- Research
- Open access
- Published:
Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator
Boundary Value Problems volume 2012, Article number: 30 (2012)
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.
1 Introduction
In this article, we consider the problem of the type
where Ω ⊂ RN(N ≥ 2) is a bounded domain with boundary of class C1. 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 C1 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)| ∈ L1(Ω) for all θ, ϑ > 0;
-
(E)
e p (x),e q (x) ∈ L∞(Ω) and ess infΩ e p (x), ess infΩ e q (x) > 0, we denote ∥e p ∥1 = ∫Ω e p (x)dx and ∥e q ∥1 = ∫Ωe q (x)dx.
It is well known that the operator -Δp(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, electro-rheological fluids, etc. see [1, 2], many results have been obtained on this kind of problems, for example [3–9]. For the special case, p(x) ≡ p(a constant), (1.1) becomes the well known p-Laplacian problem. There have been many papers on this class of problems, see [10–19] 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 [20–23], 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 C1 functional, bounded on each bounded subset of X, whose Gâteaux derivative admits a continuous inverse on X*; Ψ : X → R a C1 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 u0, u1 ∈ X such that:
(ii) Φ(u0) < r < Φ(u1);
(iii) .
Then there exists a non-empty open set Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ and every C1 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 Lp(x)(Ω), W1,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 Lp(x)(Ω) by
and (Lp(x)(Ω), | · |p(x)) becomes a Banach space, and we call it variable exponent Lebesgue space.
The space W1,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 Lp(x)(Ω) and W1,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 ∈ W1,p(x)(Ω), define
Then it is easy to see that is a norm on W1,p(x)(Ω) equivalent to ∥u∥p(x). In the following, we will use to instead of ∥ · ∥p(x)on W1,p(x)(Ω). Similarly, we use to instead of ∥ · ∥q(x)on W1,q(x)(Ω).
Proposition 1. (see [1, 5]) The conjugate space of Lp(x)(Ω) is , where . For any u ∈ Lp(x)(Ω) and , we have
Proposition 2. (see [1, 5])If we denote ρ(u) = ∫Ω |u|p(x)dx, ∀u ∈ Lp(x)(Ω), then
(i) |u|p(x)< 1(= 1; > 1) ⇔ ρ (u) < 1(= 1; > 1);
(ii)
(iii) |u|p(x)→ 0(∞) ⇔ ρ (u) → 0(∞).
From Proposition 2, the following inequalities hold:
Proposition 3.If Ω ⊂ RNis 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 W1,p(x)(Ω) × W1,q(x)(Ω) with the norm
Then X is a separable and reflexive Banach spaces. Naturally, we denote X* by the space (W1,p(x))*(Ω) × (W1,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 ∈ RN, we have the following inequality (see [30]):
Thus, we deduce that
for any z1 = (u1, v1), z2 = (u2, v2) ∈ 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) ∈ L1(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 s1,t1 ∈ R with |s1|, |t1| ≥ 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 semi-continuous 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 ≤ |s1|, |t1| such that (9) hold.
Now we set z0 = (0,0), z1 = (s1, s1) and denote , then it is easy to see that
Thus, (ii) of Theorem 1 is satisfied.
At last, by (A2) we know Ψ(z0) = 0, then
On the other way, when Φ(z) ≤ r, we have
We deduce that
and
For , then we have
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 |s2|,|t2| > 1 such that F(x,s2,t2) > 0 for any x ∈ Ω and . Set a = min{c,M}, b = min{c, N} then we have
We denote z2 = (s2,t2) 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)| ∈ L1(Ω) for all s > 0, and define for any (x,t) ∈ Ω × R, e(x) ∈ L∞(Ω) and ess infΩe(x) > 0. Assume the following conditions hold.
(B1) There exist d(x) ∈ L1(Ω) and 0 < ς < p- such that
for a.e.x ∈ Ω and t ∈ R;
(B2) There exists t3 ∈ R with |t3| ≥ 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)-2t - 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 [21–25] to the system (1).
At last, we give two examples.
Example 1. Let Ω = B(0,1) be the unit ball of RNwith N ≥ 2, set p(x) = N + e|x|,q(x) = N + 1 + ln(1 + x2), e p (x) = (1 + x2) = e q (x), G(x,u,v) = x2(u2 + v2) 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 t1 > 1, there exists s1 > 1 such that
were are positive constants and c is given by (5). Then when M ≥ s1, 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.
References
Rúzicka M: Electro-rheological Fluids: Modeling and Mathematical Theory. In Lecture Notes in Math. Volume 1784. Springer-Verlag, Berlin; 2000.
Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv 1987, 29: 33-66. 10.1070/IM1987v029n01ABEH000958
Acerbi E, Mingione G: Regularity results for a class of functionals with nonstandard growth. Arch Ration Mech Anal 2001, 156: 121-140. 10.1007/s002050100117
Fan XL: Solutions for p ( x )-Laplacian Dirichlet problems with singular coefficients. J Math Anal Appl 2005, 312: 464-477. 10.1016/j.jmaa.2005.03.057
Fan XL, Zhao D: On the spaces Lp(x)(Ω) and Wm, p(x)(Ω). J Math Anal Appl 2001, 263: 424-446. 10.1006/jmaa.2000.7617
Fan XL, Wu HQ, Wang FZ: Hartman-type results for p ( t )-Laplacian systems. Nonlinear Anal 2003, 52: 585-594. 10.1016/S0362-546X(02)00124-4
EI Hamidi A: Existence results to elliptic systems with nonstandard growth con-ditions. J Math Anal Appl 2004, 300: 30-42. 10.1016/j.jmaa.2004.05.041
Zhang QH: Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems. Nonlinear Anal 2009, 70: 305-316. 10.1016/j.na.2007.12.001
Zhang QH, Qiu ZM, Dong R: Existence and asymptotic behavior of positive solutions for a variable exponent elliptic system without variational structure. Nonlinear Anal 2010, 72: 354-363. 10.1016/j.na.2009.06.069
Alves CO, de Morais Filho DC, Souto MAS: On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal 2000, 42: 771-787. 10.1016/S0362-546X(99)00121-2
Chen CH: On positive weak solutions for a class of quasilinear elliptic systems. Nonlinear Anal 2005, 62: 751-756. 10.1016/j.na.2005.04.007
Hai DD, Shivaji R: An existence result on positive solutions of p -Laplacian systems. Nonlinear Anal 2004, 56: 1007-1010. 10.1016/j.na.2003.10.024
Hsu TS: Multiple positive solutions for a critical quasilinear elliptic system with concave-convex non-linearities. Nonlinear Anal 2009, 71: 2688-2698. 10.1016/j.na.2009.01.110
Han P: The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents. Houston J Math 2006, 32: 1241-1257.
Kristály A: Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains. Proc Edinb Math Soc II 2005, 48: 465-477.
Ricceri B: On a three critical points theorem. Arch Math (Basel) 2000, 75: 220-226. 10.1007/s000130050496
Wu TF: The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions. Nonlinear Anal 2008, 68: 1733-1745. 10.1016/j.na.2007.01.004
Yin HH, Yang ZD: Existence and nonexistence of entire positive solutions for quasilinear systems with singular and super-linear terms. Diff Equ Appl 2010, 2(2):241-249.
Yin HH, Yang ZD: Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight. Ann Pol Math 2011, 102(1):51-71. 10.4064/ap102-1-5
Afrouzi GA, Heidarkhani S: Three solutions for a Dirichlet boundary value problem involving the p -Laplacian. Nonlinear Anal 2007, 66: 2281-2288. 10.1016/j.na.2006.03.019
Bonanno G, Candito P: Three solutions to a Neumann problem for elliptic equations involving the p -Laplacian. Arch Math (Basel) 2003, 80: 424-429.
Candito P: Existence of three solutions for a nonautonomous two point boundary value problem. J Math Anal Appl 2000, 252: 532-537. 10.1006/jmaa.2000.6909
Mihăilescu M: Existence and multiplicity of solutions for a Neumann problem involving the p ( x )-Laplace operator. Nonlinear Anal 2007, 67: 1419-1425. 10.1016/j.na.2006.07.027
Li C, Tang CL: Three solutions for a class of quasilinear elliptic systems involving the ( p , q )-Laplacian. Nonlinear Anal 2008, 69: 3322-3329. 10.1016/j.na.2007.09.021
El Manouni S, Kbiri Alaoui M: A result on elliptic systems with Neumann conditions via Ricceri's three critical points theorem. Nonlinear Anal 2009, 71: 2343-2348. 10.1016/j.na.2009.01.068
Ricceri B: A three critical points theorem revisited. Nonlinear Anal 2009, 70: 3084-3089. 10.1016/j.na.2008.04.010
Bonanno G: A minimax inequality and its applications to ordinary differential equations. J Math Anal Appl 2002, 270: 210-229. 10.1016/S0022-247X(02)00068-9
Kováčik O, Rákosník J: On the spaces Lp(x)(Ω) and Wk, p(x)(Ω). Czechoslovak Math J 1991, 41: 592-618.
Sanko SG: Denseness of in the generalized Sobolev spaces Wm, p(x)( RN). Dokl Ross Akad Nauk 1999, 369(4):451-454.
Kichenassamy S, Veron L: Singular solutions of the p -Laplace equation. Math Ann 1985, 275: 599-615.
Zeider E: Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators. Springer, New York; 1990.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Yin, H., Yang, Z. Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator. Bound Value Probl 2012, 30 (2012). https://doi.org/10.1186/1687-2770-2012-30
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2012-30