Existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria

TL;DR

This paper develops new eigenvalue-based criteria for the existence of positive radial solutions to semilinear elliptic systems with gradient dependence, expanding the theoretical understanding of such systems.

Contribution

It introduces novel eigenvalue criteria linked to nonlinearities and linear operators, providing a new approach to establish solution existence for gradient-dependent elliptic systems.

Findings

Established new eigenvalue criteria for solution existence.

Linked nonlinear bounds to principal eigenvalues of integral operators.

Provided conditions ensuring solutions in specific function cones.

Abstract

In this paper new criteria are established for the existence of positive radial solutions of a semilinear elliptic system depending on the gradient. These criteria are determined by some relationships between the upper and lower bounds on suitable stripes of $\mathbb R^n$ of the nonlinearities of the system and the principal characteristic values of some associated linear Hammerstein integral operators. Moreover, using smoothing tools, the totality of the involved cone is established.

Equations

$$\{\begin{array}{ll}& -\Delta u=f_1(|x|,u,v,|\nabla u|,|\nabla v|)\text{ in }\Omega ,\\ & -\Delta v=f_2(|x|,u,v,|\nabla u|,|\nabla v|)\text{ in }\Omega ,\\ & u=0\text{ on }\partial \Omega ,\\ & v=0\text{ on }|x|=R_0\text{ and }\frac{\partial v}{\partial r}=0\text{ on }|x|=R_1,\end{array}$$

$$C_{\omega }^1[0,1]=\{w\in C[0,1]:w\text{ is continuous differentiable on }]0,1[\text{ with }\underset{t\in ]0,1[}{\sup }\omega (t)|w^{\prime }(t)|<+\infty \}.$$

$$||w|{|}_{\infty }:=\underset{t\in [0,1]}{\max }|w(t)|,\Vert w^{\prime }{\Vert }_{\omega }=\underset{t\in ]0,1[}{\sup }\omega (t)|w^{\prime }(t)|,$$

$$||w||=\max \left\{||w|{|}_{\infty },\Vert w^{\prime }{\Vert }_{\omega }\right\},$$

$${\mathcal{K}}_{\omega }:=\left\{w\in C_{\omega }^1[0,1]:w\ge 0,\underset{t\in [a,b]}{\min }w(t)\ge c||w|{|}_{\infty },||w|{|}_{\infty }\ge \Vert w^{\prime }{\Vert }_{\omega }\right\}.$$

$$C_{\omega }^1[0,1]=\overline{{\mathcal{K}}_{\omega }-{\mathcal{K}}_{\omega }}.$$

$${\mathcal{P}}_{\omega }:=\{w\in C_{\omega }^1[0,1]:w(t)\ge 0\text{ for }t\in [0,1]\}$$

$$w(t)=\beta w(t)+\gamma \Vert w{\Vert }_{\infty }-((\beta -1)w(t)+\gamma \Vert w{\Vert }_{\infty }),$$

$$\beta =\{\begin{array}{l}1,\text{ if }\Vert w^{\prime }{\Vert }_{\omega }\le \Vert w{\Vert }_{\infty }\\ \frac{\Vert w^{\prime }{\Vert }_{\omega }}{\Vert w{\Vert }_{\infty }},\text{ if }\Vert w^{\prime }{\Vert }_{\omega }>\Vert w{\Vert }_{\infty }\end{array}$$

$$\gamma =\max \left\{\frac{\Vert w^{\prime }{\Vert }_{\omega }}{\Vert w{\Vert }_{\infty }}-1,\frac{c}{1-c}\right\}\beta .$$

$$\phi (t)\ge \gamma \Vert w{\Vert }_{\infty }=\frac{\gamma }{\beta +\gamma }\Vert \phi {\Vert }_{\infty }\ge \frac{\frac{c\beta }{1-c}}{\beta +\frac{c\beta }{1-c}}\Vert \phi {\Vert }_{\infty }=c\Vert \phi {\Vert }_{\infty }$$

$$\psi (t)\ge \gamma \Vert w{\Vert }_{\infty }=\frac{\gamma }{\beta -1+\gamma }\Vert \psi {\Vert }_{\infty }\ge \frac{\frac{c\beta }{1-c}}{\beta -1+\frac{c\beta }{1-c}}\Vert \psi {\Vert }_{\infty }=\frac{c\beta }{\beta -(1-c)}\Vert \psi {\Vert }_{\infty }\ge c\Vert \psi {\Vert }_{\infty }.$$

$$\Vert {\phi }^{\prime }{\Vert }_{\omega }=\Vert w^{\prime }{\Vert }_{\omega }\le \Vert w{\Vert }_{\infty }=\frac{1}{1+\frac{c}{1-c}}\Vert \phi {\Vert }_{\infty }=(1-c)\Vert \phi {\Vert }_{\infty }<\Vert \phi {\Vert }_{\infty }$$

$$\Vert {\psi }^{\prime }{\Vert }_{\omega }=0<\Vert \psi {\Vert }_{\infty }.$$

$$\Vert {\phi }^{\prime }{\Vert }_{\omega }=\beta \Vert w^{\prime }{\Vert }_{\omega }=\frac{\Vert w^{\prime }{\Vert }_{\omega }^2}{\Vert w{\Vert }_{\infty }}={\beta }^2\Vert w{\Vert }_{\infty }\le (\beta +\gamma )\Vert w{\Vert }_{\infty }=\Vert \phi {\Vert }_{\infty }$$

$$\Vert {\psi }^{\prime }{\Vert }_{\omega }=(\beta -1)\Vert w^{\prime }{\Vert }_{\omega }={\beta }^2\Vert w{\Vert }_{\infty }-\Vert w^{\prime }{\Vert }_{\omega }\le (\beta +\gamma )\Vert w{\Vert }_{\infty }-\Vert w{\Vert }_{\infty }=(\beta +\gamma -1)\Vert w{\Vert }_{\infty }=\Vert \psi {\Vert }_{\infty }.$$

$${\mathcal{K}}_{\omega }:=\left\{w\in C_{\omega }^1[0,1]:w\ge 0,\underset{t\in [a,b]}{\min }w(t)\ge c||w|{|}_{\infty },||w|{|}_{\infty }\ge \Vert w^{\prime }{\Vert }_{\omega }\right\}$$

$${\rho }_n(x)=\frac{n}{{\int }_{\mathbb{R}}\rho (y)dy}\rho (nx);$$

$$w_n(t)=:({\rho }_n\ast w)(t)={\int }_0^{t}w(t-y){\rho }_n(y)dy={\int }_0^t{\rho }_n(t-y)w(y)dy=(w\ast {\rho }_n)(t).$$

$$w_n^{\prime }(t)={\int }_0^t{\rho }_n^{\prime }(t-y)w(y)dy+{\rho }_n(0)w(t)=({\rho }_n^{\prime }\ast w)(t)+{\rho }_n(0)w(t),$$

$$w_n^{\prime }(t)$$

$$|w_n(t)-w(t)|=\left|{\int }_0^{t}w(t-y){\rho }_n(y)dy-w(t)\right|=\left|{\int }_0^{t}w(t-y){\rho }_n(y)dy-{\int }_0^{\frac{1}{n}}w(t){\rho }_n(y)dy\right|$$

$$=\{\begin{array}{l}\left|{\int }_0^t(w(t-y)-w(t)){\rho }_n(y)dy-{\int }_t^{\frac{1}{n}}w(t){\rho }_n(y)dy\right|,\text{ if }0\le t\le \frac{1}{n}\\ \\ \left|{\int }_0^{\frac{1}{n}}(w(t-y)-w(t)){\rho }_n(y)dy\right|,\text{ if }\frac{1}{n}<t\le 1\end{array}$$

$$\le \{\begin{array}{l}{\int }_0^t|w(t-y)-w(t)|{\rho }_n(y)dy+\Vert w{\Vert }_{\infty }{\int }_t^{\frac{1}{n}}{\rho }_n(y)dy,\text{ if }0\le t\le \frac{1}{n}\\ \\ {\int }_0^{\frac{1}{n}}|w(t-y)-w(t)|{\rho }_n(y)dy,\text{ if }\frac{1}{n}<t\le 1.\end{array}$$

$$t_1,t_2\in [0,1],|t_1-t_2|<\eta \text{ implies }|w(t_1)-w(t_2)|<\frac{ϵ}{2};$$

$$|w(t-y)-w(t)|<\frac{ϵ}{2}.$$

$$\underset{n\to +\infty }{\lim }{\int }_t^{\frac{1}{n}}{\rho }_n(y)dy=0.$$

$$\Vert w{\Vert }_{\infty }{\int }_t^{\frac{1}{n}}{\rho }_n(y)dy<\frac{ϵ}{2}.$$

$$\Vert w_n-w{\Vert }_{\infty }\le \frac{ϵ}{2}{\int }_0^{\frac{1}{n}}{\rho }_n(y)dy+\frac{ϵ}{2}=ϵ.$$

$$\underset{n\to +\infty }{\lim }\Vert w_n^{\prime }-w^{\prime }{\Vert }_{\omega }=\underset{n\to +\infty }{\lim }\underset{\delta \to 0}{\lim }\underset{t\in [\delta ,1-\delta ]}{\max }\omega (t)|w_n^{\prime }(t)-w^{\prime }(t)|,$$

Full text

Partially supported by G.N.A.M.P.A. - INdAM (Italy)

1. Introduction

In this paper we establish new criteria for the existence of positive radial solutions for the system of BVPs

[TABLE]

where $\Omega=\{x\in\mathbb{R}^{n}:R_{0}<|x|<R_{1}\}$ is an annulus, $0<R_{0}<R_{1}<+\infty$, the nonlinearities $f_{i}$ are non-negative continuous functions and $\dfrac{\partial}{\partial r}$ denotes (as in   [12]) differentiation in the radial direction $r=|x|$.

 The problem of the existence of positive radial solutions of elliptic equations having nonlinearities that depend on the gradient, subject to Dirichlet or mixed boundary conditions, has been investigated, via different methods, by a number of authors, for example in  [4, 5, 6, 7, 8, 11, 31]. We seek radial solutions of the system (1.1) by means of an auxiliary system of nonlinear Hammerstein integral equations using the fixed point index theory and the invariance properties of the involved cone.

The existence of solutions for nonlinear Hammerstein integral equations or systems with nonlinearities with dependence on the first derivative has been studied in  [1, 3, 6, 7, 13, 14, 16, 18, 29, 30, 36, 38].

In the recent paper  [6] in collaboration with Pietramala, we worked in the Banach space $C_{\omega_{1}}^{1}[0,1]\times C_{\omega_{2}}^{1}[0,1]$, where the weights $\omega_{i}$ are suitable nonnegative and continuous functions on $[0,1]$. In the special case $\omega(t)=t(1-t)$, the space $C_{\omega}^{1}[0,1]$ is utilized by Agarwal and others in  [1].

We defined a cone similar to the one defined in  [1] and, applying the index fixed point theory, we gave some conditions that assure the existence of positive solutions of the system (1.1). These conditions relate the upper and lower bounds of the nonlinearities $f_{i}$ on suitable stripes and some computable constants depending on the kernels of the associate Hammerstein integral operator and on the intervals in which the kernels are strictly positive.

In this paper, via spectral theory, we establish new existence results involving the principal characteristic values of auxiliary linear Hammerstein integral operators.

In this direction some results were obtained by Erbe  [10] and Liu and Li  [28] in the case in which the kernel is symmetric. In 2006, Webb and Lan  [35] generalized these results using the permanence property of the fixed point index and requiring the uniqueness of the positive eigenvalues. In 2009, Lan  [23] obtained results for semipositone Hammerstein integral equations where the permanence property and the uniqueness of the positive eigenvalues are not used, but the results on the index being $1$ are obtained only for some open subsets. The first principal eigenvalue was also used by Li  [27] in the space $L^{2}$ requiring that the linear operator is normal and by Zhang and Sun  [37] for the $m-$point boundary value problems.

In 2011, Lan and Lin studied, via spectral theory, the existence of positive solutions of systems of Hammerstein integral equations in  [25, 26] and Lan  [24] proved a new result for the existence of positive solutions of systems of second order elliptic boundary value problems. In 2015, Infante and Pietramala in  [17] established some criteria for the existence of solutions of systems of Hammerstein integral equations that involve a comparison with the spectral radii of some associated linear operators.

The Krein-Rutman Theorem, that is a celebrated result of the spectral theory, requires the totality of the cone. In Section 2, using smoothing tools as the convolution operator and a sequence of mollifiers, we prove that the involve cone is total.

Finally, an example shows that the results here obtained are applicable when the results proved in  [6] fail.

2. The totality of the cone

Let $\omega$ be a nonnegative continuous function from $[0,1]$ in $[0,1]$ and let $C_{\omega}^{1}[0,1]$ be the functions space defined by

[TABLE]

Set

[TABLE]

it can be verified that $C_{\omega}^{1}[0,1]$, equipped with the norm

[TABLE]

is a Banach space (the proof follows as in  [1]).

Fixed $[a,b]\subseteq[0,1]$, $0<c<1$, let $\mathcal{K}_{\omega}$ be the cone in $C_{\omega}^{1}[0,1]$ defined, in a similar way as in  [1], by

[TABLE]

Note that the functions in $\mathcal{K}_{\omega}$ are strictly positive on the sub-interval $[a,b]$ and that, for $w\in\mathcal{K}_{\omega}$, the equality $\|w\|=\|w\|_{\infty}$ holds.

We prove here that the cone $\mathcal{K}_{\omega}$ is total, i.e.

[TABLE]

To the best of our knowledge, this property has not been investigated.

Let

[TABLE]

be the positive cone in $C_{\omega}^{1}[0,1]$. Firstly we prove that

Lemma 2.1**.**

$\mathcal{P}_{\omega}=\mathcal{K}_{\omega}-\mathcal{K}_{\omega}$.

Proof.

An element $w\in\mathcal{P}_{\omega}/\{0\}$ can be rewritten, for $t\in[0,1]$, as:

[TABLE]

where the constants $\beta$ and $\gamma$, depending on $w$, are defined by

[TABLE]

and

[TABLE]

Set, for $t\in[0,1]$, $\varphi(t):=\beta w(t)+\gamma\|w\|_{\infty}$ and $\psi(t):=(\beta-1)w(t)+\gamma\|w\|_{\infty}$, then $w=\varphi-\psi$; now we prove that $\varphi,\psi\in\mathcal{K}_{\omega}$.

It is clear that $\|\varphi\|_{\infty}=(\beta+\gamma)\|w\|_{\infty}$ and $\|\psi\|_{\infty}=(\beta-1+\gamma)\|w\|_{\infty}$. Since the function $\displaystyle{\frac{\gamma}{\beta+\gamma}}$ is nondecreasing with respect to $\gamma$, we have, for $t\in[0,1]$,

[TABLE]

and

[TABLE]

Now we prove the conditions on the derivatives.

In the case $\|w^{\prime}\|_{\omega}\leq\|w\|_{\infty}$, one has $\beta=1$, $\gamma=\displaystyle\frac{c}{1-c}.$ Then we have

[TABLE]

and

[TABLE]

In the case $\|w^{\prime}\|_{\omega}>\|w\|_{\infty}$, one has $\beta=\displaystyle\frac{\|w^{\prime}\|_{\omega}}{\|w\|_{\infty}}$ and $\beta^{2}\leq\beta+\gamma$; then we have

[TABLE]

and

[TABLE]

∎

To prove that $\mathcal{K}_{\omega}$ is total, for each fixed $w\in C_{\omega}^{1}[0,1]$ it is need to construct two sequences $(\varphi_{n})_{n\in\mathbb{N}},(\psi_{n})_{n\in\mathbb{N}}\in\mathcal{K}_{\omega}$ such that $\varphi_{n}-\psi_{n}$ converges to $w$ in $C_{\omega}^{1}[0,\,1]$. To do this, two tools are used: a sequence of mollifiers and the convolution operation $*$.

The convolution can be viewed as a smoothing operation: in fact, the convolution of two functions is differentiable as many times as the two functions are.

A sequence $(\rho_{n})_{n\in\mathbb{N}}$ of nonnegative functions in the space $C_{c}^{\infty}(\mathbb{R})$ of the functions with compact support is said a sequence of mollifiers if the support of $\rho_{n}$ is contained in $\left[-\frac{1}{n},\frac{1}{n}\right]$ and $\displaystyle\int_{\mathbb{R}}\rho_{n}=1$. Since our functions $w$ are defined in $[0,1]$, we construct a sequence of mollifiers starting by a nonnegative function $\rho\in C_{c}^{\infty}(\mathbb{R})$ with support in $[0,1]$.

Theorem 2.2**.**

The cone

[TABLE]

is total.

Proof.

Let $\rho\in C_{c}^{\infty}(\mathbb{R})$ be a nonnegative function with support in $[0,1]$ and therefore $\rho(0)=0$. Let $(\rho_{n})_{n\in\mathbb{N}}\subseteq C_{c}^{\infty}(\mathbb{R})$ be a sequence of mollifiers defined by

[TABLE]

note that the support of $\rho_{n}$ is contained in $\left[0,\frac{1}{n}\right]$.

Let $w\in C_{\omega}^{1}[0,1]\setminus\{0\}$ be fixed; we construct a sequence $(w_{n})_{n\in\mathbb{N}}\subseteq C^{1}[0,1]$ converging to $w$ in $C^{1}_{\omega}[0,1]$.

We discuss two cases.

Case I. $w(0)=0$.

For $n\in\mathbb{N}$, we define, for $t\in[0,1]$,

[TABLE]

The function $w_{n}$ is continuous on $[0,1]$ and from (2.2) it follows that

[TABLE]

i.e. $w_{n}\in C^{1}[0,1]$.

Moreover, integrating by parts, it follows that

[TABLE]

From the last equality, it follows that the improper integrals are finite.

Now the proof is divided in more steps.

Step 1. $\displaystyle\lim_{n\to+\infty}\|w_{n}-w\|_{\infty}=0$.

Since $\displaystyle\int_{0}^{\frac{1}{n}}\rho_{n}(x)=1$ and the support of $\rho_{n}$ is in $\displaystyle\left[0,\frac{1}{n}\right]$, one has

[TABLE]

[TABLE]

[TABLE]

From the uniform continuity of $w$ in $[0,1]$, it follows that for each $\epsilon>0$ fixed there exists $\eta>0$ such that

[TABLE]

consequently, for $t\in[0,1]$, y\in\big{[}0,\frac{1}{n}\big{]} and $n\geq\eta^{-1}$, one has

[TABLE]

Moreover, since $\rho(0)=0$, by (2.1) and the Mean Integral Theorem it follows that, for $t\in\left[0,\frac{1}{n}\right]$,

[TABLE]

Then there exists $\overline{n}\in\mathbb{N}$ such that, for $n\geq\overline{n}$,

[TABLE]

Set $n_{0}:=\displaystyle\max\left\{[\eta^{-1}]+1,\overline{n}\right\}$, we have, for $n\geq n_{0}$,

[TABLE]

Step 2. $\displaystyle\lim_{n\to+\infty}\|w^{\prime}_{n}-w^{\prime}\|_{\omega}=0$.

Since

[TABLE]

we evaluate, for $t\in[\delta,1-\delta]$,

[TABLE]

[TABLE]

[TABLE]

Let $\epsilon>0$ be fixed; since $w^{\prime}$ is uniformly continuous on compact intervals, there exists $\eta_{1}>0$ such that

[TABLE]

moreover, there exists $\tilde{n}\in\mathbb{N}$ such that, for $n\geq\tilde{n}$,

[TABLE]

Then, set $n_{1}:=\displaystyle\max\left\{[\eta_{1}^{-1}]+1,\tilde{n}\right\}$, for $n\geq n_{1}$, as above, one obtains

[TABLE]

i.e. for every $\delta\in]0,1[$

[TABLE]

Since the function $\displaystyle\max_{t\in[\delta,1-\delta]}\omega(t)|w^{\prime}_{n}(t)-w^{\prime}(t)|$ is nonincreasing with respect to $\delta$, by Dini Theorem it follows that the limit in (2.3) is uniform in the compact subintervals of $]0,1[$.

Moreover, for every $n\in\mathbb{N}$ there exists

[TABLE]

then, by the Inversion Limit Theorem, for $t$ in a compact subinterval of $]0,1[$ one has

[TABLE]

and consequently $\displaystyle\lim_{n\to+\infty}\|w^{\prime}_{n}-w^{\prime}\|_{\omega}=0$.

Step 3. $w_{n}\in\mathcal{K}_{\omega}-\mathcal{K}_{\omega}$.

The function $w$ can be rewritten as $w=w^{+}-w^{-}$, where $w^{+}$ and $w^{-}$ are respectively the positive and negative parts of $w$; then, for $n\in\mathbb{N}$

[TABLE]

where

[TABLE]

Since the functions $w_{n}^{+}$ and $w_{n}^{-}$ belong to $C^{1}[0,1]$ and are positive, they belong to the cone $\mathcal{P}_{\omega}$ and therefore there exist $\varphi_{n}^{+},\varphi_{n}^{-},\psi_{n}^{+},\psi_{n}^{-}\in\mathcal{K}_{\omega}$ such that

[TABLE]

By the identity (2.4) the assert follows.

Case II. Case $w(0)\not=0$.

With a translation we return to the previous case. In fact, set $\tilde{w}=w-w(0)$, the sequence $\tilde{w}_{n}$ defined as in (2.2) converges in $C^{1}_{\omega}[0,1]$ to $\tilde{w}$ and the sequence $\overline{w}_{n}=\tilde{w}_{n}+w(0)\to w$ in $C^{1}_{\omega}[0,1]$. Let $\tilde{\phi}_{n},\tilde{\psi}_{n}\in\mathcal{K}_{\omega}$ be such that $\tilde{w}_{n}=\tilde{\phi}_{n}-\tilde{\psi}_{n}$; then $\overline{w}_{n}=\phi_{n}-\psi_{n}$ where

[TABLE]

∎

3. Auxiliar results

In this section we recall notations and results of  [6] that will be useful in the sequel.

By a radial solution of the elliptic system

[TABLE]

we mean a solution of the associate system of ODEs

[TABLE]

where, for $t\in[0,1]$, $g_{i}$ is the nonnegative function given by

[TABLE]

$p$ is defined by

[TABLE]

and $r$ is defined by (see  [8, 9])

[TABLE]

with

[TABLE]

Fix

[TABLE]

consider the product space $C_{\omega_{1}}^{1}[0,1]\times C_{\omega_{2}}^{1}[0,1]$ equipped with the norm (with an abuse of notation)

[TABLE]

We search the solutions of the system (3.1) as fixed points of the compact operator $\mathcal{T}$ in $C_{\omega_{1}}^{1}[0,1]\times C_{\omega_{2}}^{1}[0,1]$ defined by

[TABLE]

where the Green’s functions $k_{i}$ are given by

[TABLE]

Now we resume some known properties of the functions $k_{i}$ that will be use in the sequel.

• (1)

The kernel $k_{1}$ is positive and continuous in $[0,1]\times[0,1]$. Moreover, for $[a_{1},b_{1}]\subset(0,1)$, take

[TABLE]

[TABLE]

• (2)

The function $k_{1}(\cdot,s)$ is derivable in $\tau\in[0,1]$, with

[TABLE]

for $\tau\in[0,1]$

[TABLE]

and

[TABLE]

• (3)

The kernel $k_{2}$ is positive and continuous in $[0,1]\times[0,1]$. Moreover, for $[a_{2},b_{2}]\subset(0,1]$, take

[TABLE]

[TABLE]

• (4)

The function $k_{2}(\cdot,s)$ is derivable in $\tau\in[0,1]$, with

[TABLE]

and for $\tau\in[0,1]$

[TABLE]

Moreover

[TABLE]

By direct calculation we obtained

[TABLE]

[TABLE]

[TABLE]

The following existence result for the system (1.1) is established in  [6].

Theorem 3.1**.**

[6]** Suppose that, for $i=1,2$, there exist $\rho_{i},s_{i}\in(0,+\infty)$, with $\rho_{i}<c_{i}\,s_{i}$, such that the following conditions hold

[TABLE]

and

[TABLE]

where

[TABLE]

Then the system (1.1) has at least one positive radial solution.

The following theorem follows from classical results about fixed point index (more details can be seen, for example, in  [2, 15]).

Theorem 3.2**.**

Let $K$ be a cone in an ordered Banach space $X$. Let $\Omega$ be an open bounded subset with $0\in\Omega\cap K$ and $\overline{\Omega\cap K}\neq K$. Let $\Omega^{1}$ be open in $X$ with $\overline{\Omega^{1}}\subset\Omega\cap K$. Let $F:\overline{\Omega\cap K}\rightarrow K$ be a compact map. Suppose that

• (1)

$Fx\neq\mu x$* for all $x\in\partial(\Omega\cap K)$ and for all $\mu\geq 1$.*

• (2)

There exists $h\in K\setminus\{0\}$ such that $x\neq Fx+\lambda h$ for all $x\in\partial(\Omega^{1}\cap K)$ and all $\lambda\geq 0$.

*Then $F$ has at least one fixed point $x\in(\Omega\cap K)\setminus\overline{(\Omega^{1}\cap K)}$.

Denoting by $i_{K}(F,U)$ the fixed point index of $F$ in some $U\subset X$,*

[TABLE]

The same result holds if

[TABLE]

Now we fix

[TABLE]

and we consider the cone

[TABLE]

in $C_{\omega_{1}}^{1}[0,1]\times C_{\omega_{2}}^{1}[0,1]$, where $\mathcal{K}_{\omega_{i}}$ is

[TABLE]

We have showed in Section 2 that the cone $\mathcal{K}$ is total and in  [6] that is $\mathcal{T}-$invariant.

A positive solution of the system (1.1) means a solution $(u,v)\in\mathcal{K}$ of (3.1) such that $\|(u,v)\|>0$.

In order to use the fixed point index, we utilize the open bounded sets (relative to $\mathcal{K}$), for $\rho_{1},\rho_{2}>0$,

[TABLE]

for which holds the property:

$(w_{1},w_{2})\in\partial K_{\rho_{1},\rho_{2}}$ if and only if $(w_{1},w_{2})\in\mathcal{K}$ and for some $i\in\{1,2\}$ $\|w_{i}\|_{\infty}=\rho_{i}$ and $c_{i}\rho_{i}\leq w_{i}(t)\leq\rho_{i}$ for $t\in[a_{i},b_{i}]$.

4. Characteristic values of linear operators

Let $L:X\to X$ be a linear operator on a Banach space $X$. A number $\lambda$ is said an eigenvalue of $L$ with corresponding eigenfunction $\varphi$ if $\varphi\neq 0$ and ${\lambda}\varphi=L\varphi$. The reciprocals of nonzero eigenvalues are called characteristic values of $L$. The spectral radius of $L$ is given by $r(L):=\displaystyle\lim_{n\to\infty}\|L^{n}\|^{\frac{1}{n}}$ and its principal characteristic value by $\mu(L)=1/r(L)$ .

We give the statements of the main tools in this section.

Theorem 4.1**.**

*(**Krein-Rutman Theorem)  [22]

Assume that $K$ is a total cone in a real Banach space $X$ and $L:X\to X$ is a compact linear operator such that $L(K)\subset K$ and $r(L)>0$. Then there exists a nonzero element $u\in K$ such that $Lu=r(L)u$.*

Definition 4.2**.**

[20, 21] A positive bounded linear operator $L:X\to X$ is said $u_{0}-$positive relative to the cone $K$ if there exists $u_{0}\in K\setminus\{0\}$ such that for every $u\in K\setminus\{0\}$ there are constants $d_{2,u}\geq d_{1,u}>0$ such that

[TABLE]

Theorem 4.3**.**

*(**Comparison Theorem) [19]

Let $K$ be a cone in a Banach space $X$ and let $L,S$ be bounded linear operators, with $L\leq S$. Assume that at least one of the operators is $u_{0}-$positive on $K$. If there exist*

1. (1)

$u_{1}\in K\setminus\{0\}$* and $\lambda_{1}>0$ such that $Lu_{1}\geq\lambda_{1}u_{1}$;* 2. (2)

$u_{2}\in K\setminus\{0\}$* and $\lambda_{2}>0$ such that $Su_{2}\leq\lambda_{2}u_{2}$,*

then $\lambda_{1}\leq\lambda_{2}$ and, if $\lambda_{1}=\lambda_{2}$, $u_{1}$ is a scalar multiple of $u_{2}$.

In order to state the eigenvalue criteria, consider the linear Hammerstein operator $\mathcal{L}$ on $C^{1}_{\omega_{1}}[0,1]\times C^{1}_{\omega_{2}}[0,1]$, associate to the operator $\mathcal{T}$, defined by, for $t\in[0,1]$,

[TABLE]

Theorem 4.4**.**

The operator $\mathcal{L}$ is compact and map $\mathcal{P}=\mathcal{P}_{\omega_{1}}\times\mathcal{P}_{\omega_{2}}$ into $\mathcal{K}$.

Proof.

Note that the operator $\mathcal{L}$ maps $\mathcal{P}$ into $\mathcal{P}$ because the kernels are positive functions; now we show that $\mathcal{L}$ maps $\mathcal{P}$ into $\mathcal{K}$.

By (3.3) and (3.4), for every $u_{i}\in\mathcal{P}_{\omega_{i}}$ it follows

[TABLE]

and therefore

[TABLE]

On the other hand, we have

[TABLE]

Now we prove that, if $u_{1}\in\mathcal{P}_{\omega_{1}}$, it holds

[TABLE]

In fact we have

[TABLE]

and consequently (4.1) holds.

Analogously, for $u_{2}\in\mathcal{P}_{\omega_{2}}$, we obtain

[TABLE]

and therefore we have

[TABLE]

Finally, by the properties of the Green’s functions $k_{i}$ and using the Arzèla-Ascoli Theorem, it follows that the operator $\mathcal{L}$ is compact. ∎

Since $\|w\|=\|w\|_{\infty}$ for each $w\in\mathcal{K}_{\omega_{i}}$, the proof of the following theorem is analogous to the ones in  [33, 35] and is reported for completeness.

Theorem 4.5**.**

For $i=1,2$, the spectral radius of $\mathcal{L}_{i}$ is nonzero and is an eigenvalue of $\mathcal{L}_{i}$ with an eigenfunction in $\mathcal{K}_{\omega_{i}}$.

Proof.

For $u_{i}\in\mathcal{K}_{\omega_{i}}$ and $t\in[a_{i},b_{i}]$ one has

[TABLE]

Then we have

[TABLE]

and analogously we get

[TABLE]

Thus we obtain

[TABLE]

hence we have

[TABLE]

Then, by Theorem 4.1, $r(\mathcal{L}_{i})$ is an eigenvalue of $\mathcal{L}_{i}$ with $\varphi_{i}$ eigenfunction in $\mathcal{P}_{\omega_{i}}$ and, as $\mathcal{L}_{i}$ maps $\mathcal{P}_{\omega_{i}}$ in $\mathcal{K}_{\omega_{i}}$, the eigenfunction $\varphi_{i}\in\mathcal{K}_{\omega_{i}}$.

∎

Remark 4.6**.**

Since the kernels $k_{i}$ satisfy the following symmetry properties, for all $t,s\in[0,1]$,

[TABLE]

Corollary 7.5 in  [34] assures that the linear operators $\mathcal{L}_{i}$ are $u_{0}-$positive relative to the cone $\mathcal{P}_{\omega_{i}}$ and therefore Theorem 4.3 can be used in the sequel.

Moreover, by Theorems 2.2, 4.4 and 4.5, it follows that the operators $\mathcal{L}_{i}$ satisfy the hypotheses of Krein-Rutman Theorem.

5. Eigenvalue criteria for the existence of positive solutions

In this section we give some results that determine relationships between the upper and lower bounds of the nonlinearities $f_{i}$ on some stripes of $\mathbb{R}^{n}$ and the principal characteristic values of two linear operators associated to $\mathcal{T}$.

Using the principal characteristic value of $\mathcal{L}_{i}$, in the following two theorems we provide conditions assuring that the index of the operator $\mathcal{T}$ defined in (3.2) is one in some suitable sets.

Let $f_{i}$ be the nonlinearities of the system (1.1) .

Theorem 5.1**.**

Assume that

1. for $i=1,2$ there exist $\varepsilon_{i}>0$ and $\rho_{i}>0$ such that the following condition holds:

[TABLE]

where

[TABLE]

Then $i_{\mathcal{K}}(\mathcal{T},K_{\sigma_{1},\sigma_{2}})=1$ for $\sigma_{i}\leq\rho_{i},\,i=1,2$.

Proof.

Let $\sigma_{i}\leq\rho_{i}$ for $i=1,2$. In order to show $i_{\mathcal{K}}(\mathcal{T},K_{\sigma_{1},\sigma_{2}})=1$, we prove that $\mathcal{T}(u,v)\neq\lambda(u,v)$ for $(u,v)\in\partial K_{\sigma_{1},\sigma_{2}}$ and $\lambda\geq 1$. Otherwise there exist $(u,v)\in\partial K_{\sigma_{1},\sigma_{2}}$ and $\lambda\geq 1$ such that $\lambda(u,v)=\mathcal{T}(u,v)$. Thus we have, for $t\in[0,1]$,

[TABLE]

Therefore, by the monotone properties of the operator $\mathcal{L}_{1}$ we have, for $t\in[0,1]$,

[TABLE]

thus, taking the norm, we obtain

[TABLE]

and then we get

[TABLE]

a contradiction. ∎

Theorem 5.2**.**

Assume that

1. for $i=1,2$ there exist $\varepsilon_{i}>0$ and $\theta_{i}>0$ such that the following condition holds:

[TABLE]

[TABLE]

Then there exist $\tau_{1},\tau_{2}>0$ such that $i_{\mathcal{K}}(\mathcal{T},K_{\sigma_{1},\sigma_{2}})=1$ for each $\sigma_{i}>\tau_{i},$ $i=1,2$.

Proof.

Since the functions $f_{i}$ are continuous, there exist some constants $N_{i}$ depending on $\theta_{i}$ such that

[TABLE]

Hence

[TABLE]

for $r\in[R_{0},R_{1}]$ and $w_{1},w_{2},z_{1},z_{2}\geq 0$ .

Let $\operatorname{Id}$ be the identity operator. Since for $i=1,2$ the operators $(\mu(\mathcal{L}_{i})-\varepsilon_{i})\mathcal{L}_{i}$ have spectral radius less than one, the operators $(\operatorname{Id}-(\mu(\mathcal{L}_{i})-\varepsilon_{i})\mathcal{L}_{i})^{-1}$ exist and are bounded. Moreover, from the Neumann series expression,

[TABLE]

it follows that $(\operatorname{Id}-(\mu(\mathcal{L}_{i})-\varepsilon_{i})\mathcal{L}_{i})^{-1}$ map $\mathcal{P}$ into $\mathcal{P}$, since the operators $\mathcal{L}_{i}$ have this property.

Take for $i=1,2$

[TABLE]

Now we prove that, for $\sigma_{i}>\tau_{i}$, $\mathcal{T}(u,v)\neq\lambda(u,v)$ for all $(u,v)\in\partial K_{\sigma_{1},\sigma_{2}}$ and $\lambda\geq 1$, which implies $i_{\mathcal{K}}(\mathcal{T},K_{\sigma_{1},\sigma_{2}})=1$. Otherwise there exist $(u,v)\in\partial K_{\sigma_{1},\sigma_{2}}$ and $\lambda\geq 1$ such that $\lambda(u,v)=\mathcal{T}(u,v)$. Suppose that $\|u\|=\|u\|_{\infty}=\sigma_{1}$ and $\|v\|=\|v\|_{\infty}\leq\sigma_{2}$.

From the inequality (5.1), it follows that, for $t\in[0,1]$,

[TABLE]

which implies

[TABLE]

Since $(\operatorname{Id}-(\mu(\mathcal{L}_{1})-\varepsilon_{1})\mathcal{L})^{-1}$ is non-negative, it follows that, for $t\in[0,1]$,

[TABLE]

Consequently $\sigma_{1}=\|u\|_{\infty}\leq\tau_{1}$, a contradiction.

The case $\|u\|_{\infty}\leq\sigma_{1}$ and $\|v\|_{\infty}=\sigma_{2}$ is obtained in a similar way. ∎

Remark 5.3**.**

In  [32, 35], under suitable assumptions on $k$, Webb and Lan prove that the operator $L$ defined by

[TABLE]

satisfies the following inequality $m\leq\mu(L)\leq M(a,b)$, where $m$ and $M(a,b)$ are defined similary to (3.5) and (3.6).

Moreover in  [35], in the setting of the space of continuous functions, the authors obtained that

[TABLE]

the same results hold in the space $C^{1}[0,1]$.

In the zero index calculation of the operator $\mathcal{T}$ it is more convenient to use the linear operators $\overline{\mathcal{L}}_{i}:C^{1}_{\omega_{i}}[a_{i},b_{i}]\to C^{1}_{\omega_{i}}[a_{i},b_{i}]$ defined by, for $t\in[a_{i},b_{i}]$,

[TABLE]

that have the same properties of the operators $\mathcal{L}_{i}$.

Theorem 5.4**.**

Assume that

1. there exist $\varepsilon_{i}>0$ and $\rho_{i}>0$ such that the following condition holds for some $i=1,2$:

[TABLE]

where

[TABLE]

Then $i_{\mathcal{K}}(\mathcal{T},K_{\sigma_{1},\sigma_{2}})=0$ for $\sigma_{i}\leq\rho_{i},\,i=1,2.$

Proof.

Let $\sigma_{i}\leq\rho_{i}$ and let $\varphi_{i}\in\mathcal{K}_{\omega_{i}}$ be the eigenfunction of $\overline{\mathcal{L}}_{i}$ with $\|\varphi_{i}\|=\|\varphi_{i}\|_{\infty}=1$ corresponding to the eigenvalue $1/\mu(\overline{\mathcal{L}}_{i})$. Now we show that $(u,v)\neq\mathcal{T}(u,v)+\lambda(\varphi_{1},\varphi_{2})$ for all $(u,v)$ in $\partial K_{\sigma_{1},\sigma_{2}}$ and $\lambda\geq 0$ which implies that $i_{\mathcal{K}}(\mathcal{T},K_{\sigma_{1},\sigma_{2}})=0$.

Assume, on the contrary, that there exist $(u,v)\in\partial K_{\sigma_{1},\sigma_{2}}$ and $\lambda\geq 0$ such that $(u,v)=\mathcal{T}(u,v)+\lambda(\varphi_{1},\varphi_{2})$.

Two cases are distincts. Firstly we discuss the case $\lambda>0$. Suppose that (5.2) holds for $i=1$. This implies that, for $t\in[a_{1},b_{1}]$,

[TABLE]

Moreover $u(t)>\lambda\varphi_{1}(t)$ for $t\in[a_{1},b_{1}]$; then $\overline{\mathcal{L}}_{1}u(t)\geq\lambda\overline{\mathcal{L}}_{1}\varphi_{1}(t)=\dfrac{\lambda}{\mu(\overline{\mathcal{L}}_{1})}\varphi_{1}(t)$ and we obtain

[TABLE]

By iteration, it follows that, for $t\in[a_{1},b_{1}]$,

[TABLE]

a contradiction because $\|u\|\leq\sigma_{1}$.

Now we consider the case $\lambda=0$. We have, for $t\in[a_{1},b_{1}]$,

[TABLE]

and, consequently, we obtain

[TABLE]

Then, by Comparison Theorem 4.3, it follows that $r({\overline{\mathcal{L}}}_{1})\leq\dfrac{1}{\mu(\overline{\mathcal{L}}_{1})+\varepsilon_{1}}$ and thus we get

[TABLE]

a contradiction.

∎

Remark 5.5**.**

As in  [35], we obtain by direct calculations that

[TABLE]

The difference between Theorem 5.4 and the following Theorem consists in the fact that in Theorem 5.6 the lower bound of the $f_{i}$ is calculate for $w_{i}$ enough far from the zero.

Theorem 5.6**.**

Assume that

1. for $i=1,2$ there exist $\varepsilon_{i}>0$ and $\theta_{i}>0$ such that the following condition holds:

[TABLE]

where

[TABLE]

[TABLE]

Then $i_{\mathcal{K}}(\mathcal{T},K_{s_{1},s_{2}})=0$ for $s_{i}\geq\theta_{i}$, $i=1,2$.

Proof.

Let $s_{i}\geq\theta_{i}$. We prove that $(u,v)\neq\mathcal{T}(u,v)+\lambda(\varphi_{1},\varphi_{2})$ for all $(u,v)$ in $\partial K_{s_{1},s_{2}}$ and $\lambda\geq 0$, where $\varphi_{i}\in\mathcal{K}_{\omega_{i}}$ is the eigenfunction associated to $r(\overline{\mathcal{L}}_{i})$ as in Theorem 5.4, which implies that $i_{\mathcal{K}}(\mathcal{T},K_{s_{1},s_{2}})=0$.

Assume, on the contrary, that there exist $(u,v)\in\partial K_{s_{1},s_{2}}$ and $\lambda\geq 0$ such that $(u,v)=\mathcal{T}(u,v)+\lambda(\varphi_{1},\varphi_{2})$. Suppose that $\|u\|=s_{1}$ and $\|v\|\leq s_{2}$.

Then, for $t\in[a_{1},b_{1}]$, $u(t)\geq c_{1}\|u\|=c_{1}s_{1}\geq c_{1}\theta_{1}$, thus condition (5.3) holds. Hence we obtain, for $t\in[a_{1},b_{1}]$,

[TABLE]

Proceeding as in the proof of Theorem 5.4 in the the case $\lambda>0$, this implies that, for $t\in[a_{1},b_{1}]$,

[TABLE]

Then $u(t)>n\lambda\varphi_{1}(t)$ for every $n\in\mathbb{N}$, a contradiction because $\|u\|=s_{1}$.

The proof in the case $\|u\|\leq s_{1}$ and $\|v\|=s_{2}$ is analogous and the case $\lambda=0$ is treated as in Theorem 5.4. ∎

Using Theorem 5.1 and Theorem 5.6, the following existence result of positive radial solution for the system (1.1) holds.

Theorem 5.7**.**

Assume that

1. for $i=1,2$, there exist $\varepsilon_{i},\eta_{i}>0$, $\rho_{i},\theta_{i}>0$, with $\rho_{i}\leq c_{i}\theta_{i}$, such that the following conditions hold:

[TABLE]

and

[TABLE]

*where $\Omega^{\rho_{1},\rho_{2}}$ and $D_{i}^{\theta_{1},\theta_{2}}$ are as in Theorem 5.1 and Theorem 5.6.

Then the system (1.1) has at least one positive radial solution.*

The index results in this Section can be carefully combined in order to establish results on existence of multiple positive solutions for the system (1.1). We refer to  [26] for similar statements.

Example 5.8**.**

Theorem  5.7 can be applied when the nonlinearities $f_{i}$ are of the type

[TABLE]

with $h_{i}$ continuous functions bounded by a strictly positive constant, $\alpha_{i},\beta_{i}>1$ and $\delta_{i},\gamma_{i}$ suitable positive constants.

For example, one can consider the following system

[TABLE]

where $\Omega=\{x\in\mathbb{R}^{3}:1<|x|<e\}$.

By direct computation, we obtain $\displaystyle\sup_{t\in[0,1]}p(t)=e^{2}(e-1)^{2}$ and, fixed $[a_{1},b_{1}]=\left[\frac{1}{4},\frac{3}{4}\right],\,[a_{2},b_{2}]=\left[\frac{1}{2},1\right]$, we have $c_{1}=\frac{1}{4},\,c_{2}=\frac{1}{2}$,

[TABLE]

With the choice of $\rho_{1}=1/10,\,\,\rho_{2}=1/25\,,\theta_{1}=200,\,\theta_{2}=50$, we obtain

[TABLE]

consequently the nonlinearities $f_{i}$ satisfy Theorem 5.7 and the system (5.4) admits at least one positive radial solution.

We conclude by noting that, with this choice of radius, $f_{1}$ does not satisfy the hypotheses of Theorem 3.1 because

[TABLE]