# Singular quasilinear elliptic systems in $${\mathbb {R}}^{N}$$

## Abstract

The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauder’s fixed point theorem.

## Introduction

In this paper, we consider the following system of quasilinear elliptic equations:

\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p_1} u=a_1(x)f(u,v) &{}\quad \text {in } \,\,{\mathbb {R}}^N, \\ -\Delta _{p_2} v=a_2(x)g(u,v) &{}\quad \text {in } \,\,{\mathbb {R}}^N, \\ u,v>0 &{}\quad \text {in } \,\,{\mathbb {R}}^N, \end{array}\right. \end{aligned}
(P)

where $$N\ge 3$$, $$1<p_i<N$$, while $$\Delta _{p_i}$$ denotes the $$p_i$$-Laplace differential operator. Nonlinearities $$f,g:{\mathbb {R}}^+\times {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$$ are continuous and fulfill the condition

• $$(\mathrm{H}_{f,g})$$ There exist $$m_i,M_i>0$$, $$i=1,2$$, such that

\begin{aligned}&m_1s^{\alpha _1}\le f(s,t)\le M_1s^{\alpha _1}(1+t^{\beta _1}),\\&m_2t^{\beta _2}\le g(s,t)\le M_2(1+s^{\alpha _2})t^{\beta _2} \end{aligned}

for all $$s,t\in {\mathbb {R}}^+$$, with $$-1<\alpha _1,\beta _2<0<\alpha _2,\beta _1$$,

\begin{aligned} \alpha _1+\alpha _2<p_1-1,\;\;\beta _1+\beta _2<p_2-1, \end{aligned}
(1.1)

as well as

\begin{aligned} \beta _1<\frac{p_2^*}{p_1^*}\min \{p_1-1, p_1^*-p_1\},\;\; \alpha _2<\frac{p_1^*}{p_2^*}\min \{p_2-1, p_2^*-p_2\}. \end{aligned}

Here, $$p^*_i$$ denotes the critical Sobolev exponent corresponding to $$p_i$$, namely $$p^*_i:=\frac{Np_i}{N-p_i}$$. Coefficients $$a_i:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ satisfy the assumption

• $$(\mathrm{H}_a)$$$$a_i(x)>0$$ a.e. in $${\mathbb {R}}^N$$ and $$a_i\in L^1({\mathbb {R}}^N)\cap L^{\zeta _i}({\mathbb {R}}^N)$$, where

\begin{aligned} \frac{1}{\zeta _1}\le 1-\frac{p_1}{p_1^*}-\frac{\beta _1}{p_2^*}\, ,\;\;\frac{1}{\zeta _2}\le 1- \frac{p_2}{p_2^*}- \frac{\alpha _2}{p_1^*}\, . \end{aligned}

Let $$\mathcal {D}^{1,p_i}({\mathbb {R}}^N)$$ be the closure of $$C_0^\infty ({\mathbb {R}}^N)$$ with respect to the norm

\begin{aligned} \Vert w\Vert _{\mathcal {D}^{1,p_i}({\mathbb {R}}^N)}:=\Vert \nabla w\Vert _{L^{p_i}({\mathbb {R}}^N)}. \end{aligned}

Recall [12, Theorem 8.3] that

\begin{aligned} \mathcal {D}^{1,p_i}({\mathbb {R}}^N)=\{w\in L^{p_i^*}({\mathbb {R}}^N):|\nabla w|\in L^{p_i}({\mathbb {R}}^N)\}. \end{aligned}

Moreover, if $$w\in \mathcal {D}^{1,p_i}({\mathbb {R}}^N)$$, then w vanishes at infinity, i.e., the set $$\{x\in {\mathbb {R}}^N: w(x)>k\}$$ has finite measure for all $$k>0$$; see [12, p. 201].

A pair $$(u,v)\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^{N})$$ is called a (weak) solution to (P) provided $$u,v>0$$ a.e. in $${\mathbb {R}}^N$$ and

\begin{aligned} \left\{ \begin{array}{ll} \int _{{\mathbb {R}}^N}|\nabla u|^{p_{1}-2}\nabla u\nabla \varphi \, \mathrm{d}x &{} =\int _{{\mathbb {R}}^N}a_{1}f(u,v)\varphi \, \mathrm{d}x,\\ \phantom {}\\ \int _{{\mathbb {R}}^N}|\nabla v|^{p_{2}-2}\nabla v\nabla \psi \, \mathrm{d}x &{} =\int _{{\mathbb {R}}^N}a_{2}g(u,v)\psi \, \mathrm{d}x \end{array}\right. \end{aligned}

for every $$(\varphi ,\psi )\in \mathcal {D}^{1,p_{1}}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_{2}}({\mathbb {R}}^N)$$.

The most interesting aspect of the work probably lies in the fact that both f and g can exhibit singularities through $${\mathbb {R}}^{N}$$, which, without loss of generality, are located at zero. Indeed, $$-1<\alpha _1,\beta _2<0$$ by $$(\mathrm{H}_{f,g})$$. It represents a serious difficulty to overcome and is rarely handled in the literature.

As far as we know, singular systems in the whole space have been investigated only for $$p:=q:=2$$, essentially exploiting the linearity of involved differential operators. In such a context, [3, 4, 17] treat the so-called Gierer–Meinhardt system, which arises from the mathematical modeling of important biochemical processes. Nevertheless, even in the semilinear case, (P) cannot be reduced to Gierer–Meinhardt’s case once $$(\mathrm{H}_{f,g})$$ is assumed. The situation looks quite different when a bounded domain takes the place of $${\mathbb {R}}^N$$: many singular systems fitting the framework of (P) have been studied, and meaningful contributions are already available [1, 6,7,8,9,10,11, 13,14,15,16].

Here, variational methods do not work, at least in a direct way, because the Euler function associated with problem (P) is not well defined. A similar comment holds for sub-super-solution techniques, which are usually employed in the case of bounded domains. Hence, we were naturally led to apply fixed point results. An a priori estimate in $$L^\infty ({\mathbb {R}}^N)\times L^\infty ({\mathbb {R}}^N)$$ for solutions of (P) is first established (cf. Theorem 3.4) by a Moser’s type iteration procedure and an adequate truncation, which, due to singular terms, require a specific treatment. We next perturb (P) by introducing a parameter $$\varepsilon >0$$. This produces the family of regularized systems

whose study yields useful information on the original problem. In fact, the previous $$L^\infty$$- boundedness still holds for solutions to ($$\mathrm{P}_{\varepsilon }$$), regardless of $$\varepsilon$$. Thus, via Schauder’s fixed point theorem, we get a solution $$(u_\varepsilon ,v_\varepsilon )$$ lying inside a rectangle given by positive lower bounds, where $$\varepsilon$$ does not appear, and positive upper bounds, that may instead depend on $$\varepsilon$$. Finally, letting $$\varepsilon \rightarrow 0^+$$ and using the $$(\mathrm{S})_+$$-property of the negative p-Laplacian in $$\mathcal {D}^{1,p}({\mathbb {R}}^N)$$ (see Lemma 3.3) yield a weak solution to (P); cf. Theorem 5.1.

The rest of this paper is organized as follows: Section 2 deals with preliminary results. An a priori estimate of solutions to (P) is proven in Sect. 3, while the next one treats system ($$\mathrm{P}_{\varepsilon }$$). Section 5 contains our existence result for problem (P).

## Preliminaries

Let $$\Omega \subseteq {\mathbb {R}}^N$$ be a measurable set, let $$t\in {\mathbb {R}}$$, and let $$w,z\in L^p({\mathbb {R}}^N)$$. We write $$m(\Omega )$$ for the Lebesgue measure of $$\Omega$$, while $$t^\pm :=\max \{\pm t,0\}$$, $$\Omega (w\le t):=\{x\in \Omega : w(x)\le t\}$$, $$\Vert w\Vert _p:=\Vert w\Vert _{L^p({\mathbb {R}}^N)}$$. The meaning of $$\Omega (w>t)$$, etc. is analogous. By definition, $$w\le z$$ iff $$w(x)\le z(x)$$ a.e. in $${\mathbb {R}}^N$$.

Given $$1\le q<p$$, neither $$L^p({\mathbb {R}}^N)\hookrightarrow L^q({\mathbb {R}}^N)$$ nor the reverse embedding holds true. However, the situation looks better for functions belonging to $$L^1({\mathbb {R}}^N)$$. Indeed (see also [2, p. 93]),

### Proposition 2.1

Suppose $$p>1$$ and $$w\in L^1({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)$$. Then $$w\in L^q({\mathbb {R}}^N)$$ whatever $$q\in \ ]1,p[$$.

### Proof

Thanks to Hölder’s inequality, with exponents p / q and $$p/(p-q)$$, and Chebyshev’s inequality, one has

\begin{aligned} \begin{aligned} \Vert w \Vert _q^q&=\int _{{\mathbb {R}}^N(|w|\le 1)}|w|^q\mathrm{d}x+\int _{{\mathbb {R}}^N(|w|>1)}|w|^q\mathrm{d}x\\&\le \int _{{\mathbb {R}}^N(|w|\le 1)}|w|\,\mathrm{d}x+\left( \int _{{\mathbb {R}}^N(|w|>1)}|w|^p\mathrm{d}x\right) ^{q/p} [m({\mathbb {R}}^N(|w|>1))]^{1-q/p}\\&\le \int _{{\mathbb {R}}^N}|w|\,\mathrm{d}x+\left( \int _{{\mathbb {R}}^N}|w|^p\mathrm{d}x\right) ^{q/p}\left( \int _{{\mathbb {R}}^N}|w|^p\mathrm{d}x\right) ^{1-q/p}\\&=\Vert w\Vert _1+\Vert w\Vert _p^p. \end{aligned} \end{aligned}

This completes the proof. $$\square$$

The summability properties of $$a_i$$ collected below will be exploited throughout the paper.

### Remark 2.1

Let assumption $$(\mathrm{H}_a)$$ be fulfilled. Then, for any $$i=1,2$$,

• $$(\mathrm{j}_1)$$$$a_i\in L^{(p_i^*)'}({\mathbb {R}}^N)$$.

• $$(\mathrm{j}_2)$$$$a_i\in L^{\gamma _i}({\mathbb {R}}^N)$$, where $$\gamma _i:=1/(1-t_i)$$, with

\begin{aligned} t_1:=\frac{\alpha _1+1}{p_1^*}+\frac{\beta _1}{p_2^*},\quad t_2:=\frac{\alpha _2}{p_1^*}+\frac{\beta _2+1}{p_2^*}. \end{aligned}
• $$(\mathrm{j}_3)$$$$a_i\in L^{\delta _i}({\mathbb {R}}^N)$$, for $$\delta _i:=1/(1-s_i)$$ and

\begin{aligned} s_1:=\frac{\alpha _1+1}{p_1^*},\quad s_2:=\frac{\beta _2+1}{p_2^*}. \end{aligned}
• $$(\mathrm{j}_4)$$$$a_i\in L^{\xi _i}({\mathbb {R}}^N)$$, where $$\xi _i\in \ ]p_i^*/(p_i^*-p_i),\zeta _i[$$.

To verify $$(\mathrm{j}_1)$$$$(\mathrm{j}_4)$$, we simply note that $$\zeta _i >\max \{(p_i^*)',\gamma _i,\delta _i,\xi _i\}$$ and apply Proposition 2.1.

Let us next show that the operator $$-\Delta _p$$ is of type $$(\mathrm{S})_+$$ in $$\mathcal {D}^{1,p}({\mathbb {R}}^N)$$.

### Proposition 2.2

If $$1<p<N$$ and $$\{u_n\}\subseteq \mathcal {D}^{1,p}({\mathbb {R}}^N)$$ satisfies

\begin{aligned}&u_n\rightharpoonup u\;\;\text {in}\;\;\mathcal {D}^{1,p}({\mathbb {R}}^N), \end{aligned}
(2.1)
\begin{aligned}&\limsup _{n\rightarrow \infty }\left\langle -\Delta _p u_n,u_n-u\right\rangle \le 0, \end{aligned}
(2.2)

then $$u_n\rightarrow u$$ in $$\mathcal {D}^{1,p}({\mathbb {R}}^N)$$.

### Proof

By monotonicity, one has

\begin{aligned} \left\langle -\Delta _p u_n-(-\Delta _p u),u_n-u\right\rangle \ge 0\quad \forall \, n\in \mathbb {N}, \end{aligned}

which evidently entails

\begin{aligned} \liminf _{n\rightarrow \infty }\left\langle -\Delta _p u_n-(-\Delta _p u),u_n-u\right\rangle \ge 0. \end{aligned}

Via (2.1)–(2.2), we then get

\begin{aligned} \limsup _{n\rightarrow \infty }\left\langle -\Delta _p u_n-(-\Delta _p u),u_n-u\right\rangle \le 0. \end{aligned}

Therefore,

\begin{aligned} \lim _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}\left( |\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u\right) (\nabla u_n-\nabla u)\, \mathrm{d}x=0. \end{aligned}
(2.3)

Since [18, Lemma A.0.5] yields

\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^N}\left( |\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u\right) (\nabla u_n-\nabla u)\, \mathrm{d}x \\&\qquad \ge \left\{ \begin{array}{ll} C_p\int _{{\mathbb {R}}^N}\frac{\vert \nabla (u_n-u)\vert ^2}{(|\nabla u_n|+|\nabla u|)^{2-p}}\, \mathrm{d}x &{} \quad \text { if }\,\,1<p<2,\\ \\ C_{p}\int _{{\mathbb {R}}^N}\vert \nabla (u_n-u)\vert ^p\, \mathrm{d}x &{} \quad \text {otherwise} \end{array} \right. \quad \forall \, n\in \mathbb {N}, \end{aligned} \end{aligned}

the desired conclusion, namely

\begin{aligned} \lim _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}|\nabla (u_n-u)|^p\, \mathrm{d}x=0, \end{aligned}

directly follows from (2.3) once $$p\ge 2$$. If $$1<p<2$$, then Hölder’s inequality and (2.1) lead to

\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}|\nabla (u_n-u)|^p\, \mathrm{d}x&=\int _{{\mathbb {R}}^N}\frac{|\nabla (u_n-u)|^p}{(|\nabla u_n|+|\nabla u|)^{\frac{p(2-p)}{2}}}\, (|\nabla u_n|+|\nabla u|)^{\frac{p(2-p)}{2}} \mathrm{d}x\\&\le \left( \int _{{\mathbb {R}}^N}\frac{|\nabla (u_n-u)|^2}{(|\nabla u_n|+|\nabla u|)^{2-p}}\, \mathrm{d}x\right) ^{\frac{p}{2}} \left( \int _{{\mathbb {R}}^N}(|\nabla u_n|+|\nabla u|)^p \mathrm{d}x\right) ^{\frac{2-p}{2}}\\&\le C\left( \int _{{\mathbb {R}}^N}\frac{|\nabla (u_n-u)|^2}{(|\nabla u_n|+|\nabla u|)^{2-p}}\, \mathrm{d}x\right) ^{\frac{p}{2}},\quad n\in \mathbb {N}, \end{aligned} \end{aligned}

with appropriate $$C>0$$. Now, the argument goes on as before. $$\square$$

## Boundedness of solutions

The main result of this section, Theorem 3.4 below, provides an $$L^\infty ({\mathbb {R}}^N)$$—a priori estimate for weak solutions to (P). Its proof will be performed into three steps.

### Lemma 3.1

($$L^{p_i^*}({\mathbb {R}}^N)$$—uniform boundedness) There exists $$\rho >0$$ such that

\begin{aligned} \max \left\{ \Vert u\Vert _{p_1^*},\,\Vert v\Vert _{p_2^*}\right\} \le \rho \end{aligned}
(3.1)

for every $$(u,v)\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$ solving problem (P).

### Proof

Multiply both equations in (P) by u and v, respectively, integrate over $${\mathbb {R}}^N$$, and use $$(\mathrm{H}_{f,g})$$ to arrive at

\begin{aligned} \Vert \nabla u\Vert _{p_1}^{p_1}= & {} \int _{{\mathbb {R}}^N} a_1 f(u,v)u\, \mathrm{d}x\le M_1\int _{{\mathbb {R}}^N} a_1 u^{\alpha _1+1}(1+v^{\beta _1})\, \mathrm{d}x, \\ \Vert \nabla v\Vert _{p_2}^{p_2}= & {} \int _{{\mathbb {R}}^N} a_2 g(u,v)v\, \mathrm{d}x\le M_2\int _{{\mathbb {R}}^N} a_2 (1+u^{\alpha _2})v^{\beta _2+1}\, \mathrm{d}x. \end{aligned}

Through the embedding $$\mathcal {D}^{1,p_i}({\mathbb {R}}^N)\hookrightarrow L^{p_i^*}({\mathbb {R}}^N)$$, besides Hölder’s inequality, we obtain

\begin{aligned} \Vert \nabla u\Vert _{p_1}^{p_1}\le & {} M_1\left( \Vert a_1\Vert _{\delta _1}\Vert u\Vert _{p_1^*}^{\alpha _1+1} +\Vert a_1\Vert _{\gamma _1}\Vert u\Vert _{p_1^*}^{\alpha _1+1}\Vert v\Vert _{p_2^*}^{\beta _1}\right) \\\le & {} C_1\Vert \nabla u\Vert _{p_1}^{\alpha _1+1}\left( \Vert a_1\Vert _{\delta _1}+\Vert a_1\Vert _{\gamma _1} \Vert \nabla v\Vert _{p_2}^{\beta _1}\right) ; \end{aligned}

cf. also Remark 2.1. Likewise,

\begin{aligned} \Vert \nabla v\Vert _{p_2}^{p_2}\le C_2\Vert \nabla v\Vert _{p_2}^{\beta _2+1}\left( \Vert a_2\Vert _{\delta _2} +\Vert a_2\Vert _{\gamma _2}\Vert \nabla u\Vert _{p_1}^{\alpha _2}\right) . \end{aligned}

Thus, a fortiori,

\begin{aligned} \Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1}\le & {} C_1\left( \Vert a_1\Vert _{\delta _1}+\Vert a_1\Vert _{\gamma _1} \Vert \nabla v\Vert _{p_2}^{\beta _1}\right) , \nonumber \\ \Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _2}\le & {} C_2\left( \Vert a_2\Vert _{\delta _2}+\Vert a_2\Vert _{\gamma _2} \Vert \nabla u\Vert _{p_1}^{\alpha _2}\right) , \end{aligned}
(3.2)

which imply

\begin{aligned} \begin{aligned}&\Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1}+\Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _2}\\&\quad \le C_1\left( \Vert a_1\Vert _{\delta _1}+\Vert a_1\Vert _{\gamma _1}\Vert \nabla v\Vert _{p_2}^{\beta _1}\right) +C_2\left( \Vert a_2\Vert _{\delta _2}+\Vert a_2\Vert _{\gamma _2}\Vert \nabla u\Vert _{p_1}^{\alpha _2}\right) . \end{aligned} \end{aligned}

Rewriting this inequality as

\begin{aligned} \begin{aligned}&\Vert \nabla u\Vert _{p_1}^{\alpha _2} \left( \Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1-\alpha _2}-C_2\Vert a_2\Vert _{\gamma _2}\right) +\,\Vert \nabla v\Vert _{p_2}^{\beta _1}\left( \Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _1-\beta _2}-C_1\Vert a_1\Vert _{\gamma _1}\right) \\&\quad \quad \qquad \le C_1\Vert a_1\Vert _{\delta _1}+C_2\Vert a_2\Vert _{\delta _2}, \end{aligned} \end{aligned}
(3.3)

four situations may occur. If

\begin{aligned} \Vert \nabla u \Vert _{p_1}^{p_1-1-\alpha _1-\alpha _2}\le C_2\Vert a_2 \Vert _{\gamma _2}\, ,\quad \Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _1-\beta _2}\le C_1\Vert a_1\Vert _{\gamma _1} \end{aligned}

then (3.1) follows from $$(\mathrm{j}_2)$$ of Remark 2.1, conditions (1.1), and the embedding $$\mathcal {D}^{1,p_i}({\mathbb {R}}^N)\hookrightarrow L^{p_i^*}({\mathbb {R}}^N)$$. Assume next that

\begin{aligned} \Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1-\alpha _2}>C_2\Vert a_2\Vert _{\gamma _2}\, ,\quad \Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _1-\beta _2}>C_1\Vert a_1\Vert _{\gamma _1}\, . \end{aligned}
(3.4)

Thanks to (3.3), one has

\begin{aligned} \Vert \nabla u\Vert _{p_1}^{\alpha _2}(\Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1-\alpha _2}-C_2\Vert a_2\Vert _{\gamma _2})\le C_1\Vert a_1\Vert _{\delta _1}+C_2\Vert a_2\Vert _{\delta _2}, \end{aligned}

whence, on account of (3.4),

\begin{aligned} \begin{aligned} \Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1-\alpha _2}&\le \frac{C_1\Vert a_1\Vert _{\delta _1}+C_2\Vert a_2\Vert _{\delta _2}}{\Vert \nabla u\Vert _{p_1}^{\alpha _2}} +C_2\Vert a_2\Vert _{\gamma _2} \\&\le \frac{C_1\Vert a_1\Vert _{\delta _1}+C_2\Vert a_2\Vert _{\delta _2}}{\Vert a_2\Vert _{\gamma _2}^{\frac{\alpha _2}{p_1-1-\alpha _1-\alpha _2}}}+C_2\Vert a_2\Vert _{\gamma _2}. \end{aligned} \end{aligned}

A similar inequality holds true for v. So, (3.1) is achieved reasoning as before. Finally, if

\begin{aligned} \Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1-\alpha _2}\le C_2\Vert a_2 \Vert _{\gamma _2}\, ,\quad \Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _1-\beta _2}> C_1 \Vert a_1\Vert _{\gamma _1} \end{aligned}
(3.5)

then (3.2) and (3.5) entail

\begin{aligned} \Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _2}\le C_2\left[ \Vert a_2 \Vert _{\delta _2}+ \Vert a_2 \Vert _{\gamma _2} \left( C_2\Vert a_2\Vert _{\gamma _2}\right) ^{\frac{\alpha _2}{p_1-1-\alpha _1-\alpha _2}}\right] . \end{aligned}

By (1.1) again, we thus get

\begin{aligned} \max \{\Vert \nabla u\Vert _{p_1}, \Vert \nabla v\Vert _{p_2}\}\le C_3, \end{aligned}

where $$C_3>0$$. This yields (3.1), because $$\mathcal {D}^{1,p_i}({\mathbb {R}}^N)\hookrightarrow L^{p_i^*}({\mathbb {R}}^N)$$. The last case, i.e.,

\begin{aligned} \Vert \nabla u\Vert _{p_1}^{p_1-1-\alpha _1-\alpha _2}>C_2\Vert a_2 \Vert _{\gamma _2}\, ,\quad \Vert \nabla v\Vert _{p_2}^{p_2-1-\beta _1-\beta _2}\le C_1 \Vert a_1\Vert _{\gamma _1} \end{aligned}

is analogous. $$\square$$

To shorten notation, write

\begin{aligned} {\mathcal {D}}^{1,p_i}({\mathbb {R}}^N)_+:=\{ w\in {\mathcal {D}}^{1,p_i}({\mathbb {R}}^N): w\ge 0\;\text { a.e. in }\;{\mathbb {R}}^N\}. \end{aligned}

### Lemma 3.2

(Truncation) Let $$(u,v)\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$ be a weak solution of (P). Then

\begin{aligned} \int _{{\mathbb {R}}^N(u>1)}\vert \nabla u\vert ^{p_1-2}\nabla u\nabla \varphi \, \mathrm{d}x\le M_1\int _{{\mathbb {R}}^N(u>1)} a_1(1+v^{\beta _1})\varphi \, \mathrm{d}x, \end{aligned}
(3.6)
\begin{aligned} \int _{{\mathbb {R}}^N(v>1)}\vert \nabla v\vert ^{p_2-2}\nabla v\nabla \psi \, \mathrm{d}x\le M_2\int _{{\mathbb {R}}^N(v>1)} a_2(1+u^{\alpha _2})\psi \, \mathrm{d}x \end{aligned}
(3.7)

for all $$(\varphi ,\psi )\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)_+\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)_+$$.

### Proof

Pick a $$C^1$$ cutoff function $$\eta :{\mathbb {R}}\rightarrow [0,1]$$ such that

\begin{aligned} \eta (t)=\left\{ \begin{array}{ll} 0 &{}\quad \text { if }\,\,t\le 0, \\ 1 &{}\quad \text { if }\,\,t\ge 1, \end{array} \right. \quad \eta '(t)\ge 0\quad \forall \, t\in [0,1], \end{aligned}

and, given $$\delta >0$$, define $$\eta _\delta (t):=\eta \left( \frac{t-1}{\delta }\right)$$. If $$w\in \mathcal {D}^{1,p_i}({\mathbb {R}}^N)$$, then

\begin{aligned} \eta _{\delta }\circ w\in {\mathcal {D}}^{1,p_i}({\mathbb {R}}^N),\quad \nabla (\eta _{\delta }\circ w)=(\eta _{\delta }'\circ w)\nabla w, \end{aligned}
(3.8)

as a standard verification shows.

Now, fix $$(\varphi ,\psi )\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)_+\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)_+$$. Multiply the first equation in (P) by $$(\eta _\delta \circ u)\varphi$$, integrate over $${\mathbb {R}}^N$$ and use $$(\mathrm{H}_{f,g})$$ to achieve

\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla u|^{p_1-2}\nabla u\nabla ((\eta _\delta \circ u)\varphi )\, \mathrm{d}x \le M_1\int _{{\mathbb {R}}^N} a_1u^{\alpha _1}(1+v^{\beta _1})(\eta _\delta \circ u)\varphi \, \mathrm{d}x. \end{aligned}

By (3.8), we have

\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}&|\nabla u|^{p_1-2}\nabla u\nabla ((\eta _\delta \circ u)\varphi )\, \mathrm{d}x\\&=\int _{{\mathbb {R}}^N}|\nabla u|^{p_1}(\eta _\delta '\circ u)\varphi \,\mathrm{d}x +\int _{{\mathbb {R}}^N}(\eta _\delta \circ u)|\nabla u|^{p_1-2}\nabla u\nabla \varphi \,\mathrm{d}x, \end{aligned} \end{aligned}

while $$\eta _\delta '\circ u\ge 0$$ in $${\mathbb {R}}^N$$. Therefore,

\begin{aligned} \int _{{\mathbb {R}}^N}(\eta _\delta \circ u)|\nabla u|^{p_1-2}\nabla u\nabla \varphi \,\mathrm{d}x\le M_1\int _{{\mathbb {R}}^N} a_1u^{\alpha _1}(1+v^{\beta _1})(\eta _\delta \circ u)\varphi \, \mathrm{d}x. \end{aligned}

Letting $$\delta \rightarrow 0^+$$ produces (3.6). The proof of (3.7) is similar. $$\square$$

### Lemma 3.3

(Moser’s iteration) There exists $$R>0$$ such that

\begin{aligned} \max \{\Vert u\Vert _{L^\infty (\Omega _1)},\Vert v\Vert _{L^\infty (\Omega _2)}\}\le R, \end{aligned}
(3.9)

where

\begin{aligned} \Omega _1:={\mathbb {R}}^N(u>1)\quad \text {and}\quad \Omega _2:={\mathbb {R}}^N(v>1), \end{aligned}

for every $$(u,v)\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$ solving problem (P).

### Proof

Given $$w\in L^p(\Omega _1)$$, we shall write $$\Vert w\Vert _p$$ in place of $$\Vert w\Vert _{L^p(\Omega _1)}$$ when no confusion can arise. Observe that $$m(\Omega _1)<+\infty$$ and define, provided $$M>1$$,

\begin{aligned} u_M(x):=\min \{u(x),M\},\quad x\in {\mathbb {R}}^N. \end{aligned}

Choosing $$\varphi :=u_M^{\kappa p_1+1}$$, with $$\kappa \ge 0$$, in (3.6) gives

\begin{aligned}&(\kappa p_1+1)\int _{\Omega _1(u\le M)} u_M^{\kappa p_1}\vert \nabla u\vert ^{p_1-2}\nabla u\nabla u_M\, \mathrm{d}x \nonumber \\&\qquad \le M_1\int _{\Omega _1} a_1(1+v^{\beta _1}) u_M^{\kappa p_1+1}\,\mathrm{d}x. \end{aligned}
(3.10)

Through the Sobolev embedding theorem, one has

\begin{aligned} \begin{aligned}&(\kappa p_1+1)\int _{\Omega _1(u\le M)} u_M^{\kappa p_1}\vert \nabla u\vert ^{p_1-2}\nabla u\nabla u_M\, \mathrm{d}x \\&\quad =(\kappa p_1+1)\int _{\Omega _1(u\le M)}(|\nabla u|u^{\kappa })^{p_1}\mathrm{d}x =\frac{\kappa p_1+1}{(\kappa +1)^{p_1}}\int _{\Omega _1(u\le M)}|\nabla u^{\kappa +1}|^{p_1} \mathrm{d}x \\&\quad =\frac{\kappa p_1+1}{(\kappa +1)^{p_1}}\int _{\Omega _1}|\nabla u_M^{\kappa +1}|^{p_1}\mathrm{d}x \ge C_1\frac{\kappa p_1+1}{(\kappa +1)^{p_1}}\Vert u_M^{\kappa +1}\Vert _{p_1^*}^{p_1} \end{aligned} \end{aligned}

for appropriate $$C_1>0$$. By Remark 2.1, Hölder’s inequality entails

\begin{aligned} \begin{aligned} \int _{\Omega _1} a_1(1+v^{\beta _1})u_M^{\kappa p_1+1} \mathrm{d}x&\le \int _{\Omega _1}a_1(1+v^{\beta _1})u^{\kappa p_1+1}\mathrm{d}x\\&\le \left( \Vert a_1 \Vert _{\xi _1}+ \Vert a_1 \Vert _{\zeta _1} \Vert v \Vert _{p_2^*}^{\beta _1}\right) \Vert u \Vert _{(\kappa p_1+ 1) \xi _1'}^{\kappa p_1+ 1}. \end{aligned} \end{aligned}

Hence, (3.10) becomes

\begin{aligned} \frac{\kappa p_1+1}{(\kappa +1)^{p_1}}\Vert u_M^{\kappa +1}\Vert _{p_1^*}^{p_1}\le C_2\left( \Vert a_1\Vert _{\xi _1}+ \Vert a_1\Vert _{\zeta _1}\Vert v\Vert _{p_2^*}^{\beta _1}\right) \Vert u\Vert _{(\kappa p_1+1)\xi _1'}^{\kappa p_1+1}. \end{aligned}

Since $$u(x)=\displaystyle {\lim _{M\rightarrow \infty }}u_M(x)$$ a.e. in $${\mathbb {R}}^N$$, using the Fatou lemma we get

\begin{aligned} \frac{\kappa p_1+1}{(\kappa +1)^{p_1}}\Vert u\Vert _{(\kappa +1)p_1^*}^{(\kappa +1)p_1} \le C_2\left( \Vert a_1\Vert _{\xi _1}+\Vert a_1\Vert _{\zeta _1}\Vert v\Vert _{p_2^*}^{\beta _1}\right) \Vert u\Vert _{(\kappa p_1+1)\xi _1'}^{\kappa p_1+1}, \end{aligned}

namely

\begin{aligned} \Vert u\Vert _{(\kappa +1)p_1^*}\le C_3^{\eta (\kappa )}\sigma (\kappa ) \left( 1+\Vert v\Vert _{p_2^*}^{\beta _1}\right) ^{\eta (\kappa )} \Vert u\Vert _{(\kappa p_1+1)\xi _1'}^{\frac{\kappa p_1+1}{(\kappa +1)p_1}}, \end{aligned}
(3.11)

where $$C_3>0$$, while

\begin{aligned} \eta (\kappa ):=\frac{1}{(\kappa +1)p_1},\quad \sigma (\kappa ):=\left[ \frac{\kappa +1}{(\kappa p_1+1)^{1/p_1}}\right] ^{\frac{1}{\kappa +1}}. \end{aligned}

Let us next verify that

\begin{aligned} (\kappa +1) p_1^* >(\kappa p_1+1) \xi _1'\quad \forall \, \kappa \in {\mathbb {R}}^+_0, \end{aligned}

which clearly means

\begin{aligned} \frac{1}{\xi _1}<1-\frac{\kappa p_1+ 1}{(\kappa + 1) p_1^*},\quad \kappa \in {\mathbb {R}}^+_0\, . \end{aligned}
(3.12)

Indeed, the function $$\kappa \mapsto \frac{\kappa p_1+1}{(\kappa + 1) p_1^*}$$ is increasing on $${\mathbb {R}}^+_0$$ and tends to $$\frac{p_1}{p_1^*}$$ as $$k\rightarrow \infty$$. So, (3.12) holds true, because $$\frac{1}{\xi _1}<1-\frac{p_1}{p_1^*}$$; see Remark 2.1. Now, Moser’s iteration can start. If there exists a sequence $$\{\kappa _n\}\subseteq {\mathbb {R}}^+_0$$ fulfilling

\begin{aligned} \lim _{n\rightarrow \infty }\kappa _n=+\infty ,\quad \Vert u\Vert _{(\kappa _n+1)p_1^*}\le 1\;\;\forall \, n\in {\mathbb {N}}\end{aligned}

then $$\Vert u\Vert _{L^\infty (\Omega _1)}\le 1$$. Otherwise, with appropriate $$\kappa _0>0$$, one has

\begin{aligned} \Vert u\Vert _{(\kappa + 1)p_1^*}>1\;\;\text {for any}\;\;\kappa >\kappa _0,\;\;\text {besides}\;\;\Vert u \Vert _{(\kappa _0+ 1) p_1^*} \le 1. \end{aligned}
(3.13)

Inequality (3.12) evidently forces $$\frac{\kappa _0 p_1+ 1}{(\kappa _0+ 1) p_1^*}<\frac{1}{\xi _1'}$$. Pick $$\kappa _1>\kappa _0$$ such that $$(\kappa _1 p_1+1)\xi _1'=(\kappa _0+1) p_1^*$$, set $$\kappa :=\kappa _1$$ in (3.11), and use (3.13) to arrive at

\begin{aligned} \Vert u\Vert _{(\kappa _1 +1)p_1^*}\le & {} C_3^{\eta (\kappa _1)}\sigma (\kappa _1)\left( 1+\Vert v\Vert _{p_2^*}^{\beta _1}\right) ^{\eta (\kappa _1)} \Vert u\Vert _{(\kappa _0+1)p_1^*}^{\frac{\kappa _1 p_1+1}{(\kappa _1+1)p_1}}\nonumber \\\le & {} C_3^{\eta (\kappa _1)}\sigma (\kappa _1)\left( 1+\Vert v\Vert _{p_2^*}^{\beta _1}\right) ^{\eta (\kappa _1)}. \end{aligned}
(3.14)

Choose next $$\kappa _2>\kappa _0$$ satisfying $$(\kappa _2 p_1+1)\xi _1'= (\kappa _1+1)p_1^*$$. From (3.11), written for $$\kappa := \kappa _2$$, as well as (3.13)–(3.14), it follows

\begin{aligned} \begin{aligned} \Vert u\Vert _{(\kappa _2 +1)p_1^*}&\le C_3^{\eta (\kappa _2)}\sigma (\kappa _2)\left( 1+\left\| v\right\| _{p_2^*}^{\beta _1}\right) )^{\eta (\kappa _2)} \Vert u\Vert _{(\kappa _1+1)p_1^*}^{\frac{\kappa _2 p_1+1}{(\kappa _2 +1)p_1}}\\&\le C_3^{\eta (\kappa _2)}\sigma (\kappa _2)\left( 1+\left\| v\right\| _{p_2^*}^{\beta _1}\right) ^{\eta (\kappa _2)} \Vert u\Vert _{(\kappa _1+1)p_1^*} \\&\le C_3^{\eta (\kappa _2)+\eta (\kappa _1)}\sigma (\kappa _2)\sigma (\kappa _1)\left( 1+\Vert v\Vert _{p_2^*}^{\beta _1}\right) ^{\eta (\kappa _2)+\eta (\kappa _1)}. \end{aligned} \end{aligned}

By induction, we construct a sequence $$\{\kappa _n\}\subseteq (\kappa _0,+\infty )$$ enjoying the properties below:

\begin{aligned}&(\kappa _n p_1+ 1)\xi _1'=(\kappa _{n-1}+ 1)p_1^*\, ,\quad n\in {\mathbb {N}}; \end{aligned}
(3.15)
\begin{aligned}&\Vert u\Vert _{(k_n+1)p_1^*}\le C_3^{\sum _{i=1}^{n}\eta (\kappa _i)} \prod _{i=1}^{n}\sigma (\kappa _i) \left( 1+\Vert v\Vert _{p_2^*}^{\beta _1}\right) ^{\sum _{i=1}^{n}\eta (\kappa _i)} \end{aligned}
(3.16)

for all $$n\in {\mathbb {N}}$$. A simple computation based on (3.15) yields

\begin{aligned} \kappa _n+1=(\kappa _0+1)\left( \frac{p^*_1}{p_1\xi '_1}\right) ^n +\frac{1}{p'_1}\sum _{i=0}^{n-1}\left( \frac{p^*_1}{p_1\xi '_1}\right) ^i, \end{aligned}

where $$\frac{p_1^*}{p_1\xi _1'}>1$$ due to $$(\mathrm{j}_4)$$ of Remark 2.1. Hence,

\begin{aligned} \kappa _n+1\simeq C^* \left( \frac{p_1^*}{p_1 \xi _1'}\right) ^n\;\;\text {as}\;\; n\rightarrow \infty , \end{aligned}
(3.17)

with appropriate $$C^*>0$$. Further, if $$C_4>0$$ satisfies

\begin{aligned} 1<\left[ \frac{t+1}{(t p_1+1)^{1/p_1}}\right] ^{\frac{1}{\sqrt{t +1}}}\le C_4\, ,\quad t\in {\mathbb {R}}^+_0, \end{aligned}

(cf. [5, p. 116]), then

\begin{aligned} \prod _{i=1}^{n}\sigma (\kappa _i)= & {} \prod _{i=1}^{n}\left[ \frac{\kappa _i+1}{(\kappa _i p_1+1)^{1/p_1}}\right] ^{\frac{1}{\kappa _i +1}}\\= & {} \prod _{i=1}^{n}\left\{ \left[ \frac{\kappa _i+1}{(\kappa _i p_1+1)^{1/p_1}}\right] ^{\frac{1}{\sqrt{\kappa _i +1}}} \right\} ^{\frac{1}{\sqrt{\kappa _i +1}}}\le C_4^{\sum _{i=1}^{n}\frac{1}{\sqrt{\kappa _i +1}}}. \end{aligned}

Consequently, (3.16) becomes

\begin{aligned} \Vert u\Vert _{(k_n+1)p_1^*} \le C_3^{\sum _{i=1}^{n}\eta (\kappa _i)} C_4^{\sum _{i=1}^{n} \frac{1}{\sqrt{\kappa _i+1}}} \left( 1+\Vert v \Vert _{p_2^*}^{\beta _1}\right) ^{\sum _{i=1}^{n}\eta (\kappa _i)}. \end{aligned}

Since, by (3.17), both $$\kappa _n+1\rightarrow +\infty$$ and $$\frac{1}{\kappa _n+1}\simeq \frac{1}{C^*} \left( \frac{p_1\xi _1'}{p_1^*}\right) ^n$$, while (3.1) entails $$\Vert v\Vert _{p^*_2}\le \rho$$, there exists a constant $$C_5>0$$ such that

\begin{aligned} \Vert u\Vert _{(\kappa _n+1)p_1^*}\le C_5\quad \forall \, n\in {\mathbb {N}}, \end{aligned}

whence $$\Vert u\Vert _{L^\infty (\Omega _1)}\le C_5$$. Thus, in either case, $$\Vert u\Vert _{L^\infty (\Omega _1)}\le R$$, with $$R:=\max \{1,C_5\}$$. A similar argument applies to v. $$\square$$

Using (3.9), besides the definition of sets $$\Omega _i$$, we immediately infer the following

### Theorem 3.4

Under assumptions $$(\mathrm{H}_{f,g})$$ and $$(\mathrm{H}_a)$$, one has

\begin{aligned} \max \{\Vert u\Vert _\infty ,\Vert v\Vert _\infty \}\le R \end{aligned}
(3.18)

for every weak solution $$(u,v)\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^{N})$$ to problem (P). Here, R is given by Lemma 3.3.

## The regularized system

Assertion $$(\mathrm{j}_1)$$ of Remark 2.1 ensures that $$a_i\in L^{(p_i^*)'}({\mathbb {R}}^N)$$. Therefore, thanks to Minty–Browder’s theorem [2, Theorem V.16], the equation

\begin{aligned} -\Delta _{p_i} w_i=a_i(x)\quad \text {in}\quad {\mathbb {R}}^N \end{aligned}
(4.1)

possesses a unique solution $$w_i\in \mathcal {D}^{1,p_i}({\mathbb {R}}^N)$$, $$i=1,2$$. Moreover,

• $$w_i>0$$, and

• $$w_i\in L^\infty ({\mathbb {R}}^N)$$.

Indeed, testing (4.1) with $$\varphi :=w_i^-$$ yields $$w_i\ge 0$$, because $$a_i>0$$ by $$(\mathrm{H}_a)$$. Through the strong maximum principle, we obtain

\begin{aligned} {\hbox {ess inf}}_{B_r(x)} w_i>0\;\;\text {for any } r> 0,\, x\in {\mathbb {R}}^N. \end{aligned}

Hence, $$w_i>0$$. Moser’s iteration technique then produces $$w_i\in L^\infty ({\mathbb {R}}^N)$$.

Next, fix $$\varepsilon \in \ ]0,1[$$ and define

\begin{aligned} (\underline{u},\underline{v})= & {} \left( [m_1(R+1)^{\alpha _1}]^{\frac{1}{p_1-1}}w_1, [m_2(R+1)^{\beta _2}]^{\frac{1}{p_2-1}}w_2\right) ,\nonumber \\ (\overline{u}_\varepsilon ,\overline{v}_\varepsilon )= & {} \left( [ M_1\varepsilon ^{\alpha _1}(1+R^{\beta _1})]^{\frac{1}{p_1-1}}w_1, [M_2\varepsilon ^{\beta _2}(1+R^{\alpha _2})]^{\frac{1}{p_2-1}}w_2\right) , \end{aligned}
(4.2)

where $$R>0$$ comes from Lemma 3.3, as well as

\begin{aligned} \mathcal {K}_\varepsilon :=\left\{ (z_1,z_2)\in L^{p_1^*}({\mathbb {R}}^N)\times L^{p_2^*}({\mathbb {R}}^N):\underline{u}\le z_1\le \overline{u}_\varepsilon \, ,\;\underline{v}\le z_2\le \overline{v}_\varepsilon \right\} . \end{aligned}

Obviously, $$\mathcal {K}_\varepsilon$$ is bounded, convex, closed in $$L^{p_1^*}({\mathbb {R}}^N)\times L^{p_2^*}({\mathbb {R}}^N)$$. Given $$(z_1,z_2)\in \mathcal {K}_\varepsilon$$, write

\begin{aligned} \tilde{z}_i:=\min \{z_i, R\},\quad i=1,2. \end{aligned}
(4.3)

Since, on account of (4.3), hypothesis $$(\mathrm{H}_{f,g})$$ entails

\begin{aligned} a_1 m_1 (R+1)^{\alpha _1}\le & {} a_1 f(\tilde{z}_1+\varepsilon ,\tilde{z}_2)\le a_1 M_1\varepsilon ^{\alpha _1}(1+R^{\beta _1}),\nonumber \\ a_2 m_2(R+1)^{\beta _2}\le & {} a_2 g(\tilde{z}_1,\tilde{z}_2+\varepsilon )\le a_2 M_2 (1+R^{\alpha _2})\varepsilon ^{\beta _2}, \end{aligned}
(4.4)

while, recalling Remark 2.1, $$a_i\in L^{(p_i^*)'}({\mathbb {R}}^N)$$, the functions

\begin{aligned} x\mapsto a_1(x) f(\tilde{z}_1(x)+\varepsilon ,\tilde{z}_2(x)),\quad x\mapsto a_2(x) g(\tilde{z}_1(x),\tilde{z}_2(x)+\varepsilon ) \end{aligned}

belong to $$\mathcal {D}^{-1,p_1'}({\mathbb {R}}^N)$$ and $$\mathcal {D}^{-1,p_2'}({\mathbb {R}}^N)$$, respectively. Consequently, by Minty–Browder’s theorem again, there exists a unique weak solution $$(u_\varepsilon ,v_\varepsilon )$$ of the problem

\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p_1} u=a_1(x) f(\tilde{z}_1(x)+\varepsilon ,\tilde{z}_2(x)) &{}\quad \text {in }\,\,{\mathbb {R}}^N, \\ -\Delta _{p_2} v=a_2(x) g(\tilde{z}_1(x),\tilde{z}_2(x)+\varepsilon ) &{}\quad \text {in }\,\,{\mathbb {R}}^N, \\ u_\varepsilon ,v_\varepsilon >0 &{}\quad \text {in }\,\,{\mathbb {R}}^N. \end{array} \right. \end{aligned}
(4.5)

Let $$\mathcal {T}:\mathcal {K}_\varepsilon \rightarrow L^{p_1^*}({\mathbb {R}}^N)\times L^{p_2^*}({\mathbb {R}}^N)$$ be defined by $$\mathcal {T}(z_1,z_2)=(u_\varepsilon ,v_\varepsilon )$$ for every $$(z_1,z_2)\in \mathcal {K}_\varepsilon$$.

### Lemma 4.1

One has $$\underline{u}\le u_\varepsilon \le \overline{u}_\varepsilon$$ and $$\underline{v}\le v_\varepsilon \le \overline{v}_\varepsilon$$. So, in particular, $$\mathcal {T}(\mathcal {K}_\varepsilon )\subseteq \mathcal {K}_\varepsilon$$.

### Proof

Via (4.2), (4.1), (4.5), and (4.4), we get

\begin{aligned} \begin{aligned}&\langle -\Delta _{p_1}\underline{u}-(-\Delta _{p_1}u_\varepsilon ),(\underline{u}-u_\varepsilon )^+\rangle \\&\quad =\langle -\Delta _{p_1}[m_1 (R+1)^{\alpha _1}]^{\frac{1}{p_1-1}}w_1-(-\Delta _{p_1}u_\varepsilon ), (\underline{u}-u_\varepsilon )^+\rangle \\&\quad =\int _{{\mathbb {R}}^N} a_1\left( (m_1(R+1)^{\alpha _1}-f(\tilde{z}_1+\varepsilon ,\tilde{z}_2)\right) (\underline{u}-u_\varepsilon )^+ \mathrm{d}x\le 0, \end{aligned} \end{aligned}

while Lemma A.0.5 of  furnishes

\begin{aligned} \begin{aligned}&\langle -\Delta _{p_1}\underline{u}-(-\Delta _{p_1}u_\varepsilon ),(\underline{u}-u_\varepsilon )^+\rangle \\&\quad =\int _{{\mathbb {R}}^N}\left( |\nabla \underline{u}|^{p_1-2}\nabla \underline{u}-|\nabla u_\varepsilon |^{p_1-2}\nabla u_\varepsilon \right) \nabla (\underline{u}-u_\varepsilon )^+ \mathrm{d}x\ge 0. \end{aligned} \end{aligned}

Now, arguing as in the proof of Proposition 2.2, one has $$(\underline{u}-u_\varepsilon )^+=0$$, i.e., $$\underline{u}\le u_\varepsilon$$. The remaining inequalities can be verified similarly. $$\square$$

### Lemma 4.2

The operator $$\mathcal {T}$$ is continuous and compact.

### Proof

Pick a sequence $$\{(z_{1,n},z_{2,n})\}\subseteq \mathcal {K}_\varepsilon$$ such that

\begin{aligned} (z_{1,n},z_{2,n})\rightarrow (z_1,z_2)\quad \text {in}\quad L^{p_1^*}({\mathbb {R}}^N)\times L^{p_2^*}({\mathbb {R}}^N). \end{aligned}

If $$(u_n,v_n):=\mathcal {T}(z_{1,n},z_{2,n})$$ and $$(u,v):=\mathcal {T}(z_1,z_2)$$, then

\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla u_n|^{p_1-2}\nabla u_n\nabla \varphi \, \mathrm{d}x= & {} \int _{{\mathbb {R}}^N } a_1 f(\tilde{z}_{1,n}+\varepsilon ,\tilde{z}_{2,n})\varphi \, \mathrm{d}x,\end{aligned}
(4.6)
\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla v_n|^{p_2-2}\nabla v_n\nabla \psi \, \mathrm{d}x= & {} \int _{{\mathbb {R}}^N} a_2 g(\tilde{z}_{1,n},\tilde{z}_{2,n}+\varepsilon )\psi \, \mathrm{d}x,\\ \int _{{\mathbb {R}}^N}|\nabla u|^{p_1-2}\nabla u\nabla \varphi \, \mathrm{d}x= & {} \int _{{\mathbb {R}}^N} a_1 f(\tilde{z}_1+\varepsilon ,\tilde{z}_2)\varphi \, \mathrm{d}x,\nonumber \\ \int _{{\mathbb {R}}^N}|\nabla v|^{p_2-2}\nabla v\nabla \psi \, \mathrm{d}x= & {} \int _{{\mathbb {R}}^N} a_2 g(\tilde{z}_1,\tilde{z}_2+\varepsilon )\psi \, \mathrm{d}x\nonumber \end{aligned}
(4.7)

for every $$(\varphi ,\psi )\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$. Set $$\varphi :=u_n$$ in (4.6). From (4.4), it follows after using Hölder’s inequality,

\begin{aligned} \begin{aligned} \Vert \nabla u_n\Vert _{p_1}^{p_1}&=\int _{{\mathbb {R}}^N} a_1 f(\tilde{z}_{1,n}+\varepsilon ,\tilde{z}_{2,n}) u_n\, \mathrm{d}x \\&\le M_1\int _{{\mathbb {R}}^N} a_1 \varepsilon ^{\alpha _1}(1+R^{\beta _1})u_n\, \mathrm{d}x\le C_\varepsilon \int _{{\mathbb {R}}^N}a_1 u_n\, \mathrm{d}x \\&\le C_\varepsilon \Vert a_1\Vert _{(p_1^*)'} \Vert u_n\Vert _{p_1^*}\le C_\varepsilon \Vert a_1\Vert _{(p_1^*)'} \Vert \nabla u_n\Vert _{p_1}\;\;\forall \, n\in {\mathbb {N}}, \end{aligned} \end{aligned}

where $$C_\varepsilon :=M_1\varepsilon ^{\alpha _1}(1+R^{\beta _1})$$. This actually means that $$\{u_n\}$$ is bounded in $$\mathcal {D}^{1,p_1}({\mathbb {R}}^N)$$, because $$p_1>1$$. By (4.7), an analogous conclusion holds for $$\{v_n\}$$. Along subsequences if necessary, we may thus assume

\begin{aligned} (u_n,v_n)\rightharpoonup (u,v)\;\;\text {in}\;\;\mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N). \end{aligned}
(4.8)

So, $$\{(u_n,v_n)\}$$ converges strongly in $$L^{q_1}(B_{r_1})\times L^{q_2}(B_{r_2})$$ for any $$r_i>0$$ and any $$1\le q_i\le p_i^*$$, whence, up to subsequences again,

\begin{aligned} (u_n,v_n)\rightarrow (u,v)\;\;\text {a.e. in}\;\;{\mathbb {R}}^N. \end{aligned}
(4.9)

Now, combining Lemma 4.1 with Lebesgue’s dominated convergence theorem, we obtain

\begin{aligned} (u_n,v_n)\rightarrow (u,v)\;\;\text {in}\;\; L^{p_1^*}({\mathbb {R}}^N)\times L^{p_2^*}({\mathbb {R}}^N), \end{aligned}
(4.10)

as desired. Let us finally verify that $$\mathcal {T}(\mathcal {K}_\varepsilon )$$ is relatively compact. If $$(u_n,v_n):=\mathcal {T}(z_{1,n},z_{2,n})$$, $$n\in {\mathbb {N}}$$, then (4.6)–(4.7) can be written. Hence, the previous argument yields a pair $$(u,v)\in L^{p_1^*}({\mathbb {R}}^N)\times L^{p_2^*}({\mathbb {R}}^N)$$ fulfilling (4.10), possibly along a subsequence. This completes the proof. $$\square$$

Thanks to Lemmas 4.14.2, Schauder’s fixed point theorem applies, and there exists $$(u_\varepsilon ,v_\varepsilon )\in \mathcal {K}_\varepsilon$$ such that $$(u_\varepsilon ,v_\varepsilon )=\mathcal {T}(u_\varepsilon ,v_\varepsilon )$$. Through Theorem 3.4, we next arrive at

### Theorem 4.3

Under hypotheses $$(\mathrm{H}_{f,g})$$ and $$(\mathrm{H}_a)$$, for every $$\varepsilon >0$$ small, problem ($$\mathrm{P}_{\varepsilon }$$) admits a solution $$\left( u_\varepsilon ,v_\varepsilon \right) \in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$ complying with (3.18).

## Existence of solutions

We are now ready to establish the main result of this paper.

### Theorem 5.1

Let $$(\mathrm{H}_{f,g})$$ and $$(\mathrm{H}_a)$$ be satisfied. Then, (P) has a weak solution $$(u,v)\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$, which is essentially bounded.

### Proof

Pick $$\varepsilon :=\frac{1}{n}$$, with $$n\in {\mathbb {N}}$$ big enough. Theorem 4.3 gives a pair $$(u_n,v_n)$$, where $$u_n:=u_{\frac{1}{n}}$$ and $$v_n:=v_{\frac{1}{n}}$$, such that

\begin{aligned} \int _{{\mathbb {R}}^N}\vert \nabla u_n\vert ^{p_1-2}\nabla u_n\nabla \varphi \, \mathrm{d}x= & {} \int _{{\mathbb {R}}^N} a_1 f\left( u_n+\frac{1}{n},v_n\right) \varphi \, \mathrm{d}x, \nonumber \\ \int _{{\mathbb {R}}^N}\vert \nabla v_n\vert ^{p_2-2}\nabla v_n\nabla \psi \, \mathrm{d}x= & {} \int _{{\mathbb {R}}^N}a_2 g\left( u_n,v_n+\frac{1}{n}\right) \psi \, \mathrm{d}x \end{aligned}
(5.1)

for every $$(\varphi ,\psi )\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$, as well as (cf. Lemma 4.1)

\begin{aligned} 0<\underline{u}\le u_n\le R,\quad 0<\underline{v}\le v_n\le R. \end{aligned}
(5.2)

Thanks to $$(\mathrm{H}_{f,g})$$, (5.2), and $$(\mathrm{H}_a)$$, choosing $$\varphi :=u_n$$, $$\psi :=v_n$$ in (5.1) easily entails

\begin{aligned} \begin{aligned} \Vert \nabla u_n\Vert _{p_1}^{p_1}&\le M_1\int _{{\mathbb {R}}^N} a_1 u_n^{\alpha _1+1}(1+v_n^{\beta _1}) \mathrm{d}x \le M_1 R^{\alpha _1+1}(1+R^{\beta _1})\Vert a_1\Vert _1\, ,\\ \Vert \nabla v_n\Vert _{p_2}^{p_2}&\le M_2\int _{{\mathbb {R}}^N} a_2 (1+u_n^{\alpha _2})v_n^{\beta _2+1} \mathrm{d}x \le M_2(1+R^{\alpha _2})R^{\beta _2+1}\Vert a_2\Vert _1, \end{aligned} \end{aligned}

whence both $$\{u_n\}\subseteq \mathcal {D}^{1,p_1}({\mathbb {R}}^N)$$ and $$\{v_n\}\subseteq \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$ are bounded. Along subsequences if necessary, we thus have (4.8)–(4.9). Let us next show that

\begin{aligned} (u_n,v_n)\rightarrow (u,v)\quad \text {strongly in}\quad \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N). \end{aligned}
(5.3)

Testing the first equation in (5.1) with $$\varphi :=u_n-u$$ yields

\begin{aligned} \int _{{\mathbb {R}}^N}\vert \nabla u_n\vert ^{p_1-2}\nabla u_n\nabla (u_n-u) \mathrm{d}x=\int _{{\mathbb {R}}^N} a_1f\left( u_n+\frac{1}{n},v_n\right) (u_n-u) \mathrm{d}x. \end{aligned}
(5.4)

The right-hand side of (5.4) goes to zero as $$n\rightarrow \infty$$. Indeed, by $$(\mathrm{H}_{f,g})$$, (5.2), and $$(\mathrm{H}_a)$$ again,

\begin{aligned} \left| a_1f\left( u_n+\frac{1}{n},v_n\right) (u_n-u)\right| \le 2M_1R^{\alpha _1+1}(1+R^{\beta _1})a_1\quad \forall \, n\in {\mathbb {N}}, \end{aligned}

so that, recalling (4.9), Lebesgue’s dominated convergence theorem applies. Through (5.4), we obtain $$\displaystyle {\lim _{n\rightarrow \infty }}\langle -\Delta _{p_1}u_n,u_n-u\rangle =0$$. Likewise, $$\langle -\Delta _{p_2}v_n,v_n-v\rangle \rightarrow 0$$ as $$n\rightarrow \infty$$, and (5.3) directly follows from Proposition 2.2. On account of (5.1), besides (5.3), the final step is to verify that

\begin{aligned} \lim _{n\rightarrow \infty }\int _{{\mathbb {R}}^N} a_1 f\left( u_n+\frac{1}{n},v_n\right) \varphi \, \mathrm{d}x=\int _{{\mathbb {R}}^N} a_1 f(u,v)\varphi \, \mathrm{d}x,\end{aligned}
(5.5)
\begin{aligned} \lim _{n\rightarrow \infty }\int _{{\mathbb {R}}^N} a_2 g\left( u_n,v_n+\frac{1}{n}\right) \psi \, \mathrm{d}x=\int _{{\mathbb {R}}^N} a_2 g(u,v)\psi \, \mathrm{d}x \end{aligned}
(5.6)

for all $$(\varphi ,\psi )\in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)\times \mathcal {D}^{1,p_2}({\mathbb {R}}^N)$$. If $$\varphi \in \mathcal {D}^{1,p_1}({\mathbb {R}}^N)$$, then $$(\mathrm{j}_1)$$ in Remark 2.1 gives $$a_1\varphi \in L^1({\mathbb {R}}^N)$$. Since, as before,

\begin{aligned} \left| a_1f\left( u_n+\frac{1}{n},v_n\right) \varphi \right| \le M_1R^{\alpha _1+1}(1+R^{\beta _1})a_1|\varphi |,\quad n\in {\mathbb {N}}, \end{aligned}

assertion (5.5) stems from Lebesgue’s dominated convergence theorem. The proof of (5.6) is similar at all. $$\square$$

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## Acknowledgements

This work is performed within the 2016–2018 Research Plan—Intervention Line 2: ‘Variational Methods and Differential Equations’ and partially supported by GNAMPA of INDAM.

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Correspondence to Salvatore A. Marano.