# Existence results for classes of infinite semipositone problems

## Abstract

We consider the problem

$\left\{\begin{array}{cc}-{\mathrm{\Delta }}_{p}u=\frac{a{u}^{p-1}-b{u}^{\gamma -1}-c}{{u}^{\alpha }},\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$

where ${\mathrm{\Delta }}_{p}u=div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)$, $p>1$, Ω is a smooth bounded domain in ${\mathbb{R}}^{n}$, $a>0$, $b>0$, $c\ge 0$, $\gamma >p$ and $\alpha \in \left(0,1\right)$. Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case $c=0$ is presented.

## 1 Introduction

Consider the nonsingular boundary value problem:

$\left\{\begin{array}{cc}-\mathrm{\Delta }u=au-b{u}^{2}-ch\left(x\right),\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$
(1)

where Ω is a smooth bounded domain in ${\mathbb{R}}^{n}$, $a>0$, $b>0$, $c\ge 0$, $\mathrm{\Delta }u=div\left(\mathrm{\nabla }u\right)$ is the Laplacian of u and $h:\overline{\mathrm{\Omega }}\to R$ is a ${C}^{1}\left(\overline{\mathrm{\Omega }}\right)$ function satisfying $h\left(x\right)\ge 0$ for $x\in \mathrm{\Omega }$, $h\left(x\right)\not\equiv 0$, ${max}_{x\in \overline{\mathrm{\Omega }}}h\left(x\right)=1$ and $h\left(x\right)=0$ for $x\in \partial \mathrm{\Omega }$. Existence of positive solutions of problem (1) was studied in . In particular, it was proved that given an $a>{\lambda }_{1}$ and $b>0$ there exists a ${c}^{\ast }\left(a,b,\mathrm{\Omega }\right)>0$ such that for $c<{c}^{\ast }$ (1) has positive solutions. Here, ${\lambda }_{1}$ is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when $a\le {\lambda }_{1}$. Later in , these results were extended to the case of the p-Laplacian operator, ${\mathrm{\Delta }}_{p}$, where ${\mathrm{\Delta }}_{p}u=div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)$, $p>1$. Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity $f\left(s,x\right)=as-b{s}^{2}-ch\left(x\right)$ satisfies $f\left(0,x\right)<0$ for some $x\in \mathrm{\Omega }$. See  for some existence results for semipositone problems.

In this paper, we study positive solutions to the singular boundary value problem:

$\left\{\begin{array}{cc}-{\mathrm{\Delta }}_{p}u=\frac{a{u}^{p-1}-b{u}^{\gamma -1}-c}{{u}^{\alpha }},\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$
(2)

where ${\mathrm{\Delta }}_{p}u=div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)$, $p>1$, Ω is a smooth bounded domain in ${\mathbb{R}}^{n}$, $a>0$, $b>0$, $c\ge 0$, $\alpha \in \left(0,1\right)$, $p>1$, and $\gamma >p$. In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity $f\left(s\right)=\frac{a{s}^{p-1}-b{s}^{\gamma -1}-c}{{s}^{\alpha }}$ satisfies ${lim}_{s\to {0}^{+}}f\left(s\right)=-\mathrm{\infty }$. One can refer to , and  for some recent existence results of infinite semipositone problems. We establish the following theorem.

Theorem 1.1 Given $a,b>0$, $\gamma >p$, and $\alpha \in \left(0,1\right)$, there exists a constant ${c}_{1}={c}_{1}\left(a,b,\alpha ,p,\gamma ,\mathrm{\Omega }\right)>0$ such that for $c<{c}_{1}$, (2) has a positive solution.

Remark 1.1 In the nonsingular case ($\alpha =0$), positive solutions exist only when $a>{\lambda }_{1}$ (the principal eigenvalue) (see [1, 2]). But in the singular case, we establish the existence of a positive solution for any given $a>0$.

Next, we study positive radial solutions to the problem:

(3)

where $\mathrm{\Omega }=\left\{x\in {\mathbb{R}}^{n}||x|>{r}_{0}\right\}$ is an exterior domain, $n>p$, $a>0$, $b>0$, $c\ge 0$, $\alpha \in \left(0,1\right)$, $p>1$, $\gamma >p$ and $K:\left[{r}_{0},\mathrm{\infty }\right)\to \left(0,\mathrm{\infty }\right)$ belongs to a class of continuous functions such that ${lim}_{r\to \mathrm{\infty }}K\left(r\right)=0$. By using the transformation: $r=|x|$ and $s={\left(\frac{r}{{r}_{0}}\right)}^{\frac{-n+p}{p-1}}$, we reduce (3) to the following boundary value problem:

$\left\{\begin{array}{c}-{\left({|{u}^{\prime }|}^{p-2}{u}^{\prime }\right)}^{\mathrm{\prime }}=h\left(s\right)\left(\frac{a{u}^{p-1}-b{u}^{\gamma -1}-c}{{u}^{\alpha }}\right),\phantom{\rule{1em}{0ex}}0
(4)

where $h\left(s\right)={\left(\frac{p-1}{n-p}\right)}^{p}{r}_{0}^{p}{s}^{\frac{-p\left(n-1\right)}{n-p}}K\left({r}_{0}{s}^{\frac{-\left(p-1\right)}{n-p}}\right)$. We assume:

(${H}_{1}$) $K\in C\left(\left[{r}_{0},\mathrm{\infty }\right),\left(0,\mathrm{\infty }\right)\right)$ and satisfies $K\left(r\right)<\frac{1}{{r}^{n+\theta }}$ for $r\gg 1$, and for some θ such that $\left(\frac{n-p}{p-1}\right)\alpha <\theta <\frac{n-p}{p-1}$.

With the condition (${H}_{1}$), h satisfies: (5)

We note that if $\theta \ge \frac{n-p}{p-1}$ then $h\left(s\right)$ is nonsingular at 0 and $h\in C\left(\left[0,1\right],\left(0,\mathrm{\infty }\right)\right)$. In this case, problem (4) can be studied using ideas in the proof of Theorem 1.1. Hence, our focus is on the case when $\theta <\frac{n-p}{p-1}$ in which, h may be singular at 0. Note that in this case $\stackrel{ˆ}{h}={inf}_{s\in \left(0,1\right)}h\left(s\right)>0$.

Remark 1.2 Note that $\rho +\alpha <1$ since $\theta >\left(\frac{n-p}{p-1}\right)\alpha$.

We then establish the following theorem.

Theorem 1.2 Given $a,b>0$, $\gamma >p$, $\alpha \in \left(0,1\right)$, and assume (${H}_{1}$) holds. Then there exists a constant ${c}_{2}={c}_{2}\left(a,b,\alpha ,p,\gamma \right)>0$ such that for $c<{c}_{2}$, (3) has a positive radial solution.

Finally, we prove a bifurcation result for the problem

(6)

where Ω is a smooth bounded domain in ${\mathbb{R}}^{n}$, a is a positive parameter, $b,\alpha >0$, $p>1+\alpha$ and $\gamma >p$. We prove the following.

Theorem 1.3 The boundary value problem (6) has a branch of positive solutions bifurcating from the trivial branch of solutions $\left(a,0\right)$ at $\left(0,0\right)$ (as shown in Figure 1).

Our results are obtained via the method of sub-super solutions. By a subsolution of (2), we mean a function $\psi \in {W}^{1,p}\left(\mathrm{\Omega }\right)\cap C\left(\overline{\mathrm{\Omega }}\right)$ that satisfies

and by a supersolution we mean a function $Z\in {W}^{1,p}\left(\mathrm{\Omega }\right)\cap C\left(\overline{\mathrm{\Omega }}\right)$ that satisfies:

where . The following lemma was established in .

Lemma 1.4 (see [13, 18])

Let ψ be a subsolution of (2) and Z be a supersolution of (2) such that $\psi \le Z$ in Ω. Then (2) has a solution u such that $\psi \le u\le Z$ in Ω.

Finding a positive subsolution, ψ, for such infinite semipositone problems is quite challenging since we need to construct ψ in such a way that ${lim}_{x\to \partial \mathrm{\Omega }}-{\mathrm{\Delta }}_{p}\psi =-\mathrm{\infty }$ and $-{\mathrm{\Delta }}_{p}\psi >0$ in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form $\psi =k{\varphi }_{1}^{\beta }$, where k is an appropriate positive constant, $\beta \in \left(1,\frac{p}{p-1}\right)$ and ${\varphi }_{1}$ is the eigenfunction corresponding to the first eigenvalue of $-{\mathrm{\Delta }}_{p}\varphi =\lambda {|\varphi |}^{p-2}\varphi$ in Ω, $\varphi =0$ on Ω.

In Sections 2, 3, and 4, we provide proofs of our results. Section 5 is concerned with providing some exact bifurcation diagrams of positive solutions of (2) when $\mathrm{\Omega }=\left(0,1\right)$ and $p=2$.

## 2 Proof of Theorem 1.1

We first construct a subsolution. Consider the eigenvalue problem $-{\mathrm{\Delta }}_{p}\varphi =\lambda {|\varphi |}^{p-2}\varphi$ in Ω, $\varphi =0$ on Ω. Let ${\varphi }_{1}$ be an eigenfunction corresponding to the first eigenvalue ${\lambda }_{1}$ such that ${\varphi }_{1}>0$ and ${\parallel {\varphi }_{1}\parallel }_{\mathrm{\infty }}=1$. Also, let $\delta ,m,\mu >0$ be such that $|\mathrm{\nabla }{\varphi }_{1}|\ge m$ in ${\mathrm{\Omega }}_{\delta }$ and ${\varphi }_{1}\ge \mu$ in $\mathrm{\Omega }-{\mathrm{\Omega }}_{\delta }$, where ${\mathrm{\Omega }}_{\delta }=\left\{x\in \mathrm{\Omega }|d\left(x,\partial \mathrm{\Omega }\right)\le \delta \right\}$. Let $\beta \in \left(1,\frac{p}{p-1+\alpha }\right)$ be fixed. Here, note that since $\alpha \in \left(0,1\right)$, $\frac{p}{p-1+\alpha }>1$. Choose a $k>0$ such that $2b{k}^{\gamma -p}+{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }\le a$. Define ${c}_{1}=min\left\{{k}^{p-1+\alpha }{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right){m}^{p},\frac{1}{2}{k}^{p-1}{\mu }^{\beta \left(p-1\right)}\left(a-{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }\right)\right\}$. Note that ${c}_{1}>0$ by the choice of k and β. Let $\psi =k{\varphi }_{1}^{\beta }$. Then

$-{\mathrm{\Delta }}_{p}\psi ={k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}-{k}^{p-1}{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right)\frac{{|\mathrm{\nabla }{\varphi }_{1}|}^{p}}{{\varphi }_{1}^{p-\beta \left(p-1\right)}}.$

To prove ψ is a subsolution, we need to establish: (7)

in Ω if $c<{c}_{1}$. To achieve this, we split the term ${k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}$ into three, namely,

$\begin{array}{rcl}{k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}& =& a{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}-\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}\right)\\ -\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}\right).\end{array}$

Now to prove (7) holds in Ω, it is enough to show the following three inequalities: (8) (9) (10)

From the choice of k, $-\left(a-{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }\right)\le -2b{k}^{\gamma -p}$, hence,

$\begin{array}{rl}-\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}\right)& \le -b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\\ \le -b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(\gamma -1-\alpha \right)}.\end{array}$
(11)

Using ${\varphi }_{1}\ge \mu$ in $\mathrm{\Omega }-{\mathrm{\Omega }}_{\delta }$ and $c<\frac{1}{2}{k}^{p-1}{\mu }^{\beta \left(p-1\right)}\left(a-{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }\right)$

$\begin{array}{rcl}-\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}\right)& \le & \frac{-{k}^{p-1}{\varphi }_{1}^{\beta \left(p-1\right)}\left(a-{k}^{\alpha }{\lambda }_{1}{\beta }^{p-1}\right)}{2{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }}\\ \le & \frac{-c}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }}.\end{array}$
(12)

Finally, since $|\mathrm{\nabla }{\varphi }_{1}|\ge m$, in ${\mathrm{\Omega }}_{\delta }$, and $c<{k}^{p-1+\alpha }{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right){m}^{p}$,

$\begin{array}{rcl}-{k}^{p-1}{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right)\frac{{|\mathrm{\nabla }{\varphi }_{1}|}^{p}}{{\varphi }_{1}^{p-\beta \left(p-1\right)}}& \le & \frac{-{k}^{p-1+\alpha }{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right){m}^{p}}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\varphi }_{1}^{p-\beta \left(p-1\right)-\alpha \beta }}\\ \le & \frac{-c}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\varphi }_{1}^{p-\beta \left(p-1+\alpha \right)}}.\end{array}$

Since $p-\beta \left(p-1+\alpha \right)>0$,

$-{k}^{p-1}{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right)\frac{{|\mathrm{\nabla }{\varphi }_{1}|}^{p}}{{\varphi }_{1}^{p-\beta \left(p-1\right)}}\le \frac{-c}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }}.$
(13)

From (11), (12) and (13) we see that equation (7) holds in Ω, if $c<{c}_{1}$. Next, we construct a supersolution. Let e be the solution of $-{\mathrm{\Delta }}_{p}e=1$ in $\mathrm{\Omega },e=0$ on Ω. Choose $\overline{M}>0$ such that $\frac{a{u}^{p-1}-b{u}^{\gamma -1}-c}{{u}^{\alpha }}\le {\overline{M}}^{p-1}$ $\mathrm{\forall }u>0$ and $\overline{M}e\ge \psi$. Define $Z=\overline{M}e$. Then Z is a supersolution of (2). Thus, Theorem 1.1 is proven.

## 3 Proof of Theorem 1.2

We begin the proof by constructing a subsolution. Consider

$\begin{array}{r}-{\left({|{\varphi }^{\prime }|}^{p-2}{\varphi }^{\prime }\right)}^{\mathrm{\prime }}=\lambda {|\varphi |}^{p-2}\varphi ,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),\\ \varphi \left(0\right)=\varphi \left(1\right)=0.\end{array}$
(14)

Let ${\varphi }_{1}$ be an eigenfunction corresponding to the first eigenvalue of (14) such that ${\varphi }_{1}>0$ and ${\parallel {\varphi }_{1}\parallel }_{\mathrm{\infty }}=1$. Then there exist ${d}_{1}>0$ such that $0<{\varphi }_{1}\left(t\right)\le {d}_{1}t\left(1-t\right)$ for $t\in \left(0,1\right)$. Also, let $ϵ<{ϵ}_{1}$ and $m,\mu >0$ be such that $|{\varphi }_{1}^{\prime }|\ge m$ in $\left(0,ϵ\right]\cup \left[1-ϵ,1\right)$ and ${\varphi }_{1}\ge \mu$ in $\left(ϵ,1-ϵ\right)$. Let $\beta \in \left(1,\frac{p-\rho }{p-1+\alpha }\right)$ be fixed and choose $k>0$ such that $2b{k}^{\gamma -p}+\frac{{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }}{\stackrel{ˆ}{h}}\le a$. Define ${c}_{2}=min\left\{\frac{{k}^{p-1+\alpha }{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right){m}^{p}}{{d}_{1}^{\rho }},\frac{1}{2}{k}^{p-1}{\mu }^{\beta \left(p-1\right)}\left(a-\frac{{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }}{\stackrel{ˆ}{h}}\right)\right\}$. Then ${c}_{2}>0$ by the choice of k and β. Let $\psi =k{\varphi }_{1}^{\beta }$. This implies that:

$-{\left({|{\psi }^{\prime }|}^{p-2}{\psi }^{\prime }\right)}^{\mathrm{\prime }}={k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}-{k}^{p-1}{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right)\frac{{|{\varphi }_{1}^{\prime }|}^{p}}{{\varphi }_{1}^{p-\beta \left(p-1\right)}}.$

To prove ψ is a subsolution, we need to establish: (15)

Here, we note that the term ${k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}=\frac{\stackrel{ˆ}{h}{k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}}{\stackrel{ˆ}{h}}$$h\left(t\right)\left(a{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}-\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-\frac{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}}{\stackrel{ˆ}{h}}\right)-\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-\frac{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}}{\stackrel{ˆ}{h}}\right)\right)$, where $\stackrel{ˆ}{h}={inf}_{s\in \left(0,1\right)}h\left(s\right)$. Now to prove (15) holds in $\left(0,1\right)$, it is enough to show the following three inequalities: (16) (17) (18)

From the choice of k, $-\left(a-\frac{{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }}{\stackrel{ˆ}{h}}\right)\le -2b{k}^{\gamma -p}$, hence,

$\begin{array}{rcl}-\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-\frac{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}}{\stackrel{ˆ}{h}}\right)& \le & -b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\\ \le & -b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(\gamma -1-\alpha \right)}.\end{array}$
(19)

Using ${\varphi }_{1}\ge \mu$ in $\left(ϵ,1-ϵ\right)$ and $c<\frac{1}{2}{k}^{p-1}{\mu }^{\beta \left(p-1\right)}\left(a-\frac{{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }}{\stackrel{ˆ}{h}}\right)$

$\begin{array}{rcl}-\frac{1}{2}{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-\frac{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}}{\stackrel{ˆ}{h}}\right)& \le & \frac{-{k}^{p-1}{\varphi }_{1}^{\beta \left(p-1\right)}\left(a-\frac{{k}^{\alpha }{\lambda }_{1}{\beta }^{p-1}}{\stackrel{ˆ}{h}}\right)}{2{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }}\\ \le & \frac{-c}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }}.\end{array}$
(20)

Next, we prove (18) holds in $\left(0,ϵ\right]$. Since $|{\varphi }_{1}^{\prime }|\ge m$, and $p-\beta \left(p-1\right)>\alpha \beta +\rho$

$\begin{array}{rcl}-{k}^{p-1}{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right)\frac{{|{\varphi }_{1}^{\prime }|}^{p}}{{\varphi }_{1}^{p-\beta \left(p-1\right)}}& \le & \frac{-{k}^{p-1+\alpha }{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right){m}^{p}}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\varphi }_{1}^{\rho }}\\ \le & \frac{-{k}^{p-1+\alpha }{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right){m}^{p}}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{d}_{1}^{\rho }{t}^{\rho }}.\end{array}$

Since $h\left(t\right)\le \frac{1}{{t}^{\rho }}$ in $\left(0,ϵ\right]$, and $c<\frac{{k}^{p-1+\alpha }{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right){m}^{p}}{{d}_{1}^{\rho }}$,

$-{k}^{p-1}{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right)\frac{{|{\varphi }_{1}^{\prime }|}^{p}}{{\varphi }_{1}^{p-\beta \left(p-1\right)}}\le \frac{-ch\left(t\right)}{{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }}.$
(21)

Proving (18) holds in $\left[1-ϵ,1\right)$ is straightforward since h is not singular at $t=1$. Thus, from equations (19), (20) and (21), we see that (15) holds in $\left(0,1\right)$. Hence, ψ is a subsolution. Let $Z=\overline{M}e$ where e satisfies $-{\left({|{e}^{\prime }|}^{p-2}{e}^{\prime }\right)}^{\mathrm{\prime }}=h\left(t\right)$ in $\left(0,1\right)$, $e\left(0\right)=e\left(1\right)=0$ and $\overline{M}$ is such that $\frac{a{u}^{p-1}-b{u}^{\gamma -1}-c}{{u}^{\alpha }}\le {\overline{M}}^{p-1}$ $\mathrm{\forall }u>0$ and $\overline{M}e\ge \psi$. Then Z is a supersolution of (4) and there exists a solution u of (4) such that $u\in \left[\psi ,Z\right]$. Thus, Theorem 1.2 is proven.

## 4 Proof of Theorem 1.3

We first prove (6) has a positive solution for every $a>0$. We begin by constructing a subsolution. Let ${\varphi }_{1}$ be as in the proof of Theorem 1.1 (see Section 2). Let $\beta \in \left(1,\frac{p}{p-1}\right)$, and choose a $k>0$ such that $b{k}^{\gamma -p}+{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }\le a$. Let $\psi =k{\varphi }_{1}^{\beta }$. Then

$-{\mathrm{\Delta }}_{p}\psi ={k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}-{k}^{p-1}{\beta }^{p-1}\left(\beta -1\right)\left(p-1\right)\frac{{|\mathrm{\nabla }{\varphi }_{1}|}^{p}}{{\varphi }_{1}^{p-\beta \left(p-1\right)}}.$

To prove ψ is a subsolution, we will establish:

${k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}\le a{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}-b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(\gamma -1-\alpha \right)}$
(22)

in Ω. To achieve this, we rewrite the term ${k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}$ as ${k}^{p-1}{\beta }^{p-1}{\lambda }_{1}{\varphi }_{1}^{\beta \left(p-1\right)}=a{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}-{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}\right)$. Now to prove (22) holds in Ω, it is enough to show $-{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}\right)\le -b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(\gamma -1-\alpha \right)}$. From the choice of k, $-\left(a-{\beta }^{p-1}{\lambda }_{1}{k}^{\alpha }\right)\le -b{k}^{\gamma -p}$, hence,

$\begin{array}{rcl}-{k}^{p-1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\left(a-{k}^{\alpha }{\varphi }_{1}^{\alpha \beta }{\beta }^{p-1}{\lambda }_{1}\right)& \le & -b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(p-1-\alpha \right)}\\ \le & -b{k}^{\gamma -1-\alpha }{\varphi }_{1}^{\beta \left(\gamma -1-\alpha \right)}.\end{array}$

Thus, ψ is a subsolution. It is easy to see that $Z={\left(\frac{a}{b}\right)}^{\frac{1}{\gamma -p}}$ is a supersolution of (6). Since k, can be chosen small enough, $\psi \le Z$. Thus, (6) has a positive solution for every $a>0$. Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, $u\equiv 0$ is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions $\left(a,0\right)$ at $\left(0,0\right)$.

## 5 Numerical results

Consider the boundary value problem

$\left\{\begin{array}{c}-{u}^{″}\left(x\right)=\frac{au-b{u}^{2}-c}{{u}^{\alpha }},\phantom{\rule{1em}{0ex}}x\in \left(0,1\right),\hfill \\ u\left(0\right)=0=u\left(1\right),\hfill \end{array}$
(23)

where $a,b>0$, $c\ge 0$ and $\alpha \in \left(0,1\right)$. Using the quadrature method (see ), the bifurcation diagram of positive solutions of (23) is given by

$G\left(\rho ,c\right)={\int }_{0}^{\rho }\frac{ds}{\sqrt{\left[2\left(F\left(\rho \right)-F\left(s\right)\right)\right]}}=\frac{1}{2},$
(24)

where $F\left(s\right):={\int }_{0}^{s}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ where $f\left(t\right)=\frac{at-b{t}^{2}-c}{{t}^{\alpha }}$ and $\rho =u\left(\frac{1}{2}\right)={\parallel u\parallel }_{\mathrm{\infty }}$. We plot the exact bifurcation diagram of positive solutions of (23) using Mathematica. Figure 2 shows bifurcation diagrams of positive solutions of (23) when $a=8$ ($<{\lambda }_{1}$) and $b=1$ for different values of α.

Bifurcation diagrams of positive solutions of (23) when $a=15$ ($>{\lambda }_{1}$) and $b=1$ for different values of α is shown in Figure 3.

Finally, we provide the exact bifurcation diagram for (6) when $p=2$, and $\mathrm{\Omega }=\left(0,1\right)$. Consider

$\left\{\begin{array}{c}-{u}^{″}\left(x\right)=\frac{au-b{u}^{2}}{{u}^{\alpha }},\phantom{\rule{1em}{0ex}}x\in \left(0,1\right),\hfill \\ u\left(0\right)=0=u\left(1\right),\hfill \end{array}$
(25)

where $a,b,\alpha >0$. The bifurcation diagram of positive solutions of (25) is given by

$\stackrel{˜}{G}\left(\rho ,a\right)={\int }_{0}^{\rho }\frac{ds}{\sqrt{\left[2\left(\stackrel{˜}{F}\left(\rho \right)-\stackrel{˜}{F}\left(s\right)\right)\right]}}=\frac{1}{2},$
(26)

where $\stackrel{˜}{F}\left(s\right):={\int }_{0}^{s}\stackrel{˜}{f}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ where $\stackrel{˜}{f}\left(t\right)=\frac{at-b{t}^{2}}{{t}^{\alpha }}$ and $\rho =u\left(\frac{1}{2}\right)={\parallel u\parallel }_{\mathrm{\infty }}$. The bifurcation diagram of positive solutions of (25) as well as the trivial solution branch are shown in Figure 4 when $\alpha =0.5$ and $b=1$.

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## Acknowledgements

EK Lee was supported by 2-year Research Grant of Pusan National University.

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Correspondence to R Shivaji.

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Goddard, J., Lee, E.K., Sankar, L. et al. Existence results for classes of infinite semipositone problems. Bound Value Probl 2013, 97 (2013). https://doi.org/10.1186/1687-2770-2013-97

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### Keywords

• Bifurcation Diagram
• Existence Result
• Exterior Domain
• Principal Eigenvalue
• Singular Boundary 