Positive solutions for a class of superlinear semipositone systems on exterior domains

Abstract

We study the existence of a positive radial solution to the nonlinear eigenvalue problem $-\mathrm{\Delta }u=\lambda K_1(|x|)f(v)$ in ${\mathrm{\Omega }}_e$, $-\mathrm{\Delta }v=\lambda K_2(|x|)g(u)$ in ${\mathrm{\Omega }}_e$, $u(x)=v(x)=0$ if $|x|=r_0$ (>0), $u(x)\to 0$, $v(x)\to 0$ as $|x|\to \mathrm{\infty }$, where $\lambda >0$ is a parameter, $\mathrm{\Delta }u=div(\mathrm{\nabla }u)$ is the Laplace operator, ${\mathrm{\Omega }}_e=\{x\in {\mathbb{R}}^n\mid |x|>r_0,n>2\}$, and $K_i\in C^1([r_0,\mathrm{\infty }),(0,\mathrm{\infty }))$; $i=1,2$ are such that $K_i(|x|)\to 0$ as $|x|\to \mathrm{\infty }$. Here $f,g:[0,\mathrm{\infty })\to \mathbb{R}$ are $C^1$ functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for $\lambda \gg 1$ for the single equations case.

MSC: 34B16, 34B18.

1 Introduction

We consider the nonlinear elliptic boundary value problem

$$\begin{array}{ll}-\mathrm{\Delta }u=\lambda K_1(|x|)f(v)& \text{in }{\mathrm{\Omega }}_e,\\ -\mathrm{\Delta }v=\lambda K_2(|x|)g(u)& \text{in }{\mathrm{\Omega }}_e,\\ u(x)=v(x)=0& \text{if }|x|=r_0(>0),\\ u(x)\to 0,v(x)\to 0& \text{as }|x|\to \mathrm{\infty },\end{array}\}$$

(1.1)

where $\lambda >0$ is a parameter, $\mathrm{\Delta }u=div(\mathrm{\nabla }u)$ is the Laplace operator, and ${\mathrm{\Omega }}_e=\{x\in {\mathbb{R}}^n\mid |x|>r_0,n>2\}$ is an exterior domain. Here the nonlinearities $f,g:[0,\mathrm{\infty })\to \mathbb{R}$ are $C^1$ functions which satisfy:

(H₁):$f(0)<0$ and $g(0)<0$ (semipositone).

(H₂): For $i=1,2$ there exist $b_i>0$ and $q_i>1$ such that ${lim}_{s\to \mathrm{\infty }}\frac{f(s)}{s^{q_1}}=b_1$, and ${lim}_{s\to \mathrm{\infty }}\frac{g(s)}{s^{q_2}}=b_2$.

Further, for $i=1,2$, the weight functions $K_i\in C^1([r_0,\mathrm{\infty }),(0,\mathrm{\infty }))$ are such that $K_i(|x|)\to 0$ as $|x|\to \mathrm{\infty }$. In particular, we are interested in the challenging case, where $K_i$ do not decay too fast. Namely, we assume

(H₃): There exist $\widetilde{d_1}>0$, $\widetilde{d_2}>0$, $\rho \in (0,n-2)$ such that for $i=1,2$

$$\frac{\widetilde{d_1}}{{|x|}^{n+\rho }}\le K_i(|x|)\le \frac{\widetilde{d_2}}{{|x|}^{n+\rho }}\text{for }|x|\gg 1.$$

We then establish the following.

Theorem 1.1

Let (H₁)-(H₃) hold. Then (1.1) has a positive radial solution$(u,v)$ ($u>0$, $v>0$in${\mathrm{\Omega }}_e$) when λ is small, and${\parallel u\parallel }_{\mathrm{\infty }}\to \mathrm{\infty }$, ${\parallel v\parallel }_{\mathrm{\infty }}\to \mathrm{\infty }$as$\lambda \to 0$.

We prove this result via the Leray-Schauder degree theory, by arguments similar to those used in [1] and [2]. The study of such eigenvalue problems with semipositone structure has been documented to be mathematically challenging (see [3], [4]), yet a rich history is developing starting from the 1980s (see [5]–[7]) until recently (see [8]–[12]). In [1], [2] the authors studied such superlinear semipositone problems on bounded domains. In particular, in [12] the authors studied the system

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

where Ω is a bounded domain in ${\mathbb{R}}^n$, $n\ge 1$, and establish an existence result when λ is small. The main motivation of this paper is to extend this study in the case of exterior domains (see Theorem 1.1).

We also discuss a non-existence result for the single equation model:

$$\begin{array}{ll}-\mathrm{\Delta }u=\lambda K_1(|x|)\tilde{f}(u)& \text{in }{\mathrm{\Omega }}_e,\\ u(x)=0& \text{if }|x|=r_0(>0),\\ u(x)\to 0& \text{as }|x|\to \mathrm{\infty },\end{array}\}$$

(1.2)

for large values of λ, when $\tilde{f}$, $K_1$ satisfy the following hypotheses:

(H₄):$\tilde{f}\in C^1([0,\mathrm{\infty }),\mathbb{R})$, ${\tilde{f}}^{\prime }(z)>0$ for all $z>0$, $\tilde{f}(0)<0$, and there exists $m_0>0$ such that ${lim}_{z\to \mathrm{\infty }}\frac{\tilde{f}(z)}{z}\ge m_0$.

(H₅): The weight function $K_1\in C^1([r_0,\mathrm{\infty }),(0,\mathrm{\infty }))$ is such that $s^{\frac{-2(n-1)}{n-2}}{K}_1(r_0{s}^{\frac{1}{2-n}})$ is decreasing for $s\in (0,1]$.

We establish the following.

Theorem 1.2

Let (H₃)-(H₅) hold. Then (1.2) has no nonnegative radial solution for$\lambda \gg 1$.

We establish Theorem 1.2 by recalling various useful properties of solutions established in [13], where the authors prove a uniqueness result for $\lambda \gg 1$ for such an equation in the case when $\tilde{f}$ is sublinear at ∞. However, the properties we recall from [13] are independent of the growth behavior of $\tilde{f}$ at ∞. Non-existence results for such superlinear semipositone problems on bounded domain also have a considerable history starting from the work in the 1980s in [14] leading to the recent work in [15]. Here we discuss such a result for the first time on exterior domains.

Finally, we note that the study of radial solutions $(u(r),v(r))$ (with $r=|x|$) of (1.1) corresponds to studying

$$\begin{array}{ll}-{(r^{n-1}{u}^{\prime }(r))}^{\prime }=\lambda r^{n-1}{K}_1(r)f(v(r))& \text{for }r>r_0,\\ -{(r^{n-1}{v}^{\prime }(r))}^{\prime }=\lambda r^{n-1}{K}_2(r)g(u(r))& \text{for }r>r_0,\\ u(r)=v(r)=0& \text{if }r=r_0(>0),\\ u(r)\to 0,v(r)\to 0& \text{as }r\to \mathrm{\infty },\end{array}\}$$

which can be reduced to the study of solutions $(u(s),v(s))$; $s\in [0,1]$ to the singular system:

$$\begin{array}{ll}-u^{″}(s)=\lambda h_1(s)f(v(s)),& 0<s<1,\\ -v^{″}(s)=\lambda h_2(s)g(u(s)),& 0<s<1,\\ u(0)=u(1)=0,v(0)=v(1)=0,\end{array}\}$$

(1.3)

via the Kelvin transformation $s={(\frac{r}{r_0})}^{2-n}$, where $h_i(s)=\frac{r_0^2}{{(n-2)}^2}{s}^{\frac{-2(n-1)}{(n-2)}}{K}_i(r_0{s}^{\frac{1}{2-n}})$, $i=1,2$ (see [16]).

Remark 1.3

The assumption (H₃) implies that ${lim}_{s\to 0^{+}}{h}_i(s)=\mathrm{\infty }$, for $i=1,2$, $\hat{h}={inf}_{t\in (0,1)}\{h_1(t),h_2(t)\}>0$, and there exist $d>0$, $\eta \in (0,1)$ such that $h_i(s)\le \frac{d}{s^{\eta }}$ for $s\in (0,1]$, and for $i=1,2$. When in addition (H₅) is satisfied, $h_1$ is decreasing in $(0,1]$.

We will prove Theorem 1.1 in Section 2 by studying the singular system (1.3), and Theorem 1.2 in Section 3 by studying the corresponding single equation

$$\begin{array}{ll}-u^{″}(s)=\lambda h_1(s)\tilde{f}(u(s)),& 0<s<1,\\ u(0)=u(1)=0.\end{array}\}$$

(1.4)

2 Existence result

We first establish some useful results for solutions to the system

$$\begin{array}{ll}-u^{″}(s)=b_1{h}_1(s){|v(s)+l|}^{q_1},& 0<s<1,\\ -v^{″}(s)=b_2{h}_2(s){|u(s)+l|}^{q_2},& 0<s<1,\\ u(0)=u(1)=0,v(0)=v(1)=0,\end{array}\}$$

(2.1)

where $l\ge 0$ is a parameter. (Clearly, any solution $(u_l,v_l)$ of (2.1) for $l>0$ must satisfy $u_l(s)>0$, $v_l(s)>0$ for $s\in (0,1)$. This is also true for any nontrivial solution when $l=0$.) We prove the following.

Lemma 2.1

• (i) There exists $l_0>0$ such that 2.1 has no solution if $l\ge l_0$.

• (ii) For each $l\in [0,l_0)$, there exists $M>0$ (independent of l) such that if $(u_l,v_l)$ is a solution of (2.1), then $max\{{\parallel u_l\parallel }_{\mathrm{\infty }},{\parallel v_l\parallel }_{\mathrm{\infty }}\}\le M$.

Proof of (i)

Let ${\lambda }_1:={\pi }^2$, ${\varphi }_1:=sin(\pi s)$. Here ${\lambda }_1$ is the principal eigenvalue and ${\varphi }_1$ a corresponding eigenfunction of $-{\varphi }^{″}(s)=\lambda \varphi (s)$ in $(0,1)$ with $\varphi (0)=0=\varphi (1)$. Let $a>\frac{{\lambda }_1}{\sqrt{b_1{b}_2}\hat{h}}$, $c>0$ be such that ${(s+l)}^{q_i}\ge as-c$ for all $s\ge 0$ and for $i=1,2$. Now let $(u_l,v_l)$ be a solution of (2.1). Multiplying (2.1) by ${\varphi }_1$ and integrating, we obtain

$${\lambda }_1{\int }_0^1{u}_l{\varphi }_{1}ds=b_1{\int }_0^1{h}_1(s){(v_l+l)}^{q_1}{\varphi }_{1}ds\ge b_1{\int }_0^1{h}_1(s)(a{v}_l-c){\varphi }_{1}ds$$

and

$${\lambda }_1{\int }_0^1{v}_l{\varphi }_{1}ds=b_2{\int }_0^1{h}_2(s){(u_l+l)}^{q_2}{\varphi }_{1}ds\ge b_2{\int }_0^1{h}_2(s)(a{u}_l-c){\varphi }_{1}ds.$$

By Remark 1.3, $\hat{h}={inf}_{t\in (0,1)}\{h_1(t),h_2(t)\}>0$, and ${\parallel h_i\parallel }_1:={\int }_0^1{h}_i(s)ds<\mathrm{\infty }$ for $i=1,2$. Then from the above inequalities we obtain

$${\int }_0^1{v}_l{\varphi }_{1}ds\le \frac{1}{a{b}_1\hat{h}}({\lambda }_1{\int }_0^1{u}_l{\varphi }_{1}ds+b_{1}c{\parallel h_1\parallel }_1)$$

and

$${\int }_0^1{u}_l{\varphi }_{1}ds\le \frac{1}{a{b}_2\hat{h}}({\lambda }_1{\int }_0^1{v}_l{\varphi }_{1}ds+b_{2}c{\parallel h_2\parallel }_1).$$

Hence we deduce that

$${\int }_0^1{u}_l{\varphi }_{1}ds\le \frac{m_1}{m}:=m_2,$$

where $m:=(a{b}_2\hat{h}-\frac{{\lambda }_1^2}{a{b}_1\hat{h}})$, and $m_1:=\frac{{\lambda }_{1}c{\parallel h_1\parallel }_1}{a\hat{h}}+b_{2}c{\parallel h_2\parallel }_1$. This implies

$${\int }_0^1{(v_l+l)}^{q_1}{\varphi }_{1}ds\le \frac{{\lambda }_1{m}_2}{b_1\hat{h}}:=m_3.$$

In particular, this implies ${\int }_{\frac{1}{4}}^{\frac{3}{4}}{l}^{q_1}\mathrm{d}s\le \frac{m_3}{{inf}_{[\frac{1}{4},\frac{3}{4}]}{\varphi }_1}$. Since $m_3$ is independent of l, clearly this is a contradiction for $l\gg 1$, and hence there must exists an $l_0>0$ such that for $l\ge l_0$, (2.1) has no solution.

Proof of (ii)

Assume the contrary. Then without loss of generality we can assume there exists $\{l_n\}\subset (0,l_0)$ such that ${\parallel u_{l_n}\parallel }_{\mathrm{\infty }}\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. Clearly $u_{l_n}^{″}(s)<0$, and $v_{l_n}^{″}(s)<0$ for all $s\in (0,1)$. Let $s_{1(l_n)}\in (0,1)$, $s_{2(l_n)}\in (0,1)$ be the points at which $u_{l_n}$ and $v_{l_n}$ attain their maximums. Now since $u_{l_n}^{″}(s)<0$ for all $s\in (0,1)$, we have

$$u_{l_n}(s)\ge \{\begin{array}{ll}\frac{s{u}_{l_n}(s_{1(l_n)})}{s_{1(l_n)}}& \text{for }s\in (0,s_{1(l_n)}),\\ \frac{(1-s)u_{l_n}(s_{1(l_n)})}{1-s_{1(l_n)}}& \text{for }s\in (s_{1(l_n)},1).\end{array}$$

Hence $u_{l_n}(s)\ge min\{\frac{s{\parallel u_{l_n}\parallel }_{\mathrm{\infty }}}{s_{1(l_n)}},\frac{(1-s){\parallel u_{l_n}\parallel }_{\mathrm{\infty }}}{1-s_{1(l_n)}}\}$, and in particular, for $s\in [\frac{1}{4},\frac{3}{4}]$,

$$u_{l_n}(s)\ge min\{\frac{1}{4}{\parallel u_{l_n}\parallel }_{\mathrm{\infty }},\frac{1}{4}{\parallel u_{l_n}\parallel }_{\mathrm{\infty }}\}=\frac{1}{4}{\parallel u_{l_n}\parallel }_{\mathrm{\infty }}.$$

Let $\widetilde{s_{l_n}},\overline{s_{l_n}}\in [\frac{1}{4},\frac{3}{4}]$ be such that ${min}_{[\frac{1}{4},\frac{3}{4}]}{u}_{l_n}(s)=u_{l_n}(\widetilde{s_{l_n}})$, and ${min}_{[\frac{1}{4},\frac{3}{4}]}{v}_{l_n}(s)=v_{l_n}(\overline{s_{l_n}})$. Now for $s\in [\frac{1}{4},\frac{3}{4}]$,

$$v_{l_n}(s)\ge b_2\hat{h}\tilde{m}{\int }_{\frac{1}{4}}^{\frac{3}{4}}{|u_{l_n}(t)+l|}^{q_2}dt,$$

where $\tilde{m}:={min}_{[\frac{1}{4},\frac{3}{4}]\times [\frac{1}{4},\frac{3}{4}]}G(s,t)$ (>0), and G is the Green’s function of $-Z^{″}$ with $Z(0)=0=Z(1)$. In particular, $v_{l_n}(\overline{s_{l_n}})\ge b_2\hat{h}\frac{\tilde{m}}{2}{(u_{l_n}(\widetilde{s_{l_n}}))}^{q_2}$. Similarly $u_{l_n}(\widetilde{s_{l_n}})\ge b_1\hat{h}\frac{m}{2}{(v_{l_n}(\overline{s_{l_n}}))}^{q_1}$. Hence, there exists a constant $A>0$ such that

$$u_{l_n}(\widetilde{s_{l_n}})\ge A{(u_{l_n}(\widetilde{s_{l_n}}))}^{q_1{q}_2}.$$

This is a contradiction since $q_1{q}_2>1$ and $u_{l_n}(\widetilde{s_{l_n}})\ge \frac{1}{4}{\parallel u_{l_n}\parallel }_{\mathrm{\infty }}\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. Thus (ii) holds. □

Proof of Theorem 1.1

We first extend f and g as even functions on ℝ by setting $f(-s)=f(s)$ and $g(-s)=g(s)$. Then we use the rescaling, $\lambda ={\gamma }^{\delta }$, $w_1=\gamma u$, and $w_2={\gamma }^{\theta }v$ with $\gamma >0$, $\theta =\frac{q_2+1}{q_1+1}$, and $\delta =\frac{q_1{q}_2-1}{q_1+1}$. With this rescaling, (1.3) reduces to

$$\begin{array}{ll}-w_1^{″}(s)=F(s,\gamma ,w_2),& 0<s<1,\\ -w_2^{″}(s)=G(s,\gamma ,w_1),& 0<s<1,\\ w_1(0)=w_1(1)=0,w_2(0)=w_2(1)=0,\end{array}\}$$

(2.2)

where

$$\begin{array}{c}F(s,\gamma ,w_2):={\gamma }^{1+\delta }{h}_1(s)(f\left(\frac{w_2}{{\gamma }^{\theta }}\right)-b_1{\left|\frac{w_2}{{\gamma }^{\theta }}\right|}^{q_1})+b_1{|w_2|}^{q_1}{h}_1(s),\text{and}\hfill \\ G(s,\gamma ,w_1):={\gamma }^{\theta +\delta }{h}_2(s)(g\left(\frac{w_1}{\gamma }\right)-b_2{\left|\frac{w_1}{\gamma }\right|}^{q_2})+b_2{|w_1|}^{q_2}{h}_2(s).\hfill \end{array}$$

Note that by our hypothesis (H₂), $F(s,\gamma ,w_2)\to b_1{|w_2|}^{q_1}{h}_1(s)$ and $G(s,\gamma ,w_1)\to b_2{|w_1|}^{q_2}{h}_2(s)$ as $\gamma \to 0$. Hence we can continuously extend $F(s,\gamma ,w_2)$ and $G(s,\gamma ,w_1)$ to $F(s,0,w_2)=b_1{|w_2|}^{q_1}{h}_1(s)$ and $G(s,0,w_1)=b_2{|w_1|}^{q_2}{h}_2(s)$, respectively. Note that proving (1.3) has a positive solution for λ small is equivalent to proving (2.2) has a solution $(w_1,w_2)$ with $w_1>0$, $w_2>0$ in $(0,1)$ for small $\gamma >0$. We will achieve this by establishing that the limiting equation (when $\gamma =0$)

$$\begin{array}{ll}-w_1^{″}(s)=F(s,0,w_2)=b_1{h}_1(s){|w_2|}^{q_1},& 0<s<1,\\ -w_2^{″}(s)=G(s,0,w_1)=b_2{h}_2(s){|w_1|}^{q_2},& 0<s<1,\\ w_1(0)=w_1(1)=0,w_2(0)=w_2(1)=0\end{array}\}$$

(2.3)

(which is the same as (2.1) with $l=0$) has a positive solution $w_1>0$, $w_2>0$ in $(0,1)$ that persists for small $\gamma >0$.

Let $X=C_0[0,1]\times C_0[0,1]$ be the Banach space equipped with ${\parallel \underline{w}\parallel }_X={\parallel (w_1,w_2)\parallel }_X=max\{{\parallel w_1\parallel }_{\mathrm{\infty }},{\parallel w_2\parallel }_{\mathrm{\infty }}\}$, where ${\parallel \cdot \parallel }_{\mathrm{\infty }}$ denotes the usual supremum norm in $C_0([0,1])$. Then for fixed $\gamma \ge 0$, we define the map $S(\gamma ,\cdot ):X\to X$ by

$$S(\gamma ,\underline{w}):=\underline{w}-(K(F(s,\gamma ,w_2)),K(G(s,\gamma ,w_1))),$$

where $K(H(s,\gamma ,Z(s)))={\int }_0^{1}G(t,s)H(t,\gamma ,Z(t))dt$. Note that $F(s,\gamma ,\cdot ),G(s,\gamma ,\cdot ):C_0([0,1])\to L^1(0,1)$ are continuous and $K:L^1(0,1)\to C_0^1([0,1])$ is compact. Hence $S(\gamma ,\cdot )$ is a compact perturbation of the identity. Clearly for $\gamma >0$, if $S(\gamma ,\underline{w})=\underline{0}$, then $\underline{w}=(w_1,w_2)$ is a solution of (2.2), and if $S(0,\underline{w})=\underline{0}$, then $\underline{w}=(w_1,w_2)$ is a solution of (2.3).

We first establish the following.

Lemma 2.2

There exists$R>0$such that$S(0,\underline{w})\ne \underline{0}$for all$\underline{w}=(w_1,w_2)\in X$with${\parallel \underline{w}\parallel }_X=R$and$deg(S(0,\cdot ),B_R(\underline{0}),\underline{0})=0$.

Proof

Define $S^l(0,\underline{w}):X\to X$ by

$$S^l(0,\underline{w}):=\underline{w}-(K(b_1{h}_1(s){|w_2+l|}^{q_1}),K(b_2{h}_2(s){|w_1+l|}^{q_2}))$$

for $l\ge 0$. (Note $S^0(0,\underline{w})=S(0,\underline{w})$.) By Lemma 2.1, if $l\ge l_0$ then $S^l(0,\underline{w})\ne \underline{0}$ and if $S^l(0,\underline{w})=\underline{0}$ for $l\in [0,l_0)$, then ${\parallel w\parallel }_X\le M$. This implies that there exists $R\gg 1$ such that $S^l(0,\underline{w})\ne \underline{0}$ for $\underline{w}\in \partial B_R(\underline{0})$ for any $l\ge 0$. Also, since (2.1) has no solution for $l\ge l_0$, $deg(S^{l_0}(0,\cdot ),B_R(\underline{0}),\underline{0})=0$. Hence, using the homotopy invariance of degree with the parameter $l\in [0,l_0]$ we get

$$deg(S(0,\cdot ),B_R(\underline{0}),\underline{0})=deg(S^{l_0}(0,\cdot ),B_R(\underline{0}),\underline{0})=0.$$

 □

Next we establish the following.

Lemma 2.3

There exists$r\in (0,R)$small enough such that$S(0,\underline{w})\ne \underline{0}$for all$\underline{w}=(w_1,w_2)\in X$with${\parallel \underline{w}\parallel }_X=r$and$deg(S(0,\cdot ),B_r(\underline{0}),\underline{0})=1$.

Proof

Define $T^{\tau }(0,\underline{w}):X\to X$ by

$$T^{\tau }(0,\underline{w}):=\underline{w}-(K(\tau b_1{h}_1(s){|w_2|}^{q_1}),K(\tau b_2{h}_2(s){|w_1|}^{q_2}))$$

for $\tau \in [0,1]$. Clearly $T^1(0,\underline{w})=S(0,\underline{w})$, and $T^0(0,\underline{w})=I$ is the identity operator. Note that $T^{\tau }(0,\underline{w})=0$ if $\underline{w}=(w_1,w_2)$ is a solution of

$$\begin{array}{ll}-w_1^{″}(s)=\tau b_1{h}_1(s){|w_2|}^{q_1},& 0<s<1,\\ -w_2^{″}(s)=\tau b_2{h}_2(s){|w_1|}^{q_2},& 0<s<1,\\ w_1(0)=w_1(1)=0,w_2(0)=w_2(1)=0,\end{array}\}$$

(2.4)

and for $\tau =1$, (2.4) coincides with (2.3). Assume to the contrary that (2.4) has a solution $\underline{w}=(w_1,w_2)$ with ${\parallel \underline{w}\parallel }_X=\tilde{r}>0$. Without loss of generality assume ${\parallel w_1\parallel }_{\mathrm{\infty }}=\tilde{r}$. Now,

$$w_1(s)=\tau {\int }_0^{1}G(s,t)b_1{h}_1(s){|w_2|}^{q_1}ds.$$

Then ${\parallel w_1\parallel }_{\mathrm{\infty }}\le \tilde{C}{\parallel w_2\parallel }_{\mathrm{\infty }}^{q_1}$ for some constant $\tilde{C}>0$ independent of $\tau \in [0,1]$. Similarly ${\parallel w_2\parallel }_{\mathrm{\infty }}\le \hat{C}{\parallel w_1\parallel }_{\mathrm{\infty }}^{q_2}$ for some constant $\hat{C}>0$. This implies that

$$\tilde{r}={\parallel w_1\parallel }_{\mathrm{\infty }}\le C{\parallel w_1\parallel }_{\mathrm{\infty }}^{q_1{q}_2}=C{\tilde{r}}^{q_1{q}_2}$$

for some constant $C>0$. But $q_1{q}_2>1$, and hence this is a contradiction if $\tilde{r}>0$ is small. Thus there exists small $r>0$ such that (2.4) has no solution $\underline{w}$ with ${\parallel \underline{w}\parallel }_X=r$ for all $\tau \in [0,1]$. Now using the homotopy invariance of degree with the parameter $\tau \in [0,1]$, in particular using the values $\tau =1$ and $\tau =0$, we obtain

$$deg(S(0,\cdot ),B_r(\underline{0}),\underline{0})=deg(T^1(0,\cdot ),B_r(\underline{0}),\underline{0})=deg(T^0(0,\cdot ),B_r(\underline{0}),\underline{0})=1.$$

 □

By Lemma 2.2 and Lemma 2.3, with $0<r<R$, we conclude that

$$deg(S(0,\cdot ),B_R(\underline{0})\setminus \overline{B_r}(\underline{0}),\underline{0})=-1,$$

and hence (2.3) has a solution $\underline{w}=(w_1,w_2)$ with $w_1>0$, $w_2>0$ in $(0,1)$, and $r<{\parallel w\parallel }_X<R$. Now we show that the solution obtained above (when $\gamma =0$) persists for small $\gamma >0$ and remains positive componentwise.

Lemma 2.4

Let R, r be as in Lemmas 2.2, 2.3, respectively. Then there exists${\gamma }_0>0$such that:

• (i) $deg(S(\gamma ,\cdot ),B_R(\underline{0})\setminus \overline{B_r}(\underline{0}),\underline{0})=-1$ for all $\gamma \in [0,{\gamma }_0]$.

• (ii) If $S(\gamma ,\underline{w})=\underline{0}$ for $\gamma \in [0,{\gamma }_0]$ with $r<{\parallel \underline{w}\parallel }_X<R$, then $w_1>0$, $w_2>0$ in $(0,1)$.

Proof of (i)

We first show that there exists ${\gamma }_0>0$ such that $S(\gamma ,\underline{w})\ne \underline{0}$ for all $\underline{w}=(w_1,w_2)\in X$ with ${\parallel \underline{w}\parallel }_X\in \{R,r\}$, for all $\gamma \in [0,{\gamma }_0]$. Suppose to the contrary that there exists $\{{\gamma }_n\}$ with ${\gamma }_n\to 0$, $S({\gamma }_n,\underline{w_n})=\underline{0}$ and ${\parallel \underline{w_n}\parallel }_X\in \{r,R\}$. Since $\underline{K}=(K,K):L^1(0,1)\times L^1(0,1)\to C_0^1([0,1])\times C_0^1([0,1])$ is compact, and $\{F(s,{\gamma }_n,w_{2n}),G(s,{\gamma }_n,w_{1n})\}$ are bounded in $L^1(0,1)\times L^1(0,1)$, ${\underline{w}}_n\to \underline{Z}=(Z_1,Z_2)\in C_0^1([0,1])\times C_0^1([0,1])$ (up to a subsequence) with ${\parallel \underline{Z}\parallel }_X=R$ or r and $S(0,\underline{Z})=\underline{0}$. This is a contradiction to Lemma 2.2 or 2.3 and hence there exists a small ${\gamma }_0>0$ satisfying the assertions. Now, by the homotopy invariance of degree with respect to $\gamma \in [0,{\gamma }_0]$,

$$deg(S(\gamma ,\cdot ),B_R(\underline{0})\mathrm{\setminus }\overline{B_r}(\underline{0}),\underline{0})=deg(S(0,\cdot ),B_R(\underline{0})\mathrm{\setminus }\overline{B_r}(\underline{0}),\underline{0})=-1$$

for all $\gamma \in [0,{\gamma }_0]$.

Proof of (ii)

Assume to the contrary that there exists ${\gamma }_n\to 0$ and a corresponding solution $\underline{w_n}=(w_{1n},w_{2n})$ such that $r<{\parallel \underline{w_n}\parallel }_X<R$ and

$${\mathrm{\Omega }}_n:=\{x\in (0,1)\mid w_{1n}(x)\le 0\text{ or }{w}_{2n}(x)\le 0\}\ne \mathrm{\varnothing }.$$

Arguing as before, $\underline{w_n}\to \underline{Z}\in C_0^1([0,1])\times C_0^1([0,1])$ with $S(0,\underline{Z})=\underline{0}$ (up to a subsequence). Note that $\underline{Z}\not\equiv \underline{0}$ since ${\parallel \underline{Z}\parallel }_X\ge r>0$. By the strong maximum principle $Z_1>0$, $Z_2>0$, $Z_1^{\prime }(0)>0$, $Z_2^{\prime }(0)>0$, $Z_1^{\prime }(1)<0$ and $Z_2^{\prime }(1)<0$. Now suppose there exists $\{x_n\}\in (0,1)$ with $\{x_n\}\in {\mathrm{\Omega }}_n$ and $w_{1n}(x_n)\le 0$. Then $\{x_n\}$ must have a subsequence (renamed as $\{x_n\}$ itself) such that $x_n\to \tilde{x}\in [0,1]$. But $Z_1>0$ in $(0,1)$ implies that $\tilde{x}\in \{0,1\}$. Suppose $\tilde{x}=0$. Since $w_{1n}(x_n)\le 0$ and $w_{1n}(0)=0$, there exists $y_n\in (0,x_n)$ such that $w_{1n}^{\prime }(y_n)\le 0$, and hence taking the limit as $n\to \mathrm{\infty }$ we will have $Z_1^{\prime }(0)\le 0$, which is a contradiction since $Z_1^{\prime }(0)>0$. A similar contradiction follows if $\tilde{x}=1$, using the fact that $Z_1^{\prime }(1)<0$. Further, contradictions can be achieved if there exists $\{x_n\}\in \mathrm{\Omega }$ with $\{x_n\}\in {\mathrm{\Omega }}_n$ and $w_{2n}(x_n)\le 0$ using the facts that $Z_2^{\prime }(0)>0$ and $Z_2^{\prime }(1)<0$. This completes the proof of the lemma. □

We now easily conclude the proof of Theorem 1.1. From Lemma 2.4, since $\underline{w}=(w_1,w_2)$ is a positive solution of (2.2) for γ small, $(u,v)=({\gamma }^{-1}{w}_1,{\gamma }^{-\theta }{w}_2)$ with $\theta =\frac{q_2+1}{q_1+1}$ is a positive solution of (1.3) for $\lambda ={\gamma }^{\delta }$ where $\delta =\frac{q_1{q}_2-1}{q_1+1}$. Further, since $w_1>0$ and $w_2>0$ in $(0,1)$ for $\gamma \in [0,{\gamma }_0]$, ${\parallel u\parallel }_{\mathrm{\infty }}\to \mathrm{\infty }$ and ${\parallel v\parallel }_{\mathrm{\infty }}\to \mathrm{\infty }$ as $\lambda (={\gamma }^{\delta })\to 0$. This completes the proof of Theorem 1.1.  □

3 Non-existence result

We first recall from [13] that, when (H₅) is satisfied, one can prove via an energy analysis that a nonnegative solution u of (1.4) must be positive in $(0,1)$ and have a unique interior maximum with maximum value greater than θ, where θ is the unique positive zero of $\tilde{F}(s)={\int }_0^s\tilde{f}(y)dy$. Further, for $\lambda \gg 1$ and $s_1,\widehat{s_1}\in (0,1)$ such that $\widehat{s_1}>s_1$, $u(s_1)=u(\widehat{s_1})=\beta$ (see Figure 1), where $\beta >0$ is the unique zero of $\tilde{f}$, there exists a constant C such that $s_1\le C{\lambda }^{-\frac{1}{2}}$ and $(1-\widehat{s_1})\le C{\lambda }^{-\frac{1}{2}}$. Hence we can assume $(\widehat{s_1}-s_1)>\frac{1}{2}$ for $\lambda \gg 1$. Now we provide the proof of Theorem 1.2.

Figure 1

Graph of u .

Proof of Theorem 1.2

Let $v:=u-\beta$. Then $v>0$ in $(s_1,\widehat{s_1})$ and satisfies

$$\begin{array}{ll}-v^{″}=\lambda h_1(s)\frac{\tilde{f}(u)}{u-\beta }v,& s_1<s<\widehat{s_1},\\ v(s_1)=v(\widehat{s_1})=0.\end{array}\}$$

Note that $\varphi (s)=-(sin(\frac{\pi (s-s_1)}{(\widehat{s_1}-s_1)}))>0$ in $(s_1,\widehat{s_1})$, $\varphi (s_1)=\varphi (\widehat{s_1})=0$, and it satisfies $-{\varphi }^{″}=\frac{{\pi }^2}{{(\widehat{s_1}-s_1)}^2}\varphi$ in $(s_1,\widehat{s_1})$. Hence using the fact that ${\int }_{s_1}^{\widehat{s_1}}(-\varphi v^{″}+v{\varphi }^{″})ds=0$, we obtain

$${\int }_{s_1}^{\widehat{s_1}}(\lambda \frac{\tilde{f}(u)}{u-\beta }{h}_1(s)-\frac{{\pi }^2}{{(\widehat{s_1}-s_1)}^2})v\varphi ds=0.$$

In particular,

$$\lambda \frac{\tilde{f}(u(s_{\lambda }))}{u(s_{\lambda })-\beta }{h}_1(s_{\lambda })=\frac{{\pi }^2}{{(\widehat{s_1}-s_1)}^2},\text{for some }{s}_{\lambda }\in (s_1,\widehat{s_1}).$$

(3.1)

But $\hat{h}={inf}_{(0,1)}{h}_1(s)>0$, and $(\widehat{s_1}-s_1)>\frac{1}{2}$ for $\lambda \gg 1$. Thus clearly (3.1) can hold when $\lambda \to \mathrm{\infty }$, only if $Z=u(s_{\lambda })\to \mathrm{\infty }$ with $\frac{\tilde{f}(u(s_{\lambda }))}{u(s_{\lambda })-\beta }\to 0$. But by (H₄), this is not possible since ${lim}_{Z\to \mathrm{\infty }}\frac{\tilde{f}(Z)}{Z}\ge m_0>0$. Hence the nonnegative solution cannot exist for $\lambda \gg 1$. □