014) 132–145 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa On a class of anisotropic elliptic equations without Ambrosetti–Rabinowitz type conditions Nguyen Thanh Chung a,∗ , Hoang Quoc Toan b a Department of Science Management and International Cooperation, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam b Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam article info Article history: Received November 2012 Accepted 24 September 2013 abstract This article investigates a class of anisotropic elliptic equations with non-standard growth conditions N − ∂xi |∂xi u|pi (x)−2 ∂xi u = f (x, u) u =i=0 in Ω , on ∂ Ω , where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂ Ω , and pi , i = 1, 2, , N are continuous functions on Ω such that < pi (x) < N Using variational methods, we obtain some existence and multiplicity results for such problems without Ambrosetti–Rabinowitz type conditions © 2013 Elsevier Ltd All rights reserved Introduction In this paper, we are interested in the existence of solutions for the elliptic anisotropic problem with non-standard growth conditions N − ∂xi |∂xi u|pi (x)−2 ∂xi u = f (x, u) u =i=0 in Ω , (1.1) on ∂ Ω , where Ω ⊂ R (N ≥ 3) is a bounded domain with smooth boundary ∂ Ω , and pi , i = 1, 2, , N are continuous functions on Ω such that < pi (x) < N, and f : Ω × R → R is a Carathéodory function In the case when pi (x) = p(x) for any i = 1, 2, , N, the operator involved in (1.1) has similar properties to the p(x)Laplace operator, i.e., ∆p(x) u := div(|∇ u|p(x)−2 ∇ u) This differential operator is a natural generalization of the isotropic p-Laplace operator ∆p u := div(|∇ u|p−2 ∇ u), where p > is a real constant However, the p(x)-Laplace operator possesses more complicated nonlinearities than the p-Laplace operator, due to the fact that ∆p(x) is not homogeneous The study of nonlinear elliptic problems (equations and systems) involving quasilinear homogeneous type operators like the p-Laplace operator is based on the theory of standard Sobolev spaces W k,p (Ω ) in order to find weak solutions These spaces consist of functions that have weak derivatives and satisfy certain integrability conditions In the case of nonhomogeneous p(x)Laplace operators the natural setting for this approach is the use of the variable exponent Sobolev spaces Differential and N ∗ Corresponding author Tel.: +84 523863245 E-mail addresses: ntchung82@yahoo.com (N.T Chung), hq_toan@yahoo.com (H.Q Toan) 1468-1218/$ – see front matter © 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.nonrwa.2013.09.012 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 133 partial differential equations with nonstandard growth conditions have received specific attention in recent decades The interest played by such growth conditions in elastic mechanics and electrorheological fluid dynamics has been highlighted in many physical and mathematical works We refer to some interesting works [1–8] In a recent paper [9], I Fragalà et al have studied the following anisotropic quasilinear elliptic problem N − ∂xi |∂xi u|pi −2 ∂xi u = λup−1 in Ω , i =1 (1.2) in Ω , on ∂ Ω , u ≥ u=0 where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂ Ω , pi > for all i = 1, 2, , N and p > Note that if pi = for all i = 1, 2, , N then problem (1.2) reduces to the well-known semilinear equation −∆u = λup−1 By proving an embedding theorem involving the critical exponent of anisotropic type, the authors obtained some existence and nonexistence results in the case when p > p+ = max{p1 , p2 , , pN } or p < p− = min{p1 , p2 , , pN } The results in [9] have been extended by A.D Castro et al [10], in which the authors studied problem (1.2) in the case when p− < p < p+ In − → 1, p order to study the existence of solutions for (1.2) the above authors have found the solutions in the space W0 is defined as the closure of C0∞ (Ω ) with respect to the norm → ∥ u∥ − p = N (Ω ) which |∂xi u|pi , i =1 − → where p = (p1 , p2 , , pN ) and |.|pi denotes the norm in Lpi (Ω ) for all i = 1, 2, , N In [11–13], the authors developed problem (1.2) in the case when pi (x) are continuous functions in Ω , i = 1, 2, , N Since then, many authors have been interested in the existence of solutions for elliptic problems in this direction, we refer to some interesting works [11,14–19] In [14,16], the authors studied problem (1.1) in the case when the nonlinearity f verifies + the Ambrosetti–Rabinowitz type conditions ((A–R) type conditions for short), that is, there exists a positive constant µ > P+ such that < µF (x, t ) := µ t f (x, s)ds ≤ f (x, t )t (1.3) for all x ∈ Ω and |t | > M > 0, which implies that for some positive constants c , d, we have F (x, t ) ≥ c |t |µ − d for all t ∈ R (1.4) + This says that f (x, t ) is P+ -superlinear at infinity in the sense that lim F (x, t ) |t |→∞ + |t |P+ = +∞ uniformly in x ∈ Ω In [17], the author considered problem (1.1) with a particular nonlinearity f (x, t ) = t α(x)−1 − t β(x)−1 , t ≥ and x ∈ Ω − The functions α(x) and β(x) were assumed to satisfy the condition < β − ≤ β + < α − ≤ α + < P− ≤ P++ < P−,∞ This + means that f (x, t ) is P+ -sublinear at infinity Using the minimum principle combined with the mountain pass theorem, the author obtained the existence of at least two nonnegative nontrivial weak solutions The result of [17] was improved in [18], in which the author assumed that (see the condition (10) in Theorem of [18]): max sup F (x, t ) lim sup |t |→0 x∈Ω + |t |P+ sup F (x, t ) , lim sup |t |→+∞ x∈Ω − |t |P− ≤ Using the three critical points theorem by B Ricceri [20] the author obtained a multiplicity result for (1.1) Regarding the problem (1.1) with Neumann boundary conditions, we refer to the papers [15,19] + In this paper, we consider problem (1.1) in the case when the nonlinear term f (x, t ) is P+ -superlinear at infinity but does not satisfy the (A–R) type condition (1.3) as in [14,16] More precisely, motivated by the ideas firstly introduced by O.H Miyagaki et al [21] and developed by G Li et al [22], C Ji [23], the goal this paper is to prove some existence and multiplicity results for problem (1.1) in the anisotropic case To overcome the difficulties brought, we will use the mountain pass theorem in [24] and the fountain theorem in [25] with the (Cc ) condition (see Definition 3.4) 134 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 The remainder of the paper is organized as follows In Section 2, we will recall the definitions and some properties of anisotropic variable exponent Sobolev spaces The readers can consult the paper [11–13] for details on this class of functional spaces In Section we will state and prove the main results of the paper Anisotropic variable exponent Sobolev spaces We recall in what follows some definitions and basic properties of the generalized Lebesgue–Sobolev spaces Lp(x) (Ω ) and W 1,p(x) (Ω ) where Ω is an open subset of RN In that context, we refer to the book of Musielak [7], the papers of Kováčik and Rákosník [6] and Fan et al [2,3] Set C+ (Ω ) := {h; h ∈ C (Ω ), h(x) > for all x ∈ Ω } For any h ∈ C+ (Ω ) we define h+ = sup h(x) x∈Ω and h− = inf h(x) x∈Ω For any p(x) ∈ C+ (Ω ), we define the variable exponent Lebesgue space L p(x) (Ω ) = u : a measurable real-valued function such that Ω |u(x)| p(x) dx < ∞ We recall the following so-called Luxemburg norm on this space defined by the formula u(x) p(x) = inf µ > 0; dx ≤ µ Ω |u|p(x) Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if < p− ≤ p+ < ∞ and continuous functions are dense if p+ < ∞ The inclusion between Lebesgue spaces also generalizes naturally: if < |Ω | < ∞ and p1 , p2 are variable exponents so that ′ p1 (x) ≤ p2 (x) a.e x ∈ Ω then there exists the continuous embedding Lp2 (x) (Ω ) ↩→ Lp1 (x) (Ω ) We denote by Lp (x) (Ω ) the ′ conjugate space of Lp(x) (Ω ), where p(1x) + p′1(x) = For any u ∈ Lp(x) (Ω ) and v ∈ Lp (x) (Ω ) the Hölder inequality 1 + ′ − |u|p(x) |v|p′ (x) uv dx ≤ p− (p ) Ω holds true An important role in manipulating the generalized Lebesgue–Sobolev spaces is played by the modular of the Lp(x) (Ω ) space, which is the mapping ρp(x) : Lp(x) (Ω ) → R defined by ρp(x) (u) = Ω |u|p(x) dx If u ∈ Lp(x) (Ω ) and p+ < ∞ then the following relations hold + − |u|pp(x) ≤ ρp(x) (u) ≤ |u|pp(x) (2.1) provided |u|p(x) > while + − |u|pp(x) ≤ ρp(x) (u) ≤ |u|pp(x) (2.2) provided |u|p(x) < and |un − u|p(x) → ⇔ ρp(x) (un − u) → 1,p(x) Next, we define the space W0 (Ω ) as the closure of C0∞ (Ω ) under the norm ∥u∥p(x) = |∇ u|p(x) We point out that the above norm is equivalent with the following norm ∥u∥p(x) = N i =1 |∂xi u|p(x) , (2.3) N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 1,p(x) provided that p(x) ≥ for all x ∈ Ω The space W0 135 (Ω ), ∥.∥p(x) is a separable and Banach space We note that if s ∈ C+ (Ω ) and s(x) < p (x) for all Ω then the embedding ∗ 1,p(x) W0 (Ω ) ↩→ Ls(x) (Ω ) is compact and continuous, where p∗ (x) = Np(x) N −p(x) if p(x) < N or p∗ (x) = ∞ if p(x) > N We introduce a natural generalization of the variable exponent Sobolev space W 1,p(x) (Ω ) that will enable us to study − → − → problem (1.1) with sufficient accuracy Define p : Ω → RN the vectorial function p = (p1 , p2 , , pN ) We introduce − → 1, p (x) the anisotropic variable exponent Sobolev space, W0 → ∥ u∥ − p (x) = N (Ω ), as the closure of C0∞ (Ω ) with respect to the norm |∂xi u|pi (x) i =1 − → 1, p (x) (Ω ) is a reflexive and separable Banach space, see [11–13] In the case when pi are all constant functions the − → 1, p → (Ω ), where − p is the constant vector (p1 , p2 , , pN ) The theory of such − → − → + spaces has been developed in [9,10] Let us introduce P + , P − ∈ RN and P+ , P−+ , P+− , P−− ∈ R+ as − → − → + + − − P + = (p+ P − = (p− , p2 , , pN ), , p2 , , pN ) Then W0 resulting anisotropic space is denoted by W0 and + + + P+ = max{p+ , p2 , , pN }, − − + P− = max{p− , p2 , , pN }, + + − P+ = min{p+ , p2 , , pN }, − − − P− = min{p− , p2 , , pN } Throughout this paper we assume that N i =1 p− i >1 (2.4) ∗ and define P− ∈ R and P−,∞ ∈ R+ by ∗ P− = N N − pi i=1 , + P−,∞ = max{P− , P−∗ } −1 − → 1, p (x) We recall that if s ∈ C+ (Ω ) satisfies < s(x) < P−,∞ for all x ∈ Ω then the embedding W0 see for example [13, Theorem 1] (Ω ) ↩→ Ls(x) (Ω ) is compact, Main results In this section, we state and prove the main results of this paper We will use the letter Ci to denote positive constants Let us introduce the following hypotheses: (f0 ) f : Ω × R → R is a Carathéodory function and satisfies the subcritical growth condition |f (x, t )| ≤ C (1 + |t |q(x)−1 ), ∀(x, t ) ∈ Ω × R, where such that G(x, t ) ≤ G(x, s) + C∗ + for any x ∈ Ω , < t < s or s < t < 0, where G(x, t ) := tf (x, t ) − P+ F (x, t ) and F (x, t ) := (f4 ) f (x, −t ) = −f (x, t ) for all (x, t ) ∈ Ω × R t f (x, s)ds; It should be noticed that the condition (f3 ) is a consequence of the following condition, which was firstly introduced by O.H Miyagaki et al [21] and developed by G Li et al [22] and C Ji [23]: (f3′ ) There exists t0 > such that f (x,t ) + P −2 |t | + t is nondecreasing in t ≥ t0 and nonincreasing in t ≤ −t0 for any x ∈ Ω 136 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 The readers may consult the proof and comments on this assertion in the papers [21–23] and the references cited there By some simple computations, we can show that the function + f (x, t ) = |t |P+ −2 t log(1 + |t |), t ∈R satisfies our conditions (f0 )–(f4 ) but it does not satisfy the (A–R) type condition (1.3) − → 1, p (x) (Ω ) is said to be a weak solution of problem (1.1) if |∂xi u|pi (x)−2 ∂xi u · ∂xi v dx − f (x, u)v dx = Definition 3.1 A function u ∈ W0 N i=1 Ω Ω − → 1, p (x) for all v ∈ W0 (Ω ) Our main results in this paper are given by the following two theorems Theorem 3.2 Assume that the conditions (f0 )–(f3 ) are satisfied Then problem (1.1) has a non-trivial weak solution Theorem 3.3 Assume that the conditions (f0 ), (f2 )–(f4 ) are satisfied Then problem (1.1) has infinitely many weak solutions {uk } satisfying N i=1 Ω pi (x) |∂xi uk |pi (x) dx − Ω F (x, uk ) dx → +∞, k → ∞ Our Theorem 3.2 is exactly an extension from the results by O.H Miyagaki et al [21], G Li et al [22] and C Ji [23] to problem (1.1) considered in anisotropic variable exponent Sobolev spaces (note that in this paper, we not use the parameter λ as in [21–23]), while our Theorem 3.3 seems to be new even with the well-known p-Laplace operator ∆p u In order to prove the main theorems, we recall some useful concepts and results Definition 3.4 Let (X , ∥.∥) be a real Banach space, J ∈ C (X , R) We say that J satisfies the (Cc ) condition if any sequence {um } ⊂ X such that J (um ) → c and ∥J ′ (um )∥(1 + ∥um ∥) → as m → ∞ has a convergent subsequence Proposition 3.5 (See [24]) Let (X , ∥.∥) be a real Banach space, J ∈ C (X , R) satisfies the (Cc ) condition for any c > 0, J (0) = and the following conditions hold: (i) There exists a function φ ∈ X such that ∥φ∥ > ρ and J (φ) < 0; (ii) There exist two positive constants ρ and R such that J (u) ≥ R for any u ∈ X with ∥u∥ = ρ Then the functional J has a critical value c ≥ R, i.e there exists u ∈ X such that J ′ (u) = and J (u) = c In order to prove Theorem 3.3 we will use the following fountain theorem, see [25] for details Let (X , ∥.∥) be a real reflexive Banach space presenting by X = ⊕j∈N Xj with dim(Xj ) < +∞ for any j ∈ N For each k ∈ N, we set Yk = ⊕kj=0 Xj and Zk = ⊕∞ j=k Xj Proposition 3.6 (See [25]) Let (X , ∥.∥) be a real reflexive Banach space, J ∈ C (X , R) satisfies the (Cc ) condition for any c > and J is even If for each sufficiently large k ∈ N, there exist ρk > rk > such that the following conditions hold: (i) ak := inf{u∈Zk : ∥u∥=rk } J (u) → +∞ as k → ∞; (ii) bk := max{u∈Yk : ∥u∥=ρk } J (u) ≤ Then the functional J has an unbounded sequence of critical values, i.e there exists a sequence {uk } ⊂ X such that J ′ (uk ) = and J (uk ) → +∞ as k → +∞ − → 1, p In the rest of this paper we will use the letter X to denote the anisotropic variable exponent Sobolev space W0 us define the energy functional J : X → R by the formula J (u) = N i=1 Ω pi (x) |∂xi u|pi (x) dx − Ω F (x, u) dx (Ω ) Let (3.1) By the hypothesis (f0 ) and the continuous embeddings, some standard arguments assure that the functional J is well-defined on X and J ∈ C (X ) with the derivative given by J (u)(v) = ′ N i=1 pi (x)−2 Ω |∂xi u| ∂xi u · ∂xi v dx − Ω f (x, u)v dx for all u, v ∈ X Thus, weak solutions of problem (1.1) are exactly the critical points of the functional J N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 137 Lemma 3.7 Assume that the conditions (f0 )–(f2 ) are satisfied Then we have the following assertions: (i) There exists φ ∈ X , φ > such that J (t φ) → −∞ as t → +∞; (ii) There exist ρ > and R > such that J (u) ≥ R for any u ∈ X with ∥u∥ = ρ Proof (i) From (f2 ), it follows that for any M > there exists a constant CM = C (M ) > depending on M, such that + F (x, t ) ≥ M |t |P+ − CM , ∀x ∈ Ω , ∀t ∈ R (3.2) Take φ ∈ X with φ > 0, from (3.2) we get N |∂xi t φ|pi (x) dx − F (x, t φ) dx pi (x) Ω i =1 Ω + N + + t P+ |∂xi φ|pi (x) dx − Mt P+ |φ|P+ dx + CM |Ω | ≤ − J (t φ) = P− i=1 ≤t + P+ Ω Ω N − P− i =1 pi (x) Ω |∂xi φ| dx − M + P+ |φ| dx Ω + CM |Ω |, (3.3) where t > and |Ω | denotes the Lebesgue measure of Ω From (3.3), if M is large enough such that N − P− i =1 Ω |∂xi φ|pi (x) dx − M + |φ|P+ dx < 0, Ω then we have lim J (t φ) = −∞, t →+∞ which ends the proof of (i) + (ii) Since the embeddings X ↩→ LP+ (Ω ) and X ↩→ Lq(x) (Ω ) are continuous, there exist constants C1 , C2 > such that ∥u∥ P++ L P (Ω ) → ≤ C1 ∥u∥− p (x) , + Let < ϵ C1 + < , + + P −1 2P+ N + → ∥u∥Lq(x) (Ω ) ≤ C2 ∥u∥− p (x) (3.4) where C1 is given by (3.4) From (f0 ) and (f1 ), we have + F (x, t ) ≤ ϵ|t |P+ + C (ϵ)|t |q(x) , ∀(x, t ) ∈ Ω × R (3.5) → Let u ∈ X with ∥u∥− p (x) < sufficiently small For such an element u we get |∂xi u|pi (x) < for all i = 1, 2, , N Using (2.1) and some simple computations, we obtain N i =1 Ω |∂xi u|pi (x) dx ≥ N + p |∂xi u|pii (x) i=1 ≥ N P + |∂xi u|p+i (x) i=1 P++ N |∂xi u|pi (x) i=1 ≥ N N P = + + ∥ u∥ − → p (x) + N P+ −1 (3.6) 138 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 From (3.4)–(3.6) we have Jλ (u) = N |∂xi u|pi (x) F (x, u) dx dx − pi (x) Ω i=1 Ω P ≥ ≥ + + ∥ u∥ − → p (x) + + P+ −1 P+ N −ϵ Ω P + P + N P+ −1 + |u|P+ dx − C (ϵ) + P + P |u|q(x) dx Ω + − − + + − C (ϵ)C2q ∥u∥q− − ϵ C + ∥ u∥ − ∥ u∥ − → → → p (x) p ( x) p (x) + ≥ + + P+ −1 2P+ N + + − P+ q − −P + , ∥ u∥ − − C (ϵ)C2q ∥u∥− → → p (x) p (x) (3.7) + where C2 > is given by (3.4) From (3.7) and the fact that q− > P+ , we can choose R > and ρ > such that J (u) ≥ R > → for all u ∈ X with ∥u∥− = ρ The proof of Lemma 3.7 is complete p (x) Lemma 3.8 Assume that the conditions (f0 ), (f2 )–(f3 ) are satisfied Then the functional J satisfies the (Cc ) condition for any c > Proof Let {um } ⊂ X be a (Cc ) sequence of the functional J, that is, J (um ) → c > 0, → ∥J ′ (um )∥∗ (1 + ∥um ∥− p (x) ) → as m → ∞, which shows that c = J (um ) + o(1), J ′ (um )(um ) = o(1), (3.8) where o(1) → as m → ∞ We will prove that the sequence {um } is bounded in X Indeed, if {um } is unbounded in X , we may assume that um → ∥ um ∥ − , m = 1, 2, It is clear that {wm } ⊂ X p (x) → ∞ as m → ∞ We define the sequence {wm } by wm = ∥u ∥− → m p (x) → and ∥wm ∥− p (x) = for any m Therefore, up to a subsequence, still denoted by {wm }, we have {wm } converges weakly to some w ∈ X and wm (x) → w(x), a.e in Ω , m → ∞, wm → w strongly in L q(x) (3.9) (Ω ), m → ∞, + P+ wm → w strongly in L (Ω ), m → ∞ (3.10) (3.11) → Let Ω̸= := {x ∈ Ω : w(x) ̸= 0} If x ∈ Ω̸= then it follows from (3.9) that |um (x)| = |wm (x)|∥um ∥− p (x) → +∞ as m → ∞ Moreover, from (f2 ), we have lim F (x, um (x)) m→∞ |um (x)| + P+ + |wm (x)|P+ = +∞, x ∈ Ω̸= (3.12) Using the condition (f2 ), there exists t0 > such that F (x, t ) + |t |P+ >1 (3.13) for all x ∈ Ω and |t | > t0 > Since F (x, t ) is continuous on Ω × [−t0 , t0 ], there exists a positive constant C3 such that |F (x, t )| ≤ C3 (3.14) for all (x, t ) ∈ Ω × [−t0 , t0 ] From (3.13) and (3.14) there exists C4 ∈ R such that F (x, t ) ≥ C4 (3.15) for all (x, t ) ∈ Ω × R From (3.15), for all x ∈ Ω and m, we have F (x, um (x)) − C4 P + + ∥ um ∥ − → p (x) ≥0 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 139 or F (x, um (x)) |um (x)| + P+ C4 + |wm (x)|P+ − P ≥ 0, + ∀x ∈ Ω , ∀m (3.16) + ∥um ∥− → p (x) For each i ∈ {1, 2, , N } and m ∈ N, we define αi,m = + P+ − P− if |∂xi um |pi (x) < 1, if |∂xi um |pi (x) > Using (2.1), (2.2) and some simple computations, we infer that for any m, N Ω i =1 |∂xi um |pi (x) dx ≥ N α |∂xi um |pii(,mx) i=1 ≥ N − P− |∂xi um |pi (x) − − P− |∂xi um |pi (x) − |∂ + P+ xi um pi (x) | + {i:αi,m =P+ } i=1 P−− N |∂xi um |pi (x) i=1 ≥ N N P = −N − − ∥ um ∥ − → p (x) − N P − −1 − N (3.17) By (3.8) and (3.17) we have c = J (um ) + o(1) = N |∂xi um |pi (x) dx − F (x, um ) dx + o(1) pi (x) Ω i =1 Ω P ≥ − − ∥ um ∥ − → p ( x) − + P− −1 N P+ − N + P+ − Ω F (x, um ) dx + o(1) or Ω F (x, um ) dx ≥ P − P + N P− −1 − − ∥ um ∥ − −c− → p (x) + N + P+ + o(1) → +∞ as m → ∞ (3.18) Similarly, for each i ∈ {1, 2, , N } and m ∈ N, we define βi,m = − P− + P+ if |∂xi um |pi (x) < 1, if |∂xi um |pi (x) > Using (2.1), (2.2) and some simple computations, we infer that for any m, N i =1 Ω |∂xi um |pi (x) dx ≤ N β |∂xi um |pii(,mx) i=1 ≤ N P + + − P P |∂xi um |p+i (x) − |∂xi um |p−i (x) + N − {i:βi,m =P− } i=1 ≤ |∂xi um |p+i (x) − N P++ |∂xi um |pi (x) + 2N i =1 P + + = ∥um ∥− + 2N → p (x) (3.19) 140 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 Thus, we have from (3.8) and (3.19) that c = J (um ) + o(1) N |∂xi um |pi (x) = dx − F (x, um ) dx + o(1) pi (x) Ω i =1 Ω + 2N P+ + − − ≤ − ∥ um ∥ − F (x, um ) dx + o(1) → p (x) P− P− Ω or by (3.18), P + + ≥ P−− ∥ um ∥ − → p ( x) Ω − F (x, um ) dx + cP− − 2N − o(1) > for m large enough (3.20) We claim that |Ω̸= | = In fact, if |Ω̸= | ̸= 0, then by (3.12), (3.16), (3.20) and the Fatou lemma, we have +∞ = lim Ω̸= m→∞ = lim F (x, um (x)) + P+ |um (x)| F (x, um (x)) + m→∞ ≤ lim inf + |um (x)|P+ F (x, um (x)) Ω̸= m→∞ Ω F (x, um (x)) = lim inf m→∞ Ω + P+ Ω P + C4 lim m→∞ Ω̸= P P + dx + ∥um ∥− → p (x) C4 + dx + + ∥um ∥− → p (x) C4 + |wm (x)|P+ − dx + ∥um ∥P+ C4 + |wm (x)|P+ − P dx + + ∥ um ∥ − → p (x) C4 dx − lim sup m→∞ ∥ um ∥ − → p (x) F (x, um (x)) = lim inf m→∞ + |um (x)|P+ |wm (x)|P+ − |um (x)|P+ F (x, um (x)) Ω̸= m→∞ ≤ lim inf + |wm (x)|P+ dx − Ω P + dx + ∥ um ∥ − → p (x) dx + ∥ um ∥ − → p (x) F (x, um (x)) dx Ω ≤ lim inf − − m→∞ P − Ω F (x, um ) dx + cP− − 2N − o(1) (3.21) From (3.18) and (3.21), we obtain +∞ ≤ − P− , which is a contradiction This shows that |Ω̸= | = and thus w(x) = a.e in Ω Since J (tum ) is continuous in t ∈ [0, 1], for each m there exists tm ∈ [0, 1], m = 1, 2, , such that J (tm um ) := max J (tum ) (3.22) t ∈[0,1] It is clear that tm > and J (tm um ) ≥ c > = J (0) = J (0.um ) If tm < then J ′ (tm um )(tm um ) = If tm = 1, then J ′ (um )(um ) = o(1) So we always have J ′ (tm um )(tm um ) = o(1) d J dt (tum )|t =t m = which gives (3.23) → Let {Rk } be a positive sequence of real numbers such that Rk > for any k and limk→∞ Rk = +∞ Then ∥Rk wm ∥− p (x) = Rk > for any k and m Fix k, since wm → strongly in the spaces Lq(x) (Ω ) and wm (x) → a.e x ∈ Ω as m → ∞, using the condition (f0 ) and the Lebesgue dominated convergence theorem we deduce that lim m→∞ Ω F (x, Rk wm ) dx = (3.24) N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 141 k → − → Since ∥um ∥− < for m large enough Hence, using (3.17) p (x) → ∞ as m → ∞, we also have ∥um ∥ p (x) > Rk or < ∥um ∥− → p ( x) and (3.24), it follows that R J (tm um ) ≥ J Rk → ∥ um ∥ − p (x) um = J (Rk wm ) N |∂xi Rk wm |pi (x) F (x, Rk wm ) dx = dx − pi (x) Ω i =1 Ω − ≥ = ∥R w ∥P− → k m − p (x) −N − F (x, Rk wm ) dx − + P+ N P− −1 − P Rk − − P + N P− −1 − + P ≥ Ω N + P+ − Ω F (x, Rk wm ) dx − Rk − − + P− 2P+ N −1 N − (3.25) + P+ for any m large enough From (3.25), letting m, k → ∞ we have lim J (tm um ) = +∞ (3.26) m→∞ On the other hand, using the condition (f3 ) and (3.8), for all m large enough, we have J (tm um ) = J (tm um ) − = ′ J (tm um )(tm um ) + o(1) + P+ N |∂xi tm um |pi (x) dx − F (x, tm um ) dx pi (x) Ω i =1 Ω − N + P+ i =1 Ω |∂xi tm um |pi (x) dx + + P+ Ω f (x, tm um )tm um dx + o(1) = N 1 G(x, tm um ) dx + o(1) − + |∂xi tm um |pi (x) dx + + p i ( x) P+ P+ Ω i =1 Ω ≤ N 1 − + |∂xi um |pi (x) dx + + G(x, um ) + C∗ dx + o(1) p i ( x) P+ P+ Ω i =1 Ω = N |∂xi um |pi (x) dx − F (x, um ) dx p i ( x) Ω i =1 Ω − N + P+ i=1 = J ( um ) − →c+ C∗ + P+ Ω |∂xi um |pi (x) dx − Ω f (x, um )um dx + C∗ + P+ |Ω | + o(1) ′ C∗ J (um )(um ) + + |Ω | + o(1) P+ + P+ |Ω | as m → ∞ (3.27) From (3.26) and (3.27) we obtain a contradiction This shows that the sequence {um } is bounded in X Now, since the Banach space X is reflexive, there exists u ∈ X such that passing to a subsequence, still denoted by {um }, it converges weakly to u in X and converges strongly to u in the space Lq(x) (Ω ) Using the condition (f0 ) and the Hölder 142 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 inequality, we have f ( x , u )( u − u ) dx |f (x, um )||um − u| dx ≤ m m Ω Ω ≤ C (1 + |um |q(x)−1 )|um − u| dx Ω ≤ C + ∥|um |q(x)−1 ∥ q(x) ∥um − u∥Lq(x) (Ω ) L q(x)−1 (Ω ) → as m → ∞, which yields lim m→∞ Ω f (x, um )(um − u) dx = (3.28) Therefore, we have N lim m→∞ Ω i =1 |∂xi um |pi (x)−2 ∂xi um (∂xi um − ∂xi u) dx = − → 1, p (x) Since {um } converges weakly to u in W0 lim m→∞ (3.29) (Ω ), by (3.29), we find N |∂xi um |pi (x)−2 ∂xi um − |∂xi u|pi (x)−2 ∂xi u (∂xi um − ∂xi u) dx = (3.30) Ω i =1 Next, we apply the following inequality (see [26]) (|ξ |r −2 ξ − |η|r −2 η) · (ξ − η) ≥ 2−r |ξ − η|r , ξ , η ∈ RN , (3.31) − → valid for all r ≥ Relations (3.30) and (3.31) show actually { J satisfies the (Cc ) condition 1, p (x) um converges strongly to u in W0 } (Ω ) Thus, the functional Proof of Theorem 3.2 By Lemmas 3.7 and 3.8, the functional J satisfies all the assumptions of the mountain pass theorem Therefore, the functional J has a critical value c ≥ R > Hence, problem (1.1) has at least one non-trivial weak solution in X Next, because X is a reflexive and separable Banach space, there exist {ej } ⊂ X and {e∗j } ⊂ X ∗ such that X = span {ej : j = 1, 2, , }, X ∗ = span {e∗j : j = 1, 2, , }, and ei , ej = ∗ 1, 0, if i = j, if i ̸= j For convenience, we write Xj = span{ej } and define for each k ∈ N the subspaces Yk = ⊕kj=1 Xj and Zk = ⊕∞ j=k Xj The following result is useful for our arguments + Lemma 3.9 If P+ < q− ≤ q+ < P−∗ then we have → αk := sup ∥u∥Lq (x)(Ω ) : ∥u∥− p (x) = 1, u ∈ Zk → as k → ∞ Proof Obviously, for any k ∈ N, < αk+1 ≤ αk , so αk → α ≥ as k → ∞ Let uk ∈ Zk , k = 1, 2, , satisfy → ∥ uk ∥ − p (x) = and ≤ αk − ∥uk ∥Lq(x) (Ω ) < k (3.32) Then there exists a subsequence of {uk }, still denoted by {uk } such that {uk } converges weakly to u in X and e∗j , u = lim e∗j , uk , k→∞ j = 1, 2, Since Zk is a closed subspace of X , by Mazur’s theorem, we have u ∈ Zk for any k Consequently, we get u ∈ ∩∞ k=1 Zk = {0}, + and so {uk } converges weakly to in X as k → ∞ Since P+ < q− ≤ q+ < P−∗ , the embedding X ↩→ Lq(x) (Ω ) is compact, then {uk } converges strongly to in Lq(x) (Ω ) Hence, by (3.32), we have limk→∞ αk = N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 143 Lemma 3.10 Assume that the conditions (f0 ) and (f2 ) are satisfied Then there exist ρk > rk > such that (i) ak := inf{u∈Zk : ∥u∥− → =rk } J (u) → +∞ as k → ∞; p (x) (ii) bk := max{u∈Yk : ∥u∥− → =ρk } J (u) ≤ p ( x) Proof (i) By (f0 ), there exists C6 > such that |F (x, t )| ≤ C6 (|t | + |t |q(x) ) (3.33) + for all (x, t ) ∈ Ω × R, P+ < q− ≤ q+ < P−∗ From (3.17) and (3.33), for any u ∈ Zk we have J (u) = N |∂xi u|pi (x) dx − F (x, u) dx pi (x) Ω i=1 Ω P ≥ − + P − −1 P+ N P ≥ ≥ − − ∥ u∥ − → p (x) N − + P+ − C6 Ω |u|q(x) dx − C6 |u| dx Ω − − ∥ u∥ − → p (x) → − C7 ∥u∥Lq(x) (Ω ) − C8 ∥u∥− p (x) − − P− ∥ u∥ − → N p (x) → p (x) − + + P−− −1 − C7 − C8 ∥u∥− P P+ N N q(ξ ) − + P− P+ N −1 + P+ , if |u|q(x) ≤ 1, + − P− ∥ u∥ − → + + N p (x) → − C7 αkq ∥u∥q− − C ∥ u∥ − → p (x) − + − p (x) + P− −1 P+ N P ≥ ξ ∈Ω P+ if |u|q(x) > − − ∥ u∥ − → p (x) + − + P− P+ N −1 + → − C7 αkq ∥u∥q− − C8 ∥u∥− → p (x) − C9 , p (x) (3.34) where → αk := sup ∥u∥Lq(x) (Ω ) : ∥u∥− p (x) = 1, u ∈ Zk q+ − P − −1 + → Now, for any u ∈ Zk with ∥u∥− p (x) = rk = C7 q αk N P J (u) ≥ − P− −q+ we have − − ∥ u∥ − → p (x) + + − + P− P+ N −1 → − C8 ∥u∥− − C7 αkq ∥u∥q− → p (x) − C9 p (x) − = + q+ C7 q αk N 1 − P + N P − −1 − P − −1 P− − P− −q+ q+ k − C7 α + q+ C7 q αk − N P− −1 q+ − P− −q −1 + − + − C8 C7 q+ αkq N P− −1 P− −q − C9 + − = = + P+ − N P− −1 − N P− −1 + P+ − − C7 q αk q+ q+ + q+ P − N P− −1 P− − P− −q+ −1 + − + − C8 C7 q+ αkq N P− −1 P− −q − C9 − rk − − C8 rk − C9 (3.35) ∗ By Lemma 3.9, αk → as k → ∞ and P− > q+ ≥ q− > P++ ≥ P−− > we have that rk → +∞ as k → ∞ Therefore, by (3.35) it follows that ak := inf → {u∈Zk : ∥u∥− =r k } p ( x) J (u) → +∞ as k → ∞ 144 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 → (ii) By (3.2), for any ψ ∈ Yk with ∥ψ∥− p (x) = and t > 1, we have N |∂xi t ψ|pi (x) dx − F (x, t ψ) dx pi (x) Ω i =1 Ω N + + |∂xi ψ|pi (x) ≤ t P+ dx − M |t ψ|P+ dx + CM |Ω | p i ( x) Ω i=1 Ω N + + |∂xi ψ|pi (x) = t P+ dx − M |ψ|P+ dx + CM |Ω | pi (x) Ω i =1 Ω J (t ψ) = (3.36) It is clear that we can choose M > large enough such that N + |∂xi ψ|pi (x) dx − M |ψ|P+ dx < pi (x) Ω i=1 Ω For this choice, it follows from (3.36) that lim J (t ψ) = −∞ t →+∞ Hence, there exists t > rk > large enough such that J (t ψ) ≤ and thus, if we set ρk = t we conclude that bk := max → {u∈Yk : ∥u∥− =ρk } p (x) J (u) ≤ The proof of Lemma 3.10 is complete Proof of Theorem 3.3 By Lemma 3.10, the functional J satisfies all the assumptions of the fountain theorem By Lemma 3.8, the functional J satisfies the (Cc ) condition for every c > Moreover, from the condition (f4 ), J is even Hence, we can apply the fountain theorem in order to obtain a sequence of critical points {uk } ⊂ X of J such that J (uk ) → +∞ as k → ∞ The proof of Theorem 3.3 is complete Acknowledgments The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript The paper was done when the first author was working at the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, as a Research Fellow This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) References [1] N.T Chung, Q.A Ngo, Multiple solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions, Proc Roy Soc Edinburgh Sect A 140 (2010) 259–272 [2] X.L Fan, J Shen, D Zhao, Sobolev embedding theorems for spaces W k,p(x) (Ω ), J Math Anal Appl 262 (2001) 749–760 [3] X.L Fan, D Zhao, On the spaces Lp(x) (Ω ) and W m,p(x) (Ω ), J Math Anal Appl 263 (2001) 424–446 [4] Y Fu, X Zhang, Multiple solutions for a class of p(x)-Laplacian equations in involving the critical exponent, Proc R Soc Lond Ser A 466 (2010) 1667–1686 [5] M Mihăilescu, V Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc R Soc Lond Ser A 462 (2006) 2625–2641 [6] O Kováčik, J Rákosník, On spaces Lp(x) and W 1,p(x) , Czechoslovak Math J 41 (1991) 592–618 [7] J Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Mathematics, vol 1034, Springer, Berlin, 1983 [8] M Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002 [9] I Fragalà, F Gazzola, B Kawohl, Existence and nonexistence results for anisotropic quasilinear equations, Ann Inst H Poincaré, Anal Non Linéaire 21 (2004) 715–734 [10] A.D Castro, E Montefusco, Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations, Nonlinear Anal 70 (2009) 4093–4105 [11] M Mihăilescu, G Morosanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Appl Anal 89 (2010) 257–271 [12] M Mihăilescu, G Morosanu, On an eigenvalue problem for an anisotropic elliptic equation involving variable exponents, Glasg Math J 52 (2010) 517–527 [13] M Mihăilescu, P Pucci, V Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J Math Anal Appl 340 (2008) 687–698 [14] M.M Boureanu, Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese J Math 15 (2011) 2291–2310 [15] M.M Boureanu, V Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal 75 (2012) 4471–4482 [16] M.M Boureanu, P Pucci, V Rădulescu, Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent, Complex Var Elliptic Equ 56 (2011) 755–767 [17] D.S Dumitru, Multiplicity of solutions for a nonlinear degenerate problem in anisotropic variable exponent spaces, Bull Malays Math Sci Soc (2) 36 (1) (2013) 117–130 N.T Chung, H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 145 [18] D.S Dumitru, Two nontrivial solutions for a class of anisotropic variable exponent problems, Taiwanese J Math 16 (4) (2012) 1205–1219 [19] C Ji, An eigenvalue of an anisotropic quasilinear elliptic equation with variable exponent and Neumann boundary condition, Nonlinear Anal 71 (2009) 4507–4514 [20] B Ricceri, A further three critical points theorem, Nonlinear Anal 71 (2009) 4151–4157 [21] O.H Miyagaki, M.A.S Souto, Super-linear problems without Ambrosetti and Rabinowitz growth condition, J Differential Equations 245 (2008) 3628–3638 [22] G Li, C Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti–Rabinowitz condition, Nonlinear Anal 72 (2010) 4602–4613 [23] C Ji, On the superlinear problem involving the p(x)-Lapacian, Electron J Qual Theory Differ Equ 2011 (40) (2011) 1–9 [24] D.G Costa, C.A Magalhaes, Existence results for perturbations of the p-Laplacian, Nonlinear Anal 24 (1995) 409–418 [25] M Willem, Minimax Theorems, Birkhäuser, Boston, 1996 [26] J Simon, Régularité de la solution d’une équation non linéaire dans RN , in: Ph Bénilan, J Robert (Eds.), Journées d’Analyse Non Linéaire, in: Lecture Notes in Math., vol 665, Springer-Verlag, Berlin, 1978, pp 205–227 ... for a class of anisotropic variable exponent problems, Taiwanese J Math 16 (4) (2012) 1205–1219 [19] C Ji, An eigenvalue of an anisotropic quasilinear elliptic equation with variable exponent and... H.Q Toan / Nonlinear Analysis: Real World Applications 16 (2014) 132–145 133 partial differential equations with nonstandard growth conditions have received specific attention in recent decades... Science and Technology Development (NAFOSTED) References [1] N.T Chung, Q .A Ngo, Multiple solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions,