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).
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