- Open Access
Existence of positive solutions for variable exponent elliptic systems
© Ala et al; licensee Springer. 2012
Received: 30 October 2011
Accepted: 3 April 2012
Published: 3 April 2012
We consider the system of differential equations
where Ω ⊂ ℝ N is a bounded domain with C 2 boundary ∂Ω, 1 < p(x) ∈C1 is a function. is called p(x)-Laplacian. We discuss the existence of positive solution via sub-super solutions without assuming sign conditions on f(0), h(0).
MSC: 35J60; 35B30; 35B40.
is zero in general, and only under some special conditions λ p (x)> 0 (see ), 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 p-Laplacian problems. There are more difficulties in discussing the existence of solutions of variable exponent problems.
the authors give the existence of positive solutions for problem (I).
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 λ0 > 0 such that for 0 < λ < λ0, then problem (II) has no nontrivial nonnegative weak solution.
where p(x) ∈ C1 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.
Throughout the article, we will assume that:
(H1) Ω ⊂ ℝ N is an open bounded domain with C2 boundary ∂ Ω.
(H2) p(x) ∈ C1 and 1 < p- ≤ p+.
(H5) g : [0, +∞) → (0, +∞) is a continuous function such that , and
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 [, Theorems 1.10 and 1.14]).
then is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [, Theorem 3.1]).
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 , we have the following lemma.
Lemma 1.1. (Comparison Principle). Let u, v ∈ W1,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 C2 regularly, then there exists a constant δ ∈ (0, 1) such that , and |∇d(x)| ≡ 1.
we have the following result
Lemma 1.2. (see ). If positive parameter η is large enough and w is the unique solution of (1), then we have
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 (H1) - (H5), then (P) has a positive solution when λ is large enough.
for all with q ≥ 0. According to the sub-supersolution method for p(x)-Laplacian equations (see ), then (P) has a positive solution.
Step 1. We construct a subsolution of (P).
From (7) and (8), we can see that (ϕ1, ϕ2) = (ϕ, ϕ) is a subsolution of (P).
Step 2. We construct a supersolution of (P).
where . We shall prove that (z1, z2) is a supersolution for (p).
According to (11) and (12), we can conclude that (z1, z2) is a supersolution for (P).
It only remains to prove that ϕ1 ≤ z1 and ϕ2 ≤ z2.
It is a contradiction to ∇v1(x0) - ∇ϕ(x0) = 0. Thus (13) is valid.
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 ϕ2 ≤ z2. 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 (H1)-(H5), if (u, v) is a solution of (P) which has been given in Theorem 2.1, then
where μ satisfies (10).
Thus (16) and (17) are valid.
where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, δ) is small enough.
Thus (18) and (19) are valid. This completes the proof. □
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).
- Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math 2006, 66(4):1383-1406.MathSciNetView ArticleGoogle Scholar
- Ruzicka M: Electrorheological fluids: Modeling and mathematical theory. In Lecture Notes in Math. Volume 1784. Springer-Verlag, Berlin; 2000.Google Scholar
- Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv 1987, 29: 33-36.View ArticleGoogle Scholar
- Acerbi E, Mingione G: Regularity results for a class of functionals with nonstandard growth. Arch Rat Mech Anal 2001, 156: 121-140.MathSciNetView ArticleGoogle Scholar
- Chen M: On positive weak solutions for a class of quasilinear elliptic systems. Nonlinear Anal 2005, 62: 751-756.MathSciNetView ArticleGoogle Scholar
- Fan XL, Zhao D: On the spaces L p (x)(Ω) and W 1, p(x)(Ω). J Math Anal Appl 2001, 263: 424-446.MathSciNetView ArticleGoogle Scholar
- Fan XL: Global C1, αregularity for variable exponent elliptic equations in divergence form. J Di Equ 2007, 235: 397-417.View ArticleGoogle Scholar
- El Hamidi A: Existence results to elliptic systems with nonstandard growth conditions. J Math Anal Appl 2004, 300: 30-42.MathSciNetView ArticleGoogle Scholar
- Zhang QH: Existence of positive solutions for a class of p ( x )-Laplacian systems. J Math Anal Appl 2007, 333: 591-603.MathSciNetView ArticleGoogle Scholar
- Hai DD, Shivaji R: An existence result on positive solutions of p -Laplacian systems. Nonlinear Anal 2004, 56: 1007-1010.MathSciNetView ArticleGoogle Scholar
- Fan XL, Zhang QH, Zhao D: Eigenvalues of p ( x )-Laplacian Dirichlet problem. J Math Anal Appl 2005, 302: 306-317.MathSciNetView ArticleGoogle Scholar
- Zhang QH: Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems. Nonlinear Anal 2009, 70: 305-316.MathSciNetView ArticleGoogle Scholar
- Samko SG: Densness of in the generalized Sobolev spaces Wm,p(x) ( R N ). Dokl Ross Akad Nauk 1999, 369(4):451-454.MathSciNetGoogle Scholar
- Fan XL, Zhang QH: Existence of solutions for p ( x )-Laplacian Dirichlet problem. Nonlinear Anal 2003, 52: 1843-1852.MathSciNetView ArticleGoogle Scholar
- Zhang QH: A strong maximum principle for differential equations with nonstandard p ( x )-growth con-ditions. J Math Anal Appl 2005, 312(1):24-32.MathSciNetView ArticleGoogle Scholar
- Fan XL: On the sub-supersolution method for p ( x )-Laplacian equations. J Math Anal Appl 2007, 330: 665-682.MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.