A numerical method to obtain positive solution for classes of sublinear semipositone systems

Abstract

Using a numerical method based on sub–super solution, we will obtain positive solution to the coupled-system of boundary value problems of the form

$$\begin{array}{cc}\hfill & -\mathrm{\Delta }u(x)=\lambda f(x\text{,}u\text{,}v)\text{,}x\in \Omega \text{,}\hfill \\ \hfill & -\mathrm{\Delta }v(x)=\lambda g(x\text{,}u\text{,}v)\text{,}x\in \Omega \text{,}\hfill \\ \hfill & u(x)=0=v(x)\text{,}x\in \mathrm{\partial }\Omega \text{,}\hfill \end{array}$$

where f, g are C¹ functions with at least one of f(x₀, 0, 0) or g(x₀, 0, 0) being negative for some x₀ ∈ Ω (semipositone).

Introduction

Consider positive solutions to the coupled-system of boundary value problems

$$\left\{\begin{array}{cc}-\mathrm{\Delta }u(x)=\lambda f(x\text{,}u\text{,}v)\text{,}\hfill & x\in \Omega \text{,}\hfill \\ -\mathrm{\Delta }v(x)=\lambda g(x\text{,}u\text{,}v)\text{,}\hfill & x\in \Omega \text{,}\hfill \\ u(x)=0=v(x)\text{,}\hfill & x\in \mathrm{\partial }\Omega \text{,}\hfill \end{array}\right.$$

where λ > 0, Δ is the Laplacian operator, Ω is a bounded region in Rⁿ, (N ⩾ 1) with smooth boundary ∂Ω, and f, g are C¹ functions with at least one of f(x₀, 0, 0) or g(x₀, 0, 0) being negative for some x₀ ∈ Ω (semipositone).

In this paper, we want to investigate numerically positive solution of (1) by using the method of sub–super solutions. A super-solution to (1) is defined as an ordered pair of smooth functions $(\overline{u}\text{,}\overline{v})$ on Ω satisfying

$$\left\{\begin{array}{cc}-\mathrm{\Delta }\overline{u}⩾\lambda f(x\text{,}\overline{u}\text{,}\overline{v})\text{,}\hfill & x\in \Omega \text{,}\hfill \\ -\mathrm{\Delta }\overline{v}⩾\lambda g(x\text{,}\overline{u}\text{,}\overline{v})\text{,}\hfill & x\in \Omega \text{,}\hfill \\ \overline{u}⩾0\text{;}\overline{v}⩾0\text{,}\hfill & x\in \mathrm{\partial }\Omega .\hfill \end{array}\right.$$

Sub-solutions are similarly defined with inequalities reversed. Let $D=[{\underline{\rho }}_1\text{,}{\overline{\rho }}_1]\times [{\underline{\rho }}_2\text{,}{\overline{\rho }}_2]$ where ρ₁ = inf{u(x) : x ∈ ∂Ω}, ${\overline{\rho }}_1=\sup \{\overline{u}(x):x\in \mathrm{\partial }\Omega \}$, ρ₂ = inf{v(x) : x ∈ ∂Ω}, ${\overline{\rho }}_2=\sup \{\overline{v}(x):x\in \mathrm{\partial }\Omega \}$.

Theorem 1

Let $(\overline{u}\text{,}\overline{v})$, (u, v) be ordered pairs of super- and sub-solutions of (1), respectively. Suppose that

$$\frac{\mathrm{\partial }f}{\mathrm{\partial }v}\text{,}\frac{\mathrm{\partial }g}{\mathrm{\partial }u}⩾0\text{<mi mathvariant="italic" is="true">on</mi>}\overline{\Omega }\times D(\text{<mi mathvariant="italic" is="true">cooperative system</mi>})\text{.}$$

If $\underline{u}⩽\overline{u}$ and $\underline{v}⩽\overline{v}$ on $\overline{\Omega }$, then there is solutions (u, v) of (1) such that $\underline{u}⩽u⩽\overline{u}$ and $\underline{v}⩽v⩽\overline{v}$ on Ω.

In [1] for the first time in the literature, the authors consider a class of semipositone systems. In particular they extend many of the results discussed for the positive solutions of single equation in [2] to semipositone systems. It was shown positive solutions to (1) for either λ near the first eigenvalue λ₁ of the operator −Δ subject to Dirichlet boundary conditions, or for λ large exists. We consider following assumptions:

f, g are C¹ functions satisfying

$$\text{either}f(x_0\text{,}0\text{,}0)<0\text{or}g(x_0\text{,}0\text{,}0)<0\text{for some}{x}_0\in \Omega \text{,}$$

$$\underset{u\to \infty }{\lim }\frac{f(x\text{,}u\text{,}v)}{u}=0\text{uniformly in}x\text{,}v$$

and

$$\underset{u\to \infty }{\lim }\frac{g(x\text{,}u\text{,}v)}{v}=0\text{uniformly in}x\text{,}u\text{.}$$

To introduce additional hypotheses to prove existence results near λ₁, first we recall the anti-maximum principal by Clement Pletier (see [4]), namely, if Z_λ is the unique solution of

$$\left\{\begin{array}{cc}-\mathrm{\Delta }z-\lambda z=-1\text{,}\hfill & x\in \Omega \text{,}\hfill \\ z=0\text{,}\hfill & x\in \mathrm{\partial }\Omega \hfill \end{array}\right.$$

for λ ∈ (λ₁, λ₁ + δ), where λ₁ is the smallest eigenvalue of the problem

$$\left\{\begin{array}{cc}-\mathrm{\Delta }\varphi (x)=\lambda \varphi (x)\text{,}\hfill & x\in \Omega \text{,}\hfill \\ \varphi (x)=0\text{,}\hfill & x\in \mathrm{\partial }\Omega .\hfill \end{array}\right.$$

Let I = [α, γ] where α > λ₁ and γ < λ₁ + δ, and let

$$\sigma ≔\underset{\lambda \in I}{\max }\Vert z_{\lambda }\Vert \text{,}$$

where ∥·∥ denotes the supremum norm. Now assuming that there exists a m₁ > 0 such that

$$f(x\text{,}u\text{,}v)⩾u-m_1\forall x\in \overline{\Omega }\text{,}u\in [0\text{,}{m}_1\gamma \sigma ]\text{,}v⩾0$$

and exists a m₂ > 0 such that

$$g(x\text{,}u\text{,}v)⩾v-m_2\forall x\in \overline{\Omega }\text{,}v\in [0\text{,}{m}_2\gamma \sigma ]\text{,}u⩾0\text{.}$$

Finally to prove existence results for λ large, in addition to (3), (4), (5), we assume

$$\exists f_1(u)⩽f(x\text{,}u\text{,}v)\forall x\in \overline{\Omega }\text{,}u⩾0\text{,}v⩾0\text{such that}{f}_1(r_1)=0\text{,}{f}_1^{\prime }(r_1)<0\text{,}{\int }_0^{r_1}{f}_1(s)\mathrm{d}s>0\text{for some}{r}_1>0$$

and

$$\exists g_2(v)⩽g(x\text{,}u\text{,}v)\forall x\in \overline{\Omega }\text{,}u⩾0\text{,}v⩾0\text{such that}{g}_2(r_2)=0\text{,}{g}_2^{\prime }(r_2)<0\text{,}{\int }_0^{r_2}{g}_2(s)\mathrm{d}s>0\text{for some}{r}_2>0\text{.}$$