Abstract
The existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p(x)Laplace operator with Neumann boundary condition. The technical approach is mainly based on a three critical points theorem due to Ricceri.
MSC: 35D05; 35J60; 58E05.
Keywords:
p(x)Laplacian; Sobolev space; three critical points theorem1 Introduction
In this article, we consider the problem of the type
where Ω ⊂ R^{N}(N ≥ 2) is a bounded domain with boundary of class C^{1}. ν is the outer unit normal to ∂Ω, λ, μ ≥ 0 are real numbers. with , N < q^{ }≤ q^{+}, F : Ω × R × R → R is a function such that F(·, s, t) is measurable in Ω for all (s, t) ∈ R × R and F(x, ·, ·) is C^{1 }in R × R for a.e. x ∈ Ω, F_{s }denotes the partial derivative of F with respect to s. We assume G(x,s,t) and e_{p}(x),e_{q}(x) satisfy the following conditions:
(G) G : Ω × R × R → R is a Carathéodory function, sup_{{s≤θ,t≤ϑ} }G(·,s,t) ∈ L^{1}(Ω) for all θ, ϑ > 0;
(E) e_{p}(x),e_{q}(x) ∈ L^{∞}(Ω) and ess inf_{Ω }e_{p}(x), ess inf_{Ω }e_{q}(x) > 0, we denote ∥e_{p}∥_{1 }= ∫_{Ω }e_{p}(x)dx and ∥e_{q}∥_{1 }= ∫_{Ω}e_{q}(x)dx.
It is well known that the operator Δ_{p(x) }= div(∇u^{p(x)2}∇u) is called p(x)Laplacian and the corresponding problem is called a variable exponent elliptic systems. The study of differential equations and variational problems with nonstandard p(x)growth conditions has been attracting attention of many authors in the last two decades. It arises from nonlinear elasticity theory, electrorheological fluids, etc. see [1,2], many results have been obtained on this kind of problems, for example [39]. For the special case, p(x) ≡ p(a constant), (1.1) becomes the well known pLaplacian problem. There have been many papers on this class of problems, see [1019] and the reference therein.
Recently, many papers have appeared in which the technical approach adopted is based on the three critical points theorem obtained by Ricceri [16]. We cite papers [2023], where the authors established the existence of at least three weak solutions to the problems with Dirichlet or Neumann boundary value conditions. Li and Tang in [24] obtained the existence of at least three weak solutions to problem (1) when p(x) ≡ p with Dirichlet boundary value conditions. El Manouni and Kbiri Alaoui [25] obtained the existence of at least three solutions of system (1) when p(x) ≡ p in Ω by the three critical points theorem obtained by Ricceri [26].
The main purpose of the present paper is to prove the existence of at least three solutions of problem (1). We study problem (1) by using the three critical points theorem by Ricceri [26] too. On the basis of [27], we state an equivalent formulation of the three critical points theorem in [26] as follows.
Theorem 1. Let X be a reflexive real Banach space, Φ : X → R a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous C^{1 }functional, bounded on each bounded subset of X, whose Gâteaux derivative admits a continuous inverse on X*; Ψ : X → R a C^{1 }functional with compact Gâteaux derivative. Assume that
(i) lim_{∥u∥→∞}(Φ(u) + λ Ψ(u)) = ∞ for all λ > 0; and there are r ∈ R and u_{0}, u_{1 }∈ X such that:
(ii) Φ(u_{0}) < r < Φ(u_{1});
Then there exists a nonempty open set Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ and every C^{1 }functional J : X → R with compact Gâteaux derivative, there exists σ > 0 such that for each μ ∈ [0, σ], the equation
has at least three solutions in X whose norms are less than ρ.
The paper is organized as follows. In section 2, we recall some facts that will be needed in the paper. In section 3, we establish our main result.
2 Notations and preliminaries
In order to deal with p(x)Laplacian problem, we need some theories on spaces L^{p(x)}(Ω), W^{1,p(x)}(Ω) and properties of p(x)Laplacian which we will use later (see [1,5,28,29]).
We denote
We can introduce a norm on L^{p(x)}(Ω) by
and (L^{p(x)}(Ω),  · _{p(x)}) becomes a Banach space, and we call it variable exponent Lebesgue space.
The space W^{1,p(x)}(Ω) is defined by
and it can be equipped with the norm
and we call it variable exponent Sobolev space. From [5], we know that spaces L^{p(x)}(Ω) and W^{1,p(x)}(Ω) are separable, reflexive and uniform convex Banach spaces.
When e_{p}(x) satisfy (E), we define
with the norm
then is a Banach space. For any u ∈ W^{1,p(x)}(Ω), define
Then it is easy to see that is a norm on W^{1,p(x)}(Ω) equivalent to ∥u∥_{p(x)}. In the following, we will use to instead of ∥ · ∥_{p(x) }on W^{1,p(x)}(Ω). Similarly, we use to instead of ∥ · ∥_{q(x) }on W^{1,q(x)}(Ω).
Proposition 1. (see [1,5]) The conjugate space of L^{p(x)}(Ω) is , where . For any u ∈ L^{p(x)}(Ω) and , we have
Proposition 2. (see [1,5])If we denote ρ(u) = ∫_{Ω }u^{p(x)}dx, ∀u ∈ L^{p(x)}(Ω), then
(i) u_{p(x) }< 1(= 1; > 1) ⇔ ρ (u) < 1(= 1; > 1);
(iii) u_{p(x) }→ 0(∞) ⇔ ρ (u) → 0(∞).
From Proposition 2, the following inequalities hold:
Proposition 3.If Ω ⊂ R^{N }is a bounded domain, then the imbedding is compact whenever N < p^{}.
Proof. It is well know that is a continuous embedding, and the embedding is compact when N < p^{ }and Ω is bounded. So we obtain the embedding is compact whenever N < p^{}.
From now on, we denote X by W^{1,p(x)}(Ω) × W^{1,q(x)}(Ω) with the norm
Then X is a separable and reflexive Banach spaces. Naturally, we denote X* by the space (W^{1,p(x)})*(Ω) × (W^{1,q(x)})*(Ω), the dual space of X.
From Proposition 3, we know that when p^{},q^{ }> N, the embedding is compact, there exist a positive constant c such that
3 Existence of three solutions
We define Φ, Ψ, J : X → R by
Then for any (ζ,η) ∈ X,
We say that z = (u, v) ∈ X is a weak solution of problem (1) if for any (ζ, η) ∈ X
Thus, we deduce that z ∈ X is a weak solution of (1) if z is a solution of (2). It follows that we can seek for weak solutions of (1) by applying Theorem 1.
We first give the following result.
Lemma 1. If Φ is defined in (6), then (Φ')^{1 }: X* → X exists and it is continuous.
Proof. First, we show that Φ' is uniformly monotone. In fact, for any ζ, η ∈ R^{N}, we have the following inequality (see [30]):
Thus, we deduce that
for any z_{1 }= (u_{1}, v_{1}), z_{2 }= (u_{2}, v_{2}) ∈ X, i.e.,Φ' is uniformly monotone.
From (3), (4), we can see that for any z ∈ X, we have that
That's meaning Φ' is coercive on X.
By a standard argument, we know that Φ' is hemicontinuous. Therefore, the conclusion follows immediately by applying Theorem 26.A [31].
To obtain our main result, we assume the following conditions on F(x,s,t):
(A1) There exist d(x) ∈ L^{1}(Ω) and 0 < ς < p^{}, 0 < τ < q^{ }such that
for a.e.x ∈ Ω and (s,t) ∈ R × R;
(A2) F(x,0,0) = 0 for a.e.x ∈ Ω;
(A3) There exist s_{1},t_{1 }∈ R with s_{1}, t_{1} ≥ 1 such that
where c is given in (5) and
(A3)' F(x,s,t) > 0 for any x ∈ Ω and s or t large enough, and there exist M, N > 0 such that
Then we have the following main theorem.
Theorem 2. Assume (A1),(A2),(A3)(or (A3)'),(G) and (E) hold. Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ, there exists σ > 0 such that for each μ ∈ [0, σ], problem (1) has at least three weak solutions whose norms are less than ρ.
Proof. By the definitions of Φ, Ψ, J, we know that Ψ' is compact, Φ is weakly lower semicontinuous and bounded on each bounded subset of X. From lemma 1 we can see that (Φ')^{1 }is well defined, from condition (G), J is well defined and continuously Gâteaux differentiable on X, with compact derivative. Then we can use Theorem 1 to obtain the result. Now we show that the hypotheses of Theorem 1 are fulfilled.
Thanks to (A1), for each λ ≥ 0, one has that
and so the assumption (i) of Theorem 1 holds.
Now we consider in two cases:
Case (i): (A3) holds, i.e., there exist 1 ≤ s_{1}, t_{1} such that (9) hold.
Now we set z_{0 }= (0,0), z_{1 }= (s_{1}, s_{1}) and denote , then it is easy to see that
Thus, (ii) of Theorem 1 is satisfied.
At last, by (A2) we know Ψ(z_{0}) = 0, then
On the other way, when Φ(z) ≤ r, we have
We deduce that
and
By (5), we obtain
Thus, from (7), we have
From (9)(11) and the definition of r, we can see (iii) of Theorem 1 is hold.
Case (ii): (A3)' holds. Then there exist s_{2},t_{2} > 1 such that F(x,s_{2},t_{2}) > 0 for any x ∈ Ω and . Set a = min{c,M}, b = min{c, N} then we have
We denote z_{2 }= (s_{2},t_{2}) and . Then it is easy to see that
So, (ii) of Theorem 1 is satisfied.
When Φ(z) ≤ r, similar to the above arguments, we obtain that
At last, we see that
From (7) and (12), we have
From (14) and (15), we can see (iii) of Theorem 1 is still hold.
Then all the hypotheses of Theorem 1 are fulfilled. By Theorem 1, we know that there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], problem (1) has at least three weak solutions whose norms are less than ρ.
By Theorem 2, we have the following result.
Corollary 1. Let f, g : Ω × R → R be Carathéodory functions, sup_{ζ≤s }g(·, ζ) ∈ L^{1}(Ω) for all s > 0, and define for any (x,t) ∈ Ω × R, e(x) ∈ L^{∞}(Ω) and ess inf_{Ω}e(x) > 0. Assume the following conditions hold.
(B1) There exist d(x) ∈ L^{1}(Ω) and 0 < ς < p^{ }such that
for a.e.x ∈ Ω and t ∈ R;
(B2) There exists t_{3 }∈ R with t_{3} ≥ 1 such that
where c is given in (5) and
or
(B2)' F(x,t) > 0 for any x ∈ Ω and t large enough, and there exist M > 0 such that
Then there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0, σ], the problem
has at least three weak solutions whose norms are less than ρ.
Remark 1. if p(x) = p in Ω, μ = 0, problem (17) was considered in [21]. If we take f(x,t) = t^{γ(x)2}t  t with satisfies 2 < γ^{ }≤ γ^{+ }< p^{}, μ = 0, Corollary 1 becomes a version of Theorem 2 in [23]. Hence our Corollary 1 unifies and generalizes Theorem 2 in [21] and Theorem 2 in [23] and our Theorem 2 generalizes the main results of [2125] to the system (1).
At last, we give two examples.
Example 1. Let Ω = B(0,1) be the unit ball of R^{N }with N ≥ 2, set p(x) = N + e^{x},q(x) = N + 1 + ln(1 + x^{2}), e_{p}(x) = (1 + x^{2}) = e_{q}(x), G(x,u,v) = x^{2}(u^{2 }+ v^{2}) and
where M is a positive constant, i.e., we consider the following problem
where
We can see that , and it is easy to see that for any t_{1 }> 1, there exists s_{1 }> 1 such that
were are positive constants and c is given by (5). Then when M ≥ s_{1}, F(x,u,v) defined in (18) satisfies (A1)(A3) of Theorem 2, and G(x,u,v),e(x) satisfy
(G) and (E) respectively, by Theorem 2, there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], system (19) has at least three weak solutions whose norms are less than ρ.
Example 2. Assume Ω,p(x),q(x),e_{p}(x),e_{q}(x),G(x,u,v) are the same as in example 1, and suppose N ≥ 8. Let
Obviously, F(x,u,v) satisfies (A1) and (A2). By simple computation, we can see that
and
i.e., (A3)' hold for F(x,u,v) defined in (22).
Thus, there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0, σ], the system
has at least three weak solutions whose norms are less than ρ.
Remark 2. We remark that the methods used in this paper are also applicable for the cases of the other boundary value conditions, for example, Dirichlet boundary value conditions.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
The project supported by the National Natural Science Foundation of China (No. 11171092). Project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 08KJB110005).
References

Rúzicka, M: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Math, SpringerVerlag, Berlin (2000)

Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv. 29, 33–66 (1987). Publisher Full Text

Acerbi, E, Mingione, G: Regularity results for a class of functionals with nonstandard growth. Arch Ration Mech Anal. 156, 121–140 (2001). Publisher Full Text

Fan, XL: Solutions for p(x)Laplacian Dirichlet problems with singular coefficients. J Math Anal Appl. 312, 464–477 (2005). Publisher Full Text

Fan, XL, Zhao, D: On the spaces L^{p(x)}(Ω) and W^{m,p(x)}(Ω). J Math Anal Appl. 263, 424–446 (2001). Publisher Full Text

Fan, XL, Wu, HQ, Wang, FZ: Hartmantype results for p(t)Laplacian systems. Nonlinear Anal. 52, 585–594 (2003). Publisher Full Text

EI Hamidi, A: Existence results to elliptic systems with nonstandard growth conditions. J Math Anal Appl. 300, 30–42 (2004). Publisher Full Text

Zhang, QH: Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems. Nonlinear Anal. 70, 305–316 (2009). Publisher Full Text

Zhang, QH, Qiu, ZM, Dong, R: Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structure. Nonlinear Anal. 72, 354–363 (2010). Publisher Full Text

Alves, CO, de Morais Filho, DC, Souto, MAS: On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal. 42, 771–787 (2000). Publisher Full Text

Chen, CH: On positive weak solutions for a class of quasilinear elliptic systems. Nonlinear Anal. 62, 751–756 (2005). Publisher Full Text

Hai, DD, Shivaji, R: An existence result on positive solutions of pLaplacian systems. Nonlinear Anal. 56, 1007–1010 (2004). Publisher Full Text

Hsu, TS: Multiple positive solutions for a critical quasilinear elliptic system with concaveconvex nonlinearities. Nonlinear Anal. 71, 2688–2698 (2009). Publisher Full Text

Han, P: The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents. Houston J Math. 32, 1241–1257 (2006)

Kristály, A: Existence of two nontrivial solutions for a class of quasilinear elliptic variational systems on striplike domains. Proc Edinb Math Soc II. 48, 465–477 (2005)

Ricceri, B: On a three critical points theorem. Arch Math (Basel). 75, 220–226 (2000). Publisher Full Text

Wu, TF: The Nehari manifold for a semilinear elliptic system involving signchanging weight functions. Nonlinear Anal. 68, 1733–1745 (2008). Publisher Full Text

Yin, HH, Yang, ZD: Existence and nonexistence of entire positive solutions for quasilinear systems with singular and superlinear terms. Diff Equ Appl. 2(2), 241–249 (2010)

Yin, HH, Yang, ZD: Multiplicity results for a class of concaveconvex elliptic systems involving signchanging weight. Ann Pol Math. 102(1), 51–71 (2011). Publisher Full Text

Afrouzi, GA, Heidarkhani, S: Three solutions for a Dirichlet boundary value problem involving the pLaplacian. Nonlinear Anal. 66, 2281–2288 (2007). Publisher Full Text

Bonanno, G, Candito, P: Three solutions to a Neumann problem for elliptic equations involving the pLaplacian. Arch Math (Basel). 80, 424–429 (2003)

Candito, P: Existence of three solutions for a nonautonomous two point boundary value problem. J Math Anal Appl. 252, 532–537 (2000). Publisher Full Text

Mihăilescu, M: Existence and multiplicity of solutions for a Neumann problem involving the p(x)Laplace operator. Nonlinear Anal. 67, 1419–1425 (2007). Publisher Full Text

Li, C, Tang, CL: Three solutions for a class of quasilinear elliptic systems involving the (p,q)Laplacian. Nonlinear Anal. 69, 3322–3329 (2008). Publisher Full Text

El Manouni, S, Kbiri Alaoui, M: A result on elliptic systems with Neumann conditions via Ricceri's three critical points theorem. Nonlinear Anal. 71, 2343–2348 (2009). Publisher Full Text

Ricceri, B: A three critical points theorem revisited. Nonlinear Anal. 70, 3084–3089 (2009). Publisher Full Text

Bonanno, G: A minimax inequality and its applications to ordinary differential equations. J Math Anal Appl. 270, 210–229 (2002). Publisher Full Text

Kováčik, O, Rákosník, J: On the spaces L^{p(x)}(Ω) and W^{k,p(x)}(Ω). Czechoslovak Math J. 41, 592–618 (1991)

Sanko, SG: Denseness of in the generalized Sobolev spaces W^{m,p(x)}(R^{N}). Dokl Ross Akad Nauk. 369(4), 451–454 (1999)

Kichenassamy, S, Veron, L: Singular solutions of the pLaplace equation. Math Ann. 275, 599–615 (1985)

Zeider, E: Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators. Springer, New York (1990)