Now we will establish the main result of this paper: the existence of a nontrivial positive solution for (P) when \mathrm{\Omega}\subset {\mathbb{R}}^{N} is a smooth bounded domain.

### 3.1 The constants {k}_{1} and {k}_{2} in Ω

For the general problem

we define T:K\to K by

T(u,v)=({T}_{1}(u,v),{T}_{2}(u,v))=({z}_{1},{z}_{2}),

where

\{\begin{array}{l}-{\mathrm{\Delta}}_{p}{z}_{1}=w(x)f(v)\phantom{\rule{1em}{0ex}}\text{in}\mathrm{\Omega},\\ -{\mathrm{\Delta}}_{q}{z}_{2}=\rho (x)g(u)\phantom{\rule{1em}{0ex}}\text{in}\mathrm{\Omega},\\ ({z}_{1},{z}_{2})=(0,0)\phantom{\rule{1em}{0ex}}\text{on}\partial \mathrm{\Omega}.\end{array}

(13)

It is well known in *p*-Laplacian operator theory that *T* is completely continuous. Moreover, a simple maximum principle argument guarantees that {z}_{i}>0 in Ω for i=1,2.

As it has been done in the radial case, in order to obtain a result of existence for (P), we will apply Lemma 3.

Let us denote by {\varphi}_{s,h}\in {C}^{1,\alpha}(\mathrm{\Omega}) the solution of

\{\begin{array}{l}-{\mathrm{\Delta}}_{s}u=h(x)\phantom{\rule{1em}{0ex}}\text{in}\mathrm{\Omega},\\ u=0\phantom{\rule{1em}{0ex}}\text{on}\partial \mathrm{\Omega},\end{array}

where h\in {L}^{{s}^{\prime}}(\mathrm{\Omega}) and {s}^{\prime} is such that \frac{1}{s}+\frac{1}{{s}^{\prime}}=1. As *ρ* and *w* satisfy the same hypotheses as *h*, we can define

{k}_{1}=min\{{\parallel {\varphi}_{p,w}\parallel}_{\mathrm{\infty}}^{1-p},{\parallel {\varphi}_{q,\rho}\parallel}_{\mathrm{\infty}}^{1-q}\}.

(14)

Fix {x}_{0}\in \mathrm{\Omega} such that w({x}_{0})\ne 0, let R>0 be such that \overline{B({x}_{0},R)}\subset \mathrm{\Omega} and

{w}^{\ast}(s)=\{\begin{array}{ll}min\{w(y):|y-{x}_{0}|=s\}& \text{if}0s\le R,\\ w({x}_{0})& \text{if}s=0.\end{array}

(15)

Furthermore, let us define {k}_{2} by

{k}_{2}={\left({\int}_{\xi}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\frac{\xi}{2}}{\left(\frac{s}{r}\right)}^{N-1}{w}^{\ast}(s)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta \right)}^{1-p}

(16)

(where *ξ* is defined similarly as it has been done in (4) and (5)).

### 3.2 Main theorem

**Theorem 4** *Suppose that* f,g:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) *are positive*, *continuous nonlinearities satisfying* (H1) *and* (H2) (*with constants* {k}_{1} *and* {k}_{2} *defined in* (14) *and* (16), *respectively*), *and let the weight functions* w,\rho :\overline{\mathrm{\Omega}}\to \mathbb{R} *be continuous*, *nonnegative and not identically null*. *Then problem* (P) *has at least a nontrivial positive solution*. *Moreover*, *if* (u,v) *is a positive solution for* (P), *then*

\delta \le {\parallel (u,v)\parallel}_{\mathrm{\infty}}=max\{{\parallel u\parallel}_{\mathrm{\infty}},{\parallel v\parallel}_{\mathrm{\infty}}\}\le M.

*Proof* Let (u,v)\in K be such that {\parallel (u,v)\parallel}_{\mathrm{\infty}}=M. It follows from (H1) and (13) that

-{\mathrm{\Delta}}_{p}{z}_{1}=w(x)f(v)\le w(x){k}_{1}{M}^{p-1}=-{\mathrm{\Delta}}_{p}\left({k}_{1}^{\frac{1}{p-1}}M{\varphi}_{p,w}\right)\phantom{\rule{1em}{0ex}}\text{in}\mathrm{\Omega}

and

{z}_{1}=0={k}_{1}^{\frac{1}{p-1}}M{\varphi}_{p,w}\phantom{\rule{1em}{0ex}}\text{in}\partial \mathrm{\Omega}.

Therefore, by the maximum principle, we conclude that

{z}_{1}\le {k}_{1}^{\frac{1}{p-1}}M{\varphi}_{p,w}\le {k}_{1}^{\frac{1}{p-1}}M{\parallel {\varphi}_{p,w}\parallel}_{\mathrm{\infty}}\le M

and, consequently,

{\parallel {z}_{1}\parallel}_{\mathrm{\infty}}\le M.

In the same way, we obtain

{\parallel {z}_{2}\parallel}_{\mathrm{\infty}}\le M,

and, as a consequence of the last two inequalities, we have

{\parallel T(u,v)\parallel}_{\mathrm{\infty}}\le M={\parallel (u,v)\parallel}_{\mathrm{\infty}}.

It follows from Lemma 3 that

We claim that i(T,{K}_{\delta},K)=0. To prove our claim, we will show, as it has been done in the radial case, that there exists ({u}_{0},{v}_{0})\in {K}_{\delta} such that

(u,v)-T(u,v)\ne t({u}_{0},{v}_{0})\phantom{\rule{1em}{0ex}}\text{for all}(u,v)\in \partial {K}_{\delta}\text{and}t0.

For fixed {x}_{0}\in \mathrm{\Omega}, let \mathrm{\Theta}\in C(\overline{\mathrm{\Omega}}) be such that \mathrm{\Theta}(x)\in [0,1] and

\mathrm{\Theta}(x)=\mathrm{\Theta}(|x-{x}_{0}|)=\{\begin{array}{ll}1& \text{if}0\le |x-{x}_{0}|\le \frac{\xi}{2},\\ 0& \text{if}\xi \le |x-{x}_{0}|\le R.\end{array}

Consider

({u}_{0}(r),{v}_{0}(r))=(\mathrm{\Theta}(r),\mathrm{\Theta}(r)),

where r=|x-{x}_{0}|.

We claim that

(u,v)-T(u,v)\ne t(\mathrm{\Theta},\mathrm{\Theta})\phantom{\rule{1em}{0ex}}\text{for all}(u,v)\in \partial {K}_{\delta}\text{and}t0.

In fact, suppose that there exist ({u}^{\ast},{v}^{\ast})\in {K}_{\delta} and {t}_{0}\ge 0 such that

({u}^{\ast},{v}^{\ast})-({z}_{1}^{\ast},{z}_{2}^{\ast})={t}_{0}(\mathrm{\Theta},\mathrm{\Theta}),

(17)

where ({z}_{1}^{\ast},{z}_{2}^{\ast})=T({u}^{\ast},{v}^{\ast}).

Since T({u}^{\ast},{v}^{\ast})\in K, it follows that

\{\begin{array}{l}{u}^{\ast}(x)\ge {t}_{0}\mathrm{\Theta}(x),\\ {v}^{\ast}(x)\ge {t}_{0}\mathrm{\Theta}(x).\end{array}

Let {z}_{1R}^{\ast} be the solution of

\{\begin{array}{l}-{\mathrm{\Delta}}_{p}{z}_{1R}^{\ast}={w}^{\ast}(r){f}^{\ast}(r)\phantom{\rule{1em}{0ex}}\text{in}B({x}_{0},R),\\ {z}_{1R}^{\ast}=0\phantom{\rule{1em}{0ex}}\text{on}\partial B({x}_{0},R),\end{array}

where

{f}^{\ast}(s)=\{\begin{array}{ll}min\{f({v}^{\ast}(y)):|y-{x}_{0}|=s\}& \text{if}0s\le R,\\ f({v}^{\ast}({x}_{0}))& \text{if}s=0.\end{array}

(18)

Then

{z}_{1R}^{\ast}(x)={z}_{1R}^{\ast}(|x|)={\int}_{|x-{x}_{0}|}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\theta}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s){f}^{\ast}(s)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta .

Note that given x\in B({x}_{0},R), we obtain

\{\begin{array}{l}-{\mathrm{\Delta}}_{p}{z}_{1R}^{\ast}(|x|)={w}^{\ast}(s){f}^{\ast}(s)\le w(x)f({v}^{\ast}(x))=-{\mathrm{\Delta}}_{p}{z}_{1}^{\ast}\phantom{\rule{1em}{0ex}}\text{in}B({x}_{0},R),\\ {z}_{1R}^{\ast}=0{z}_{1}^{\ast}\phantom{\rule{1em}{0ex}}\text{on}\partial B({x}_{0},R),\end{array}

and by the maximum principle, we have

{z}_{1R}^{\ast}\le {z}_{1}^{\ast}\phantom{\rule{1em}{0ex}}\text{for every}x\in B({x}_{0},R).

Thus, by (17) we obtain

{u}^{\ast}(x)={z}_{1}^{\ast}(x)+{t}_{0}\mathrm{\Theta}(x)\ge {z}_{1R}^{\ast}(x)+{t}_{0}\mathrm{\Theta}(x)\phantom{\rule{1em}{0ex}}\text{for all}x\in B({x}_{0},R).

Moreover, if x\in B({x}_{0},\xi )\subset B({x}_{0},R), we have

\begin{array}{rcl}{u}^{\ast}(x)& \ge & {\int}_{|x-{x}_{0}|}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\theta}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s){f}^{\ast}(s)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta +{t}_{0}\mathrm{\Theta}(r)\\ \ge & {\int}_{\xi}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\frac{\xi}{2}}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s){f}^{\ast}(s)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta +{t}_{0}\mathrm{\Theta}(r).\end{array}

As s\in [0,\frac{\xi}{2}], we conclude from (18) that there is y\in B({x}_{0},\xi /2) such that

{f}^{\ast}(s)=f({v}^{\ast}(y)).

Noting that f({v}^{\ast}(y))\ge {k}_{2}{({v}^{\ast})}^{p-1}(y) and considering

{t}^{\ast}:=sup\{t:{u}^{\ast}(x)\ge t\mathrm{\Theta}(x)\text{and}{v}^{\ast}(x)\ge t\mathrm{\Theta}(x)\text{for all}x\in \mathrm{\Omega}\},

(19)

we have

\begin{array}{rcl}{u}^{\ast}(x)& \ge & {\int}_{\xi}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\frac{\xi}{2}}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s){f}^{\ast}(s)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta +{t}_{0}\mathrm{\Theta}(x)\\ =& {\int}_{\xi}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\frac{\xi}{2}}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s)f({v}^{\ast}(y))\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta +{t}_{0}\mathrm{\Theta}(x)\\ \ge & {\int}_{\xi}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\frac{\xi}{2}}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s){k}_{2}{\left({v}^{\ast}\right)}^{p-1}(y)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta +{t}_{0}\mathrm{\Theta}(x)\\ \ge & {k}_{2}^{\frac{1}{p-1}}{t}^{\ast}{\int}_{\xi}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\frac{\xi}{2}}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s){\mathrm{\Theta}}^{p-1}(y)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta +{t}_{0}\mathrm{\Theta}(x)\\ =& {k}_{2}^{\frac{1}{p-1}}{t}^{\ast}{\int}_{\xi}^{R}{\psi}_{{p}^{\prime}}({\int}_{0}^{\frac{\xi}{2}}{\left(\frac{s}{\theta}\right)}^{N-1}{w}^{\ast}(s)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}d\theta +{t}_{0}\mathrm{\Theta}(x)\end{array}

by the fact that \mathrm{\Theta}(y)=1 if y\in B({x}_{0},\xi /2). Repeating the same ideas of the radial case, we conclude that

{u}^{\ast}(x)\ge ({t}_{0}+{t}^{\ast})\mathrm{\Theta}(x),

which contradicts (19). By the additivity of index, it follows that

i(T,{K}_{M}\mathrm{\setminus}{K}_{\delta},K)=1

proving the theorem. As in the radial case, the regularity follows from [14] and [15]. □

**Remark 5** Due to the hypotheses on the nonlinearities and on the weight functions, simple applications of the maximum principle allow us to guarantee that if (u,v) is a solution of problem (P), then both *u* and *v* are strictly positive in Ω.