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

figurea

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 [18] 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.

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Marano, S.A., Marino, G. & Moussaoui, A. Singular quasilinear elliptic systems in \({\mathbb {R}}^{N}\). Annali di Matematica 198, 1581–1594 (2019). https://doi.org/10.1007/s10231-019-00832-1

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Keywords

  • Singular elliptic system
  • p-Laplacian
  • Schauder’s fixed point theorem
  • A priori estimate

Mathematics Subject Classification

  • 35J75
  • 35J48
  • 35J92