## Boundary Value Problems

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# Sub-super solutions for (p-q) Laplacian systems

Boundary Value Problems20112011:52

DOI: 10.1186/1687-2770-2011-52

Accepted: 2 December 2011

Published: 2 December 2011

## Abstract

In this work, we consider the system:

$\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p}u=\lambda \left[g\left(x\right)a\left(u\right)+f\left(v\right)\right]\hfill & \text{in}\mathrm{\Omega }\hfill \\ -{\mathrm{\Delta }}_{q}v=\lambda \left[g\left(x\right)b\left(v\right)+h\left(u\right)\right]\hfill & \text{in}\mathrm{\Omega }\hfill \\ u=v=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}$

where Ω is a bounded region in R N with smooth boundary ∂Ω, Δ p is the p-Laplacian operator defined by Δ p u = div (|u|p-2u), p, q > 1 and g (x) is a C1 sign-changing the weight function, that maybe negative near the boundary. f, h, a, b are C1 non-decreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0. Using the method of sub-super solutions, we prove the existence of weak solution.

## 1 Content

In this paper, we study the existence of positive weak solution for the following system:
$\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p}u=\lambda \left[g\left(x\right)a\left(u\right)+f\left(v\right)\right]\hfill & \text{in}\mathrm{\Omega }\hfill \\ -{\mathrm{\Delta }}_{q}v=\lambda \left[g\left(x\right)b\left(v\right)+h\left(u\right)\right]\hfill & \text{in}\mathrm{\Omega }\hfill \\ u=v=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}$
(1)

where Ω is a bounded region in R N with smooth boundary ∂Ω, Δ p is the p-Laplacian operator defined by Δ p u = div(|u|p-2u), p, q > 1 and g(x) is a C1 sign-changing the weight function, that maybe negative near the boundary. f, h, a, b are C1 non-decreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0.

This paper is motivated by results in [15]. We shall show the system (1) with sign-changing weight functions has at least one solution.

## 2 Preliminaries

(Al) $\mathsf{\text{lim}}\frac{f\left(M{\left(h\left(s\right)\right)}^{\frac{1}{q-1}}\right)}{{s}^{p-1}}=0$ as s → ∞, M > 0

(A2) lim f (s) = lim h (s) = ∞ as s → ∞.

(A3) $\mathsf{\text{lim}}\frac{a\left(s\right)}{{s}^{p-1}}=lim\frac{b\left(s\right)}{{s}^{q-1}}=0$ as s → ∞.

Let λ p , λ q be the first eigenvalue of -Δ p , -Δ q with Dirichlet boundary conditions and φ p , φ q be the corresponding positive eigenfunctions with ||φ p || = ||φ q || = 1.

Let m, δ, γ, μ p , μ q > 0 be such that
$\left\{\begin{array}{cc}\hfill |\nabla {\phi }_{p}{|}^{p}-{\lambda }_{p}& {\phi }_{p}\ge m\phantom{\rule{1em}{0ex}}\mathsf{\text{in}}\phantom{\rule{0.3em}{0ex}}{\overline{\mathrm{\Omega }}}_{\delta }\hfill \\ {\phi }_{p}\ge {\mu }_{p}\phantom{\rule{1em}{0ex}}\mathsf{\text{on}}\phantom{\rule{0.3em}{0ex}}\mathrm{\Omega }-{\mathrm{\Omega }}_{\delta }\hfill \end{array}\right\$
(2)
and
$\left\{\begin{array}{rr}\hfill |\nabla {\phi }_{q}{|}^{q}-{\lambda }_{q}{\phi }_{q}\ge m& \hfill \text{in}{\overline{\mathrm{\Omega }}}_{\delta }\\ \hfill {\phi }_{p}\ge {\mu }_{p}& \hfill \text{on}\mathrm{\Omega }-{\mathrm{\Omega }}_{\delta }.\end{array}$
(3)
${\overline{\mathrm{\Omega }}}_{\delta }=\left\{x\in \mathrm{\Omega };\phantom{\rule{2.77695pt}{0ex}}d\left(x,\phantom{\rule{2.77695pt}{0ex}}\partial \mathrm{\Omega }\right)\le \delta \right\}.$
We assume that the weight function g(x) take negative values in Ω δ , but it requires to be strictly positive in Ω-Ω δ . To be precise, we assume that there exist positive constants β and η such that g(x) ≥-β on ${\overline{\mathrm{\Omega }}}_{\delta }$ and g(x) ≥ η on Ω-Ω δ . Let s0 ≥ 0 such that ηa(s) + f (s) > 0, ηb(s) + h(s) > 0 for s > s0 and
${f}_{0}=\mathrm{\text{max}}\left\{0,\phantom{\rule{2.77695pt}{0ex}}-f\left(0\right)\right\},\phantom{\rule{2.77695pt}{0ex}}{h}_{0}=\mathsf{\text{max}}\left\{0,\phantom{\rule{2.77695pt}{0ex}}-h\left(\mathsf{\text{0}}\right)\right\}.$
For γ such that γr-1t > s0; t = min {α p , α q }, r = min{p, q} we define
$\begin{array}{l}A=\mathrm{max}\left[\frac{\gamma {\lambda }_{p}}{\eta a\left({\gamma }^{\frac{1}{p-1}}{\alpha }_{p}\right)+f\left({\gamma }^{\frac{1}{q-1}}{\alpha }_{q}\right)},\frac{\gamma {\lambda }_{q}}{\eta b\left({\gamma }^{\frac{1}{q-1}}{\alpha }_{q}\right)+h\left({\gamma }^{\frac{1}{p-1}}{\alpha }_{p}\right)}\right]\\ B=\mathrm{min}\left[\frac{m\gamma }{\beta a\left({\gamma }^{\frac{{}_{1}}{p-1}}\right)+{f}_{0}},\frac{m\gamma }{\beta b\left({\gamma }^{\frac{{}_{1}}{q-1}}\right)+{h}_{0}}\right]\end{array}$

where ${\alpha }_{p}=\frac{p-1}{p}{{\mu }_{p}}^{\frac{p}{p-1}}$ and ${\alpha }_{q}=\frac{q-1}{q}{\mu }_{q}\frac{q}{q-1}$.

We use the following lemma to prove our main results.

Lemma 1.1[6]. Suppose there exist sub and supersolutions (ψ1, ψ2) and (z1, z2) respectively of (1) such that (ψ1, ψ2) ≤ (z1, z2). then (1) has a solution (u, v) such that (u, v) [(ψ1, ψ2), (z1, z2)].

## 3 Main result

Theorem 3.1 Suppose that (A1)-(A3) hold, then for every λ [A, B], system (1) has at least one positive solution.

Proof of Theorem 3.1 We shall verify that (ψ1, ψ2) is a sub solution of (1.1) where
$\begin{array}{c}{\psi }_{1}\phantom{\rule{2.77695pt}{0ex}}={\gamma }^{\frac{1}{p-1}}\frac{p-1}{p}{{\phi }_{p}}^{\frac{p}{p-1}}\\ {\psi }_{2}\phantom{\rule{2.77695pt}{0ex}}={\gamma }^{\frac{1}{q-1}}\frac{q-1}{q}{{\phi }_{q}}^{\frac{q}{q-1}}.\end{array}$
Let $W\in {{H}_{0}}^{1}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{\Omega }\right)$ with w ≥ 0. Then
${\int }_{\mathrm{\Omega }}|\nabla {\psi }_{1}{|}^{p-2}\nabla {\psi }_{1}\nabla w\mathsf{\text{d}}x=\gamma {\int }_{\mathrm{\Omega }}\left({\lambda }_{p}{{\phi }_{p}}^{p}-|\nabla {\phi }_{p}{|}^{p}\right)w\mathsf{\text{d}}x$
(4)
Now, on ${\overline{\mathrm{\Omega }}}_{\delta }$ by (2),(3) we have
$\gamma \left({\lambda }_{p}{{\phi }_{p}}^{p}-|\nabla {\phi }_{p}{|}^{p}\right)\le -m\gamma$
Since λB then
$\lambda \le \frac{m\gamma }{\beta a\left({\gamma }^{\frac{1}{p-1}}\right)+{f}_{0}}.$
thus
then by (4)
A similar argument shows that
Next, on $\mathrm{\Omega }-{\overline{\mathrm{\Omega }}}_{\delta }$. Since λA, then
$\lambda \ge \frac{\gamma {\lambda }_{p}}{\eta a\left({\gamma }^{\frac{1}{p-1}}{\alpha }_{p}\right)+f\left({\gamma }^{\frac{1}{q-1}}{\alpha }_{q}\right)}$
so we have
$\begin{array}{cc}\hfill \gamma \left({\lambda }_{p}{{\phi }_{p}}^{p}-|\nabla {\phi }_{p}{|}^{p}\right)& \phantom{\rule{2.77695pt}{0ex}}\le \gamma {\lambda }_{p}\hfill \\ \le \lambda \left[\eta a\left({\gamma }^{\frac{1}{p-1}}{\alpha }_{p}\right)+f\left({\gamma }^{\frac{1}{q-1}}{\alpha }_{q}\right)\right]\hfill \\ \le \lambda \left[g\left(x\right)a\left({\psi }_{1}\right)+f\left({\psi }_{2}\right)\right],\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Omega }-\overline{{\mathrm{\Omega }}_{\delta }}\hfill \end{array}$
Then by (4) on we have
$-{\mathrm{\Delta }}_{p}{\psi }_{1}\le \lambda \left[g\left(x\right)a\left({\psi }_{1}\right)+f\left({\psi }_{2}\right)\right]\phantom{\rule{1em}{0ex}}\mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Omega }-\overline{{\mathrm{\Omega }}_{\delta }}$
A similar argument shows that
$-{\mathrm{\Delta }}_{q}{\psi }_{2}\le \lambda \left[g\left(x\right)b\left({\psi }_{2}\right)+h\left({\psi }_{1}\right)\right]$
We suppose that κ p and κ q be solutions of
$\begin{array}{c}\left\{\begin{array}{cc}\hfill -{\mathrm{\Delta }}_{p}& {\kappa }_{p}=1\phantom{\rule{1em}{0ex}}\mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Omega }\hfill \\ {\kappa }_{p}=0\phantom{\rule{1em}{0ex}}\mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \mathrm{\Omega }\hfill \end{array}\right\\\ \left\{\begin{array}{cc}\hfill -{\mathrm{\Delta }}_{q}& {\kappa }_{q}=1\phantom{\rule{1em}{0ex}}\mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Omega }\hfill \\ {\kappa }_{q}=0\phantom{\rule{1em}{0ex}}\mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \mathrm{\Omega }\hfill \end{array}\right\\end{array}$

respectively, and μ' p = ||κ p ||κ, ||κ q ||κ = μ' q .

Let
$\left({z}_{1},\phantom{\rule{2.77695pt}{0ex}}{z}_{2}\right)=\left(\frac{c}{{{\mu }^{\prime }}_{p}}{\lambda }^{\frac{1}{p-1}}{\kappa }_{p},\phantom{\rule{2.77695pt}{0ex}}{\left[2h\left(c{\lambda }^{\frac{1}{q-1}}\right)\right]}^{\frac{1}{q-1}}{\lambda }^{\frac{1}{q-1}}{\kappa }_{q}\right).$

Let $W\in {{H}_{0}}^{1}\left(\mathrm{\Omega }\right)$ with w ≥ 0.

For sufficient C large
$\frac{{{\mu }^{\prime }}_{p}{p-}^{1}\left[||g|{|}_{\infty }a\left(C{\lambda }^{\frac{1}{p-1}}\right)+f{\left(\left(2h\left(C{\lambda }^{\frac{1}{p-1}}\right)\right)}^{\frac{1}{q-1}}{\lambda }^{\frac{1}{q-1}}{{\mu }^{\prime }}_{q}\right)\right]}{{C}^{p-1}}\le 1$
then
$\begin{array}{cc}\hfill \int |\nabla {z}_{1}{|}^{p-2}\nabla {z}_{1}\nabla w\mathsf{\text{d}}x& =\lambda {\left(\frac{C}{{{\mu }^{\prime }}_{p}}\right)}^{p-1}\int \phantom{\rule{2.77695pt}{0ex}}w\mathsf{\text{d}}x\hfill \\ \ge \lambda \int \phantom{\rule{2.77695pt}{0ex}}\left[||g|{|}_{\infty }a\phantom{\rule{0.3em}{0ex}}\left(C{\lambda }^{\frac{1}{p-1}}\right)+f\left({\left(2h\phantom{\rule{0.3em}{0ex}}\left(C{\lambda }^{\frac{1}{p-1}}\right)\right)}^{\frac{1}{q-1}}{\lambda }^{\frac{1}{q-1}}{{\mu }^{\prime }}_{q}\right)\right]\phantom{\rule{0.3em}{0ex}}\mathsf{\text{d}}x\hfill \\ \ge \lambda \int \phantom{\rule{2.77695pt}{0ex}}\left[g\left(x\right)\phantom{\rule{0.3em}{0ex}}a\phantom{\rule{0.3em}{0ex}}\left(C{\lambda }^{\frac{1}{p-1}}\frac{{\kappa }_{p}}{{{\mu }^{\prime }}_{p}}\right)+f\left({\left(2h\phantom{\rule{0.3em}{0ex}}\left(C{\lambda }^{\frac{1}{p-1}}\right)\right)}^{\frac{1}{q-1}}{\lambda }^{\frac{1}{q-1}}{\kappa }_{q}\right)\right]\phantom{\rule{0.3em}{0ex}}\mathsf{\text{d}}x\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\int \phantom{\rule{2.77695pt}{0ex}}\left[g\left(x\right)\phantom{\rule{0.3em}{0ex}}a\phantom{\rule{0.3em}{0ex}}\left({z}_{1}\right)+f\left({z}_{2}\right)\right]\phantom{\rule{0.3em}{0ex}}w\mathsf{\text{d}}x\hfill \end{array}$
Similarly, choosing C large so that
then
$\begin{array}{cc}\hfill \int |\nabla {z}_{2}{|}^{q-2}\nabla {z}_{2}\nabla w\mathsf{\text{d}}x& =2\lambda h\left(C{\lambda }^{\frac{1}{p-1}}\right)\int \phantom{\rule{2.77695pt}{0ex}}w\mathsf{\text{d}}x\hfill \\ \ge \lambda \int \phantom{\rule{2.77695pt}{0ex}}\left[||g|{|}_{\infty }b\left({z}_{2}\right)+h\left({z}_{1}\right)\right]w\mathsf{\text{d}}x.\hfill \end{array}$

Hence by Lemma (1.1), there exist a positive solution (u, v) of (1) such that (ψ1, ψ2) ≤ (u, v) ≤ (z1, z2).

## Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University
(2)
Department of Mathematics, Faculty of Mathematical Sciences University of Mazandaran

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