Abstract
We consider the system of differential equations
where Ω ⊂ ℝ^{N }is a bounded domain with C^{2 }boundary ∂Ω, 1 < p(x) ∈C^{1 } is a function. is called p(x)Laplacian. We discuss the existence of positive solution via subsuper solutions without assuming sign conditions on f(0), h(0).
MSC: 35J60; 35B30; 35B40.
Keywords:
positive solutions; p(x)Laplacian problems; subsupersolution1. Introduction
The study of diferential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc., (see[13]). Many results have been obtained on this kind of problems, for example [1,38]. In [7], Fan gives the regularity of weak solutions for differential equations with variable exponent. On the existence of solutions for elliptic systems with variable exponent, we refer to [8,9]. In this article, we mainly consider the existence of positive weak solutions for the system
where Ω ⊂ ℝ^{N }is a bounded domain with C^{2 }boundary ∂Ω, 1 < p(x) ∈ C^{1 } is a function. The operator is called p(x)Laplacian. Especially, if p(x) ≡ p (a constant), (P) is the wellknown pLaplacian system. There are many articles on the existence of solutions for pLaplacian elliptic systems, for example [5,10]. Owing to the nonhomogeneity of p(x)Laplacian problems are more complicated than those of pLaplacian, many results and methods for pLaplacian are invalid for p(x)Laplacian; for example, if Ω is bounded, then the Rayleigh quotient
is zero in general, and only under some special conditions λ_{p}_{(x) }> 0 (see [11]), and maybe the first eigenvalue and the first eigenfunction of p(x)Laplacian do not exist, but the fact that the first eigenvalue λ_{p }> 0 and the existence of the first eigenfunction are very important in the study of pLaplacian problems. There are more difficulties in discussing the existence of solutions of variable exponent problems.
Hai and Shivaji [10], consider the existence of positive weak solutions for the following pLaplacian problems
the first eigenfunction is used to construct the subsolution of pLaplacian problems successfully. On the condition that λ is large enough and
the authors give the existence of positive solutions for problem (I).
Chen [5], considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:
the first eigenfunction is used to construct the subsolution of problem(II), the main results are as following
(i) If α, β ≥ 0, γ, δ > 0, θ = (p  1  α)(q  1  β)  γδ > 0, then problem (II) has a positive weak solution for each λ > 0;
(ii) If θ = 0 and pγ = q(p  1  α), then there exists λ_{0 }> 0 such that for 0 < λ < λ_{0}, then problem (II) has no nontrivial nonnegative weak solution.
On the p(x)Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)Laplacian do not exist. Even if the first eigenfunction of p(x)Laplacian exist, because of the nonhomogeneity of p(x)Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)Laplacian problems. Zhang [12] investigated the existence of positive solutions of the system
In this article, we consider the existence of positive solutions of the system
where p(x) ∈ C^{1 } is a function, F(x, u, v) = [g(x)a(u) + f(v)], G(x, u, v) = [g(x)b(v) +h(u)], λ is a positive parameter and Ω ⊂ ℝ^{N }is a bounded domain.
To study p(x)Laplacian problems, we need some theory on the spaces L^{p}^{(x)}(Ω), W^{1,p(x)}(Ω) and properties of p(x)Laplacian which we will use later (see [6,13]). If Ω ⊂ ℝ^{N }is an open domain, write
Throughout the article, we will assume that:
(H_{1}) Ω ⊂ ℝ^{N }is an open bounded domain with C^{2 }boundary ∂Ω.
(H_{2}) p(x) ∈ C^{1 } and 1 < p^{ }≤ p^{+}.
(H_{3}) a, b ∈ C^{1}([0, ∞)) are nonnegative, nondecreasing functions such that
(H_{4}) f, h : [0, +∞) → R are C^{1}, monotone functions, lim_{u}_{→+∞ }f(u) = +∞, lim_{u}_{→+∞ }h(u) = +∞, and
(H_{5}) g : [0, +∞) → (0, +∞) is a continuous function such that , and
Denote
We introduce the norm on L^{p}^{(x)}(Ω) by
and (L^{p}^{(x)}(Ω), ._{p}_{(x)}) becomes a Banach space, we call it generalized Lebesgue space. The space (L^{p}^{(x)}(Ω), ._{p}_{(x)}) is a separable, reflexive, and uniform convex Banach space (see [[6], Theorems 1.10 and 1.14]).
The space W^{1,p(x)}(Ω) is defined by W^{1,p(x)}(Ω) = {u ∈ L^{p}^{(x) }:  ∇u ∈ L^{p}^{(x)}}, and it is equipped with the norm
We denote by W_{0}^{1,p(x)}(Ω) is the closure of in W^{1,p(x)}(Ω). W^{1,p(x)}(Ω) and are separable, reflexive, and uniform convex Banach space (see [[6], Theorem 2.1] We define
then is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [[14], Theorem 3.1]).
If is called a weak solution of (P) if it satisfies
where l(x, u) is continuous on , and l(x, .) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [15], we have the following lemma.
Lemma 1.1. (Comparison Principle). Let u, v ∈ W^{1,p(x)}(Ω) satisfying Au  Av ≥ 0 in . If (i.e., u ≥ v on ∂Ω ), then u ≥ v a.e. in Ω.
Here and hereafter, we will use the notation d(x, ∂Ω) to denote the distance of x ∈ Ω to the boundary of Ω.
Denote d(x) = d(x, ∂Ω) and . Since ∂Ω is C^{2 }regularly, then there exists a constant δ ∈ (0, 1) such that , and ∇d(x) ≡ 1.
Denote
we have the following result
Lemma 1.2. (see [16]). If positive parameter η is large enough and w is the unique solution of (1), then we have
(i) For any θ∈ (0, 1) there exists a positive constant C_{1 }such that
(ii) There exists a positive constant C_{2 }such that
2. Existence results
In the following, when there be no misunderstanding, we always use C_{i }to denote positive constants.
Theorem 2.1. On the conditions of (H_{1})  (H_{5}), then (P) has a positive solution when λ is large enough.
Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (Φ_{1}, Φ_{2}) and supersolution (z_{1}, z_{2}) of (P), such that Φ_{1 }≤ z_{1 }and Φ_{2 }≤ z_{2}. That is (Φ_{1}, Φ_{2}) and (z_{1}, z_{2}) satisfies
and
for all with q ≥ 0. According to the subsupersolution method for p(x)Laplacian equations (see [16]), then (P) has a positive solution.
Step 1. We construct a subsolution of (P).
Let σ ∈ (0, δ) is small enough. Denote
It is easy to see that . Denote
By computation
From (H_{3}) and (H_{4}), there exists a positive constant M > 1 such that
If k is sufficiently large, from (2), we have
Let λζ = kα, then
from (3), then we have
Since , then there exists a positive constant C_{3 }such that
If k is sufficiently large, let λζ = kα, we have
then
Since ϕ (x) ≥ 0 and a, f are monotone, when λ is large enough, then we have
Obviously
Combining (4), (5), and (6), we can conclude that
Similarly
From (7) and (8), we can see that (ϕ_{1}, ϕ_{2}) = (ϕ, ϕ) is a subsolution of (P).
Step 2. We construct a supersolution of (P).
We consider
where . We shall prove that (z_{1}, z_{2}) is a supersolution for (p).
For with q ≥ 0, it is easy to see that
Since ,when μ is sufficiently large, combining Lemma 1.2 and (H_{3}), then we have
Hence
Also
By (H_{3}), (H_{4}), when μ is sufficiently large, combining Lemma 1.2 and (H_{3}), we have
Then
According to (11) and (12), we can conclude that (z_{1}, z_{2}) is a supersolution for (P).
It only remains to prove that ϕ_{1 }≤ z_{1 }and ϕ_{2 }≤ z_{2}.
In the definition of v_{1}(x), let . We claim that
From the definition of v_{1}, it is easy to see that
and
It only remains to prove that
Since then there exists a point such that
If v_{1}(x_{0})  ϕ(x_{0}) < 0, it is easy to see that 0 < d(x_{0}) < δ, and then
From the definition of v_{1}, we have
It is a contradiction to ∇v_{1}(x_{0})  ∇ϕ(x_{0}) = 0. Thus (13) is valid.
Obviously, there exists a positive constant C_{3 }such that
Since , according to the proof of Lemma 1.2, then there exists a positive constant C_{4 }such that
According to the comparison principle, we have
From (13) and (14), when and λ ≥ 1 is sufficiently large, we have
According to the comparison principle, when μ is large enough, we have
Combining the definition of v_{1}(x) and (15), it is easy to see that
When μ ≥ 1 and λ is large enough, from Lemma 1.2, we can see that is large enough, then is large enough. Similarly, we have ϕ_{2 }≤ z_{2}. This completes the proof. □
3. Asymptotic behavior of positive solutions
In this section, when parameter λ → +∞, we will discuss the asymptotic behavior of maximum of solutions about parameter λ, and the asymptotic behavior of solutions near boundary about parameter λ.
Theorem 3.1. On the conditions of (H_{1})(H_{5}), if (u, v) is a solution of (P) which has been given in Theorem 2.1, then
(i) There exist positive constants C_{1 }and C_{2 }such that
(ii) for any θ ∈ (0, 1), there exist positive constants C_{3 }and C_{4 }such that
where μ satisfies (10).
Proof. (i) Obviously, when 2δ ≤ d(x), we have
then there exists a positive constant C_{1 }such that
It is easy to see
then
Similarly
Thus (16) and (17) are valid.
(ii) Denote
where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, δ) is small enough.
Obviously, v_{3}(x) ∈ C^{1}(Ω_{ρ}), By computation
where
Let where ρ > 0 is small enough, it is easy to see that
where ρ > 0 is small enough, then we have
Obviously v_{3}(x) ≥ z_{1}(x) on ∂Ω_{ρ}. According to the comparison principle, we have v_{3}(x) ≥ z_{1 }(x) on Ω_{ρ. }Thus
Let when ρ > 0 is small enough, it is easy to see that
Similarly, when ρ > 0 is small enough, we have
Obviously, when d(x) < σ, we have
Thus (18) and (19) are valid. This completes the proof. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to appreciate the referees for their helpful comments and suggestions. The third author partly supported by the National Science Foundation of China (10701066 & 10971087).
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