A quasilinear elliptic system with natural growth terms

Abstract

In this paper, we prove existence of solutions for an elliptic system of the type

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}(a(x,z) \nabla u) = f, &{}\quad \text{ in } \varOmega \text{; } \\ -\mathrm{div}(b(x) \nabla z)+ h(x,z)|\nabla u|^2 = g, &{}\quad \text{ in } \varOmega \text{; } \\ \,u = 0 = z, &{}\quad \text{ on } \partial \varOmega \text{, } \end{array}\right. } \end{aligned}$$

under various assumptions on the functions $a(x,s)$ and $h(x,s)$, and on the data $f$ and $g$ (in Lebesgue spaces).

1 Introduction

In this paper, we study the existence of solutions of elliptic systems of the type

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}(a(x,z) \nabla u) = f, &{}\quad \text{ in } \varOmega \text{; } \\ -\mathrm{div}(b(x) \nabla z)+ h(x,z)|\nabla u|^2 = g, &{}\quad \text{ in } \varOmega \text{; } \\ \,u = 0 = z, &{}\quad \text{ on } \partial \varOmega \text{; } \end{array}\right. } \end{aligned}$$

(1.1)

where we assume that $\varOmega$ is a bounded open subset of $\mathbb {R}^N$, with $N > 2$, $f$ and $g$ are functions belonging to suitable Lebesgue spaces, $a(x,s) : \varOmega \times \mathbb {R}\rightarrow \mathbb {R}$ and $h(x,s) : \varOmega \times \mathbb {R}\rightarrow \mathbb {R}$ are Carathéodory functions, and $b : \varOmega \rightarrow \mathbb {R}$ is a measurable function, such that

$$\begin{aligned} \alpha \le a(x,s), \quad \alpha \le b(x) \le \beta , \quad h(x,s)\cdot s \ge 0, \end{aligned}$$

(1.2)

for some $0 < \alpha \le \beta$ in $\mathbb {R}^+$ (see the various sections of this paper for the precise assumptions on $a$, $b$ and $h$).

Our interest in studying system (1.1) is motivated by the following example of the calculus of variations. Suppose that $f$ and $g$ are functions in $L^{m}(\varOmega )$, with $m > \frac{N}{2}$, and that $s \mapsto a(x,s)$ is increasing, continuously differentiable, with $a_{s}(x,s)$ a bounded function with respect to both $x$ and $s$, and such that $a_{s}(x,0) \equiv 0$; then, if we define

$$\begin{aligned} I(v,w) = \frac{1}{2} \int _{\varOmega }a(x,|w|)|\nabla v|^{2} + \frac{1}{2} \int _{\varOmega }b(x)|\nabla w|^{2} - \int _{\varOmega }f\,v - \int _{\varOmega }g\,w, \end{aligned}$$

for every $v$ and $w$ in $H_0^{1}(\varOmega )$ such that

$$\begin{aligned} \int _{\varOmega }a(x,|w|)|\nabla v|^{2} < +\infty , \end{aligned}$$

and $I(v,w) = +\infty$ if the above integral is not finite, it is easy to see, using the assumptions on $a$ and $b$ that $I$ is both coercive and weakly lower semicontinuous on $H_0^{1}(\varOmega )\times H_0^{1}(\varOmega )$. This implies that there exists a minimum $(u,z)$ of $I$. Standard techniques of calculus of variations (together with the assumptions on $a$, $b$, $f$ and $g$) imply that $u$ and $z$ belong to $L^{\infty }(\varOmega )$ and that are weak solutions of the boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}(a(x,|z|)\nabla u) = f, &{}\quad \text{ in } \varOmega ;\\ -\mathrm{div}(b(x)\nabla z) + \frac{1}{2} a_{s}(x,|z|)\mathrm{sgn}(z)|\nabla u|^{2} = g, &{}\quad \text{ in } \varOmega ;\\ u = 0 = z, &{}\quad \text{ in } \partial \varOmega , \end{array}\right. } \end{aligned}$$

(1.3)

i.e., of a problem of type (1.1) with $h(x,s) = \frac{1}{2} a_{s}(x,|s|)\,\mathrm{sgn}(s)$. Note that the function $s \mapsto a_{s}(x,|s|)\,\mathrm{sgn}(s)$ is continuous at the origin thanks to the assumption $a_{s}(x,0) = 0$.

Thus, our interest in the study of the (general and nonvariational) system (1.1) is motivated by (1.3), more specifically, a system with an equation having a lower-order term, which has quadratic dependence with respect to the gradient and satisfies a “sign assumption” [see (1.2), or (2.2) below]. The study of nonvariational systems will allow us to weaken the summability assumptions on the data $f$ and $g$; we will see that, depending on whether $a$ is bounded from above or not, we will have existence of solutions for $L^{1}(\varOmega )$ data $f$ and $g$, or for data belonging to smaller Lebesgue spaces.

Even if our system is “natural” and motivated by the study of the very simple functional $I$, there seems to be no existence result in the literature for this kind of problems. General results on quasilinear elliptic systems, in contexts not applicable in the present case, can be found for example in [2, 9–11].

The plan of the paper is as follows: in the next section, we will study existence of solutions for (1.1) under a boundedness assumption on $a$ [see (2.1)], which will allow us to prove results for $L^{1}(\varOmega )$ data $f$ and $g$. The case of $a$ unbounded from above is dealt with in Sect. 3: after some existence results for bounded solutions $z$ (which will yield that $a(x,z)$ is bounded from above, thus giving results comparable to those of Sect. 2), we will study two particular cases (see Theorems 3.4 and 3.5), where the presence of a suitably growing, with respect to $a$, lower-order term $h(x,s)$ will help in proving existence of solutions. In Sect. 4, we will prove how the presence of a quadratic (with respect to the gradient) lower-order term in the second equation will “break” the maximum principle: if the datum of the first equation is an actual $L^{1}(\varOmega )$ function $f$, the solution $z$ of the second equation has to be zero on the set of “explosion” of $f$, even if the datum $g$ is strictly positive in $\varOmega$. The case of measure data, yielding nonexistence of solutions, will be studied in the final Sect. 5.

2 The case of $a$ bounded

In this section, we will study system (1.1) if the function $a(x,s)$ which appears in the first equation is bounded: that is, instead of (1.2), we assume that

$$\begin{aligned} \alpha \le a(x,s) \le \beta , \quad \alpha \le b(x) \le \beta , \end{aligned}$$

(2.1)

for almost every $x$ in $\varOmega$, for every $s$ in $\mathbb {R}$, with $0 < \alpha \le \beta$ in $\mathbb {R}^{+}$.

Let $h : \varOmega \times \mathbb {R}\rightarrow \mathbb {R}$ be a Carathéodory function such that $h(x,s)$ is increasing with respect to $s$ and that

$$\begin{aligned} h(x,s)\,s \ge 0, \quad \sup _{|s| \le t}\, |h(\cdot ,s)| = h_{t}(\cdot ) \in L^{\infty }(\varOmega ), \end{aligned}$$

(2.2)

for almost every $x$ in $\varOmega$, for every $s$ and $t$ in $\mathbb {R}$.

Under these assumptions, we will prove an existence result for problem (1.1). We begin with the case of bounded data $f$ and $g$.

Proposition 2.1

Assume that (2.1) and (2.2) hold, and let $f$ and $g$ be in $L^{m}(\varOmega )$, with $m > \frac{N}{2}$. Then, there exists a solution $(u,z)$ in $H_0^{1}(\varOmega )\times H_0^{1}(\varOmega )$ of system (1.1). Furthermore, both $u$ and $z$ belong to $L^{\infty }(\varOmega )$.

Sketch of the proof

We will prove the result using Schauder’s fixed-point theorem. To this aim, fix $z_0\in L^{2}(\varOmega )$ and let $u$ in $H_0^{1}(\varOmega )$ be the unique solution of the first equation; let then $z$ in $H_0^{1}(\varOmega )$ be the unique solution (see [6]) of the second equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}(a(x,z_{0}) \nabla u) = f, &{}\quad \text{ in } \varOmega \text{; } \\ -\mathrm{div}(b(x) \nabla z)+ h(x,z)|\nabla u|^2 = g, &{}\quad \text{ in } \varOmega \text{; } \\ \, u = 0 = z, &{}\quad \text{ on } \partial \varOmega \text{. } \end{array}\right. } \end{aligned}$$

Therefore, if we define $S(z_{0}) = z$, then $S$ maps $L^{2}(\varOmega )$ into itself. It is standard to prove that the ball $B_{R}$ of radius $R$ in $L^{2}(\varOmega )$ is invariant for $S$, where

$\mathcal S$ is the Sobolev constant, and $2_{*} = \frac{2N}{N+2}$.

Moreover, by Rellich-Kondrachov theorem, $\overline{S(B_{R})}$ is compact in $L^{2}(\varOmega )$.

To prove that $S$ is continuous, we begin with the observation that if $z_n$ converges to $z_{0}$ in $L^{2}(\varOmega )$, then, defining $u_n$ as the solution of

$$\begin{aligned} u_n\in H_0^{1}(\varOmega ): -\mathrm{div}(a(x,z_n) \nabla u_n) = f, \end{aligned}$$

we have that $u_n$ is strongly convergent in $H_0^{1}(\varOmega )$ to a function $u$, which is the unique solution of

$$\begin{aligned} u \in H_0^{1}(\varOmega ): -\mathrm{div}(a(x,z_{0}) \nabla u) = f. \end{aligned}$$

Therefore, the sequence $\{|\nabla u_n|^{2}\}$ is strongly convergent in $L^{1}(\varOmega )$ to $|\nabla u|^{2}$. This fact, and the stability result of [6] (or [7]), proves that the sequence $\tilde{z}_{n} = S(z_{n})$ of solutions of

$$\begin{aligned} \tilde{z_{n}} \in H_0^{1}(\varOmega ): -\mathrm{div}(b(x) \nabla \tilde{z}_{n})+ h(x,\tilde{z}_{n})|\nabla u_n|^2 = g, \end{aligned}$$

converges in $L^{2}(\varOmega )$ to the unique solution of

$$\begin{aligned} \tilde{z} \in H_0^{1}(\varOmega ): -\mathrm{div}(b(x) \nabla \tilde{z})+ h(x,\tilde{z})|\nabla u|^2 = g. \end{aligned}$$

Thus, $S(z_n)$ converges in $L^{2}(\varOmega )$ to $S(z_{0})$, and so $S$ is continuous. By Schauder’s theorem, there exists a fixed-point $z$ of $S$, and so a solution $(u,z)$ of system (1.1). The fact that both $u$ and $z$ belong to $L^{\infty }(\varOmega )$ then follows by standard elliptic results (see [13]). $\square$

We study now to the more general case of $L^{1}(\varOmega )$ data $f$ and $g$. In what follows, we will use the functions $T_{k}(s) = \max (-k,\min (s,k))$, and $G_{k}(s) = s - T_{k}(s)$, for $s$ in $\mathbb {R}$ and $k \ge 0$.

Theorem 2.2

Assume that (2.1) and (2.2) hold, and let $f$ and $g$ be $L^{1}(\varOmega )$ functions. Then, there exists a distributional solution $(u,z)$ of system (1.1), with both $u$ and $z$ belonging to $W^{1,q}_{0}(\varOmega )$, for every $q < \frac{N}{N-1}$. Moreover, $h(x,z)|\nabla u|^2 \in L^{1}(\varOmega )$.

Proof

Let $\{f_{n}\}$ and $\{g_{n}\}$ be sequences of $L^{\infty }(\varOmega )$ functions, which converge in $L^{1}(\varOmega )$ to $f$ and $g$, respectively, and are such that

$$\begin{aligned} |f_{n}| \le |f|, \quad |g_{n}| \le |g|, \end{aligned}$$

(2.3)

almost everywhere in $\varOmega$ [take for example $f_{n} = T_{n}(f)$ and $g_{n} = T_{n}(g)$]. By the result of Theorem 2.1, there exist $(u_n,z_n)$ weak solutions of

$$\begin{aligned} {\left\{ \begin{array}{ll} u_n\in H_0^{1}(\varOmega ): -\mathrm{div}(a(x,z_n) \nabla u_n) = f_{n}, \\ z_n\in H_0^{1}(\varOmega ): -\mathrm{div}(b(x) \nabla z_n)+ h(x,z_n)|\nabla u_n|^2 = g_{n}, \end{array}\right. } \end{aligned}$$

(2.4)

with both $u_n$ and $z_n$ belonging to $L^{\infty }(\varOmega )$.

We begin with some a priori estimates on the sequences $\{u_n\}$, $\{z_n\}$.

Taking $T_{k}(u_n)$ as test function in the first equation of (2.4), and using (2.1) and the first of (2.3), we easily obtain (see [1])

(2.5)

Starting from (2.5), and reasoning as in [3], we have that

$$\begin{aligned} \int _{\varOmega }|\nabla u_n|^q \le C_{q}, \quad \forall q<\frac{N}{N-1}, \end{aligned}$$

(2.6)

where $C_{q}$ is a positive constant, which tends to infinity as $q$ tends to $\frac{N}{N-1}$. Choosing $T_{1}(G_{k}(u_n))$ as test function in the first equation of (2.4), and using again (2.1) and the first of (2.3), we get

$$\begin{aligned} \alpha \int \nolimits _{\{k \le |u_n|< k+1\}}|\nabla u_n|^2 \le \int \nolimits _{\{|u_n| \ge k\}}|f|, \end{aligned}$$

(2.7)

while the choice of $T_{k}(z_n)$ as test function in the second equation of (2.4) yields [using (2.1) and the second of (2.3)]

(2.8)

As before, from (2.8), it follows that

$$\begin{aligned} \int _{\varOmega }|\nabla z_n|^q \le C_{q}, \quad \forall q < \frac{N}{N-1}, \end{aligned}$$

(2.9)

while the choice of $T_{1}(G_{k}(z_n))$ as test function in the second equation of (2.4) yields (using again (2.1), the second of (2.3) and (2.2))

$$\begin{aligned} \alpha \int \nolimits _{\{k\le |z_n|< k+1\}} |\nabla z_n|^2 \le \int \nolimits _{\{|z_n| \ge k\}}|g|, \end{aligned}$$

(2.10)

and

$$\begin{aligned} \int \nolimits _{\{|z_n| \ge k+1\}} |h(x,z_n)||\nabla u_n|^2 \le \int \nolimits _{\{|z_n| \ge k\}} |g|. \end{aligned}$$

(2.11)

Thanks to these a priori estimates, we have that (up to subsequences still denoted by $u_n$ and $z_n$) $u_n$ and $z_n$ converge to some functions $u$ and $z$ weakly in $W^{1,q}_{0}(\varOmega )$, for every $q < \frac{N}{N-1}$ strongly in $L^{1}(\varOmega )$, and almost everywhere. These convergences (together with the boundedness of $a(x,s)$) are enough to pass to the limit in the first equation, thus proving that $u$ and $z$ are such that

$$\begin{aligned} \int _{\varOmega }a(x,z)\nabla u \cdot \nabla \varphi = \int _{\varOmega }f\,\varphi , \quad \forall \varphi \in C^{1}_{c}(\varOmega ). \end{aligned}$$

(2.12)

Therefore, we only have to pass to the limit in the weak formulation of the second equation, i.e.,

$$\begin{aligned} \int _{\varOmega }b(x)\nabla z_n\cdot \nabla \varphi + \int _{\varOmega }h(x,z_n)|\nabla u_n|^{2}\varphi = \int _{\varOmega }g_{n}\varphi , \end{aligned}$$

(2.13)

for every $\varphi$ in $C^{1}_{c}(\varOmega )$.

In order to do that, the convergences proved so far (and the assumption on $g_{n}$) allow to pass to the limit in the first and the third integral, while to pass to the limit in the second integral we need to prove that

$$\begin{aligned} h(x,z_n)|\nabla u_n|^{2} \rightarrow h(x,z)|\nabla u|^{2}, \quad \text{ strongly } \text{ in } L^{1}(\varOmega )\text{. } \end{aligned}$$

(2.14)

In order to prove (2.14), we begin by following the technique of [12] to prove that

$$\begin{aligned} T_{k}(u_n) \rightarrow T_{k}(u) \quad \text{ strongly } \text{ in } H_0^{1}(\varOmega ). \end{aligned}$$

(2.15)

As in [12], we choose in the first equation the test function

$$\begin{aligned} w = T_{2k}(u_n- T_{h}(u_n) + T_{k}(u_n) - T_{k}(u)), \end{aligned}$$

where $h > k > 0$. Thanks to the assumptions on $f_{n}$, we have

$$\begin{aligned} \lim _{h \rightarrow +\infty }\,\lim _{n \rightarrow +\infty }\,\int _{\varOmega }f_{n}\,T_{2k}(u_n- T_{h}(u_n) + T_{k}(u_n) - T_{k}(u)) = 0, \end{aligned}$$

so that

$$\begin{aligned} \lim _{h \rightarrow +\infty }\,\lim _{n \rightarrow +\infty }\,\int _{\varOmega }a(x,z_n)\nabla u_n\cdot \nabla T_{2k}(u_n- T_{h}(u_n) + T_{k}(u_n) - T_{k}(u)) = 0. \end{aligned}$$

Splitting the integral on the sets where $|u_n| \ge k$ and $|u_n| < k$, and using that $a(x,z_n) \ge \alpha > 0$, we obtain (as in [12])

$$\begin{aligned} \begin{array}{l} \displaystyle \int _{\varOmega }a(x,z_n)\nabla u_n\cdot \nabla T_{2k}(u_n- T_{h}(u_n) + T_{k}(u_n) - T_{k}(u)) \\ \displaystyle \quad \ge \int _{\varOmega }a(x,z_n)\nabla T_{k}(u_n) \cdot \nabla (T_{k}(u_n) - T_{k}(u))\\ \displaystyle \qquad - \int \nolimits _{\{|u_n| \ge k\}} |a(x,z_n)||\nabla T_{4k+h}(u_n)||\nabla T_{k}(u)|. \end{array} \end{aligned}$$

Since the last integral tends to zero as $n$ tends to infinity (here, we use the boundedness of $a(x,s)$ with respect to $s$), we have adding and subtracting the term

$$\begin{aligned} \int _{\varOmega }a(x,z_n)\nabla T_{k}(u)\cdot \nabla (T_{k}(u_n) - T_{k}(u)), \end{aligned}$$

which tends to zero as $n$ tends to infinity,

$$\begin{aligned} 0 \le \lim _{n \rightarrow +\infty }\,\int _{\varOmega }a(x,z_n)|\nabla (T_{k}(u_n) - T_{k}(u)|^{2} \le 0, \end{aligned}$$

which then implies (2.15). Hence, up to subsequences still denoted by $u_n$,

$$\begin{aligned} \nabla u_n \rightarrow \nabla u \quad \text{ almost } \text{ everywhere } \text{ in } \;\varOmega . \end{aligned}$$

Therefore,

$$\begin{aligned} h(x,z_n)|\nabla u_n|^{2} \rightarrow h(x,z)|\nabla u|^{2}, \quad \text{ almost } \text{ everywhere } \text{ in }\; \varOmega . \end{aligned}$$

(2.16)

We now define, for $k \ge 0$, $j \ge 0$, and $\varepsilon > 0$, the functions $\psi _{k}(s) = |T_{1}(G_{k}(s))|$ and

$$\begin{aligned} H_{j,\varepsilon }(s)= \left\{ \begin{array}{l@{\quad }l} s/\varepsilon &{}\; 0 \le s \le \varepsilon , \\ 1 &{} \;\varepsilon < s \le j, \\ -s+j+1 &{}\; j < s \le j+1,\\ 0 &{}\; s > j+1, \\ - H_{j,\varepsilon }(-s)&{}\; s < 0, \end{array} \right. \end{aligned}$$

and choose $H_{j,\varepsilon }(z_n)\psi _k(u_n)$ as test function in the second equation. We obtain

$$\begin{aligned} \begin{array}{l} \displaystyle \int _{\varOmega }b(x) |\nabla z_n|^{2} \psi _k(u_n) H'_{j,\varepsilon }(z_n) + \int _{\varOmega }b(x) \nabla z_n\nabla u_n \psi _k'(u_n) H_{j,\varepsilon }(z_n) \\ \displaystyle \quad + \int _{\varOmega }\,h(x,z_n)|\nabla u_n|^2 H_{j,\varepsilon }(z_n)\psi _k(u_n) = \int _{\varOmega }\,g_n H_{j,\varepsilon }(z_n)\psi _k(u_n). \end{array} \end{aligned}$$

Thus, using the definition of $H_{j,\varepsilon }$, and dropping nonnegative terms [thanks to (2.2)],

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{1}{\varepsilon }\int \nolimits _{\{|z_n|<\varepsilon \}} b(x) |\nabla z_n|^{2} \psi _k(u_n) - \int \nolimits _{\{j\le |z_n|<j+1\}} b(x) |\nabla z_n|^{2} \psi _k(u_n)\\ \displaystyle \qquad + \int _{\varOmega }b(x) \nabla z_n\nabla u_n \psi _k'(u_n) H_{j,\varepsilon }(z_n) + \int \nolimits _{\left\{ \genfrac{}{}{0.0pt}{}{\varepsilon \le |z_n| < j}{|u_n| > k+1} \right\} } |h(x,z_n)||\nabla u_n|^2\\ \displaystyle \quad \le \int \nolimits _{\{|u_n| > k\}}|g_n| \end{array} \end{aligned}$$

Dropping the first term, which is positive, and letting $\varepsilon$ tend to zero, we obtain [using (2.1) and the second of (2.3)]

$$\begin{aligned} \begin{array}{l} \displaystyle \int \nolimits _{\left\{ \genfrac{}{}{0.0pt}{}{|z_n|<j}{|u_n| > k+1} \right\} } |h(x,z_n)||\nabla u_n|^2 \le \int \nolimits _{\{|u_n| > k\}} |g|\\ \displaystyle \quad +\, \beta \int \nolimits _{\{j \le |z_n| < j+1\}} |\nabla z_n|^2 + \beta \int \nolimits _{\left\{ \genfrac{}{}{0.0pt}{}{|z_n| < j+1}{k < |u_n|\le k+1} \right\} } |\nabla z_n||\nabla u_n|. \end{array} \end{aligned}$$

Now, fix $\delta >0$, and let $j^*$ be such that

$$\begin{aligned} \int \nolimits _{\{j^* \le |z_n| < j^*+1\}}|\nabla z_n|^2 \le \int \nolimits _{\{|z_n| \ge j^*\}} |g| \le \int \nolimits _{\{|z_n| \ge j^*-1\}} |g| \le \delta . \end{aligned}$$

(2.17)

Such a choice of $j^{*}$ is possible thanks to (2.10), to the strong convergence of $z_n$ in $L^{1}(\varOmega )$ and to the absolute continuity of the integral. Thus,

$$\begin{aligned} \begin{array}{l} \displaystyle \int \nolimits _{\left\{ \genfrac{}{}{0.0pt}{}{|z_n|<j^{*}}{|u_n| > k+1} \right\} } |h(x,z_n)||\nabla u_n|^2\\ \displaystyle \quad \le \int \nolimits _{\{|u_n| > k\}} |g| + \beta \delta + \beta \int \nolimits _{\{k <|u_n| \le k+1\}} |\nabla T_{j^*+1}(z_n)||\nabla u_n|, \end{array} \end{aligned}$$

which implies [using (2.8) and (2.7)]

(2.18)

Let now $k^*$ be such that

such a choice is possible thanks to the strong convergence of $u_n$ in $L^{1}(\varOmega )$ and to the absolute continuity of the integral. Therefore,

$$\begin{aligned} \int \nolimits _{\left\{ \genfrac{}{}{0.0pt}{}{|z_n|<j^{*}}{|u_n| > k^{*}+1} \right\} } |h(x,z_n)||\nabla u_n|^2 \le (2\beta +1)\delta . \end{aligned}$$

(2.19)

Now, we use Vitali Theorem. For every measurable $E\subset \varOmega$, we have, using (2.11), (2.17) and (2.19),

Recalling (2.15), it is possible to choose $\mathrm{meas}(E)$ small enough so that

uniformly with respect to $n$. Therefore, we have proved that the sequence $\{h(x,z_n)|\nabla u_n|^{2}\}$ is equiintegrable. This fact, together with the almost everywhere convergence (2.16), proves (2.14). $\square$

3 The case of $a$ unbounded

In this section, we are going to deal with system (1.1) under the assumption that the function $a(x,s)$, which “links” the two equations, may be unbounded with respect to $s$. In this case, we have to make slight changes in the proof of Theorem 2.1 also if both $f$ and $g$ are such that the solutions are bounded.

Proposition 3.1

Assume that (1.2) and (2.2) hold, and let $f$ and $g$ be in $L^{m}(\varOmega )$, with $m > \frac{N}{2}$. Then, there exists a solution $(u,z)$ in $H_0^{1}(\varOmega )\times H_0^{1}(\varOmega )$ of the system (1.1). Furthermore, both $u$ and $z$ belong to $L^{\infty }(\varOmega )$.

Proof

Let $k \ge 0$ be an integer; since $a(x,T_{k}(s))$ satisfies (2.1), by Proposition 2.1, there exists a solution $(u_{k},z_{k})$ of the system

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{k} \in H_0^{1}(\varOmega ): -\mathrm{div}(a(x,T_{k}(z_{k})) \nabla u_{k}) = f, \\ z_{k} \in H_0^{1}(\varOmega ): -\mathrm{div}(b(x) \nabla z_{k})+ h(x,z_{k})|\nabla u_{k}|^2 = g. \end{array}\right. } \end{aligned}$$

Since $g$ belongs to $L^{m}(\varOmega )$, with $m > \frac{N}{2}$, standard elliptic results, and assumption (2.2), imply that there exists a constant $M$, independent on $k$, such that

Thus, taking $\overline{k} > M$, we have that $T_{\overline{k}}(z_{\overline{k}}) = z_{\overline{k}}$ so that $(u,z) = (u_{\overline{k}},z_{\overline{k}})$ is a solution of (1.1) as desired. $\square$

Now, we turn to the case of less regular data; for example, we would like to take (as in Theorem 2.2) both $f$ and $g$ in $L^{1}(\varOmega )$. However, the fact that $a$ may be unbounded with respect to $s$, even if it allows to prove (2.5)–(2.7) (where only the ellipticity constant $\alpha$ is used), prevents us to apply the technique of [12] to prove the strong convergence of truncates (2.15), an essential tool to pass to the limit in the “quadratic” lower-order term of the second equation.

Nonetheless, if we make stronger assumptions on the data, we can easily recover existence results, as is in the next theorem.

Theorem 3.2

Assume that (1.2) and (2.2) hold, and let $f$ be an $L^{1}(\varOmega )$ function and let $g$ be a function in $L^{m}(\varOmega )$, with $m > \frac{N}{2}$. Then, there exists a distributional solution $(u,z)$ of system (1.1), with $u$ belonging to $W^{1,q}_{0}(\varOmega )$, for every $q < \frac{N}{N-1}$, and $z$ belonging to $H_0^{1}(\varOmega )\cap L^{\infty }(\varOmega )$.

Proof

We follow the lines of the proof of Theorem 2.2, approximating $f$ and $g$ with sequences of $L^{\infty }(\varOmega )$ functions such that (2.3) holds, and considering the solutions $(u_n,z_n)$, given by Theorem 3.1, of the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}(a(x,z_n) \nabla u_n) = f_{n}, &{}\quad \text{ in } \varOmega \text{; } \\ -\mathrm{div}(b(x) \nabla z_n)+ h(x,z_n)|\nabla u_n|^2 = g_{n}, &{}\quad \text{ in } \varOmega \text{; } \\ \, u_n= 0 = z_n, &{}\quad \text{ on } \partial \varOmega \text{. } \end{array}\right. } \end{aligned}$$

Taking $G_{k}(z_n)$ as test function in the second equation, and dropping the nonnegative second term, we obtain [using (2.1)]

$$\begin{aligned} \alpha \int _{\varOmega }|\nabla G_{k}(z_n)|^{2} \le \int _{\varOmega }g_{n}\,G_{k}(z_n). \end{aligned}$$

Starting from this inequality, one can prove (see [13]) that the sequence $\{z_n\}$ is bounded in $L^{\infty }(\varOmega )$ and, taking $k = 0$, that it is bounded in $H_0^{1}(\varOmega )$. Thus, the term $a(x,z_n)$ in the first equation is bounded in $L^{\infty }(\varOmega )$; this fact allows to prove the strong convergence of $T_{k}(u_n)$ in $H_0^{1}(\varOmega )$ with the same technique of [12] and to conclude the proof as in Theorem 2.2. $\square$

The result of the previous theorem is however rather unsatisfactory: The unboundedness of $a$ is “invisible” due to the boundedness of $z$; rather than dealing with the general case, we are going to prove an existence result in a particular case, which will not be variational, to show how the lower-order term in the second equation may be helpful. Before stating and proving the theorem, we need a technical lemma.

Lemma 3.3

Let $\sigma _{n}$ be a sequence of nonnegative functions in $L^{\infty }(\varOmega )$, almost everywhere convergent to some function $\sigma$, and let $\rho _{n}$ be a sequence of functions, which is weakly convergent in $(L^{2}(\varOmega ))^{N}$ to some function $\rho$. If the sequence $\sigma _{n}|\rho _{n}|^{2}$ is bounded in $L^{1}(\varOmega )$, then $\sigma |\rho |^{2}$ belongs to $L^{1}(\varOmega )$ and

$$\begin{aligned} \int _{\varOmega }\,\sigma |\rho |^{2} \le \liminf _{n \rightarrow +\infty }\,\int _{\varOmega }\,\sigma _{n}|\rho _{n}|^{2}. \end{aligned}$$

(3.1)

Proof

We begin by proving that, for every $k \ge 0$,

$$\begin{aligned} \int _{\varOmega }\,T_{k}(\sigma )|\rho |^{2}\le \liminf _{n \rightarrow +\infty }\,\int _{\varOmega }\,T_{k}(\sigma _{n})|\rho _{n}|^{2}. \end{aligned}$$

(3.2)

Indeed, starting from

$$\begin{aligned} \int _{\varOmega }T_{k}(\sigma _{n})|\rho _{n} - \rho |^{2} \ge 0, \end{aligned}$$

we have

$$\begin{aligned} 2\int _{\varOmega }T_{k}(\sigma _{n}) \rho _{n} \cdot \rho - \int _{\varOmega }T_{k}(\sigma _{n}) |\rho |^{2} \le \int _{\varOmega }T_{k}(\sigma _{n}) |\rho _{n}|^{2}. \end{aligned}$$

Since, by the assumptions on $\sigma _{n}$ and $\rho$, $T_{k}(\sigma _{n})\rho$ converges to $T_{k}(\sigma )\rho$ strongly in $(L^{2}(\varOmega ))^{N}$ by Lebesgue theorem, and $T_{k}(\sigma _{n}) |\rho |^{2}$ converges to $T_{k}(\sigma ) |\rho |^{2}$ strongly in $L^{1}(\varOmega )$ (again by Lebesgue theorem), the left-hand side passes to the limit, and (3.2) is proved. But now, since $\sigma _{n}$ is nonnegative,

$$\begin{aligned} \int _{\varOmega }\,T_{k}(\sigma )|\rho |^{2} \le \liminf _{n \rightarrow +\infty } \int _{\varOmega }\,T_{k}(\sigma _{n})|\rho _{n}|^{2} \le \liminf _{n \rightarrow +\infty } \int _{\varOmega }\,\sigma _{n}|\rho _{n}|^{2} \le C, \end{aligned}$$

and (3.1) then follows letting $k$ tend to infinity and using the monotone convergence theorem.

Theorem 3.4

Let $p > 0$ be a real number, and let $a : \varOmega \rightarrow \mathbb {R}$ be a measurable function such that

$$\begin{aligned} 0 < \alpha \le a(x) \le \beta , \quad \text{ almost } \text{ everywhere } \text{ in } \, \varOmega . \end{aligned}$$

Let $f$ be a function in $L^{2_{*}}(\varOmega )$, where $2_{*} = \frac{2N}{N+2}$, and let $g$ be a function in $L^{1}(\varOmega )$. Then, there exists a solution $(u,z)$ of the system say

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}([a(x) + |z|^{p}] \nabla u) = f &{}\quad \text{ in } \, \varOmega , \\ -\mathrm{div}(b(x) \nabla z)+ z|z|^{2p-1}\,|\nabla u|^2 = g &{}\quad \text{ in }\, \varOmega , \\ \, u = 0 = z &{}\quad \text{ on }\, \partial \varOmega , \end{array}\right. } \end{aligned}$$

with $u$ in $H_0^{1}(\varOmega )$ and $z$ in $W^{1,q}_{0}(\varOmega )$, for every $q < \frac{N}{N-1}$.

Proof

As before, we begin by approximating the system: let $f_{n}$ and $g_{n}$ be sequences of $L^{\infty }(\varOmega )$ functions converging to $f$ and $g$ in $L^{2_{*}}(\varOmega )$ and $L^{1}(\varOmega )$, respectively, and such that (2.3) holds, and let $(u_n,z_n)$ be a solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}([a(x)+|z_n|^p] \nabla u_n) = f_{n} &{}\quad \text{ in } \varOmega , \\ -\mathrm{div}(b(x) \nabla z_n)+z_n|z_n|^{2p-1}|\nabla u_n|^2 = g_{n} &{}\quad \text{ in } \varOmega , \\ \, u_n= 0 = z_n&{}\quad \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}$$

(3.3)

which exists by Theorem 3.2. Choosing $u_n$ as test function in the first equation, we obtain

Starting from this inequality, and using Sobolev embedding, we have that $u_n$ is bounded in $H_0^{1}(\varOmega )$. Hence, up to subsequences still denoted by $u_n$, it converges weakly in $H_0^{1}(\varOmega )$, strongly in $L^{2}(\varOmega )$ and almost everywhere to some function $u$ in $H_0^{1}(\varOmega )$.

Using, as in the proof of Theorem 2.2, the fact that the lower-order term of the second equation has the same sign of $z_n$, we have that both (2.9) and (2.11) hold true. In particular, (2.11) becomes

$$\begin{aligned} \int \nolimits _{\{|z_n| \ge k+1\}} |z_n|^{2p}|\nabla u_n|^2 \le \int \nolimits _{\{|z_n| \ge k\}} |g|. \end{aligned}$$

(3.4)

The boundedness of $z_n$ in $W^{1,q}_{0}(\varOmega )$, for every $q < \frac{N}{N-1}$, implies that, up to subsequences still denoted by the same name, $z_n$ converges weakly in $W^{1,q}_{0}(\varOmega )$, for every $q < \frac{N}{N-1}$, strongly in $L^{1}(\varOmega )$ and almost everywhere in $\varOmega$, to some function $z$ which belongs to $W^{1,q}_{0}(\varOmega )$, for every $q < \frac{N}{N-1}$.

If we take $k = 0$ in (3.4), we obtain

$$\begin{aligned} \int \nolimits _{\{|z_n| \ge 1\}}\,|z_n|^{2p}|\nabla u_n|^{2} \le \int _{\varOmega }|g_{n}| \le C. \end{aligned}$$

Since $u_n$ is bounded in $H_0^{1}(\varOmega )$, from this inequality we obtain

$$\begin{aligned} \int _{\varOmega }|z_n|^{2p}|\nabla u_n|^{2} \le \int \nolimits _{\{|z_n| \le 1\}} |\nabla u_n|^{2} + \int \nolimits _{\{|z_n| \ge 1\}}\,|z_n|^{2p}|\nabla u_n|^{2} \le C. \end{aligned}$$

(3.5)

Applying Lemma 3.3 with $\sigma _{n} = |z_n|^{2p}$ and $\rho _{n} = \nabla u_n$, we have, from (3.5),

$$\begin{aligned} \int _{\varOmega }|z|^{2p}|\nabla u|^{2} \le \liminf _{n \rightarrow +\infty } \, \int _{\varOmega }|z_n|^{2p}|\nabla u_n|^{2} \le C, \end{aligned}$$

so that $|z|^{2p}|\nabla u|^{2}$ belongs to $L^{1}(\varOmega )$.

Now, we prove that $|z_n|^{p} \nabla u_n$ weakly converges to $|z|^{p}\nabla u$ in $(L^{2}(\varOmega ))^{N}$. If $\Psi$ is a function in $(L^{2}(\varOmega ))^{N}$, then

$$\begin{aligned} \displaystyle \int _{\varOmega }|z_n|^{p}\nabla u_n\cdot \Psi&= \displaystyle \int \nolimits _{\{|z_n| \le k\}} |z_n|^{p}\nabla u_n\cdot \Psi + \int \nolimits _{\{|z_n| \ge k\}} |z_n|^{p}\nabla u_n\cdot \Psi \\&= \displaystyle \int \nolimits _{\{|z| \le k\}} |z|^{p}\nabla u \cdot \Psi + \varepsilon _{n} + \int \nolimits _{\{|z_n| \ge k\}} |z_n|^{p}\nabla u_n\cdot \Psi , \end{aligned}$$

where $\varepsilon _{n}$ is a quantity which tends to zero as $n$ tends to infinity. On the other hand,

$$\begin{aligned} \displaystyle \int _{\varOmega }|z|^{p}\nabla u \cdot \Psi&= \displaystyle \int \nolimits _{\{|z_n| \le k\}} |z|^{p}\nabla u \cdot \Psi + \int \nolimits _{\{|z_n| \ge k\}} |z|^{p}\nabla u \cdot \Psi \\&= \displaystyle \int \nolimits _{\{|z| \le k\}} |z|^{p}\nabla u \cdot \Psi + \varepsilon _{n} + \int \nolimits _{\{|z_n| \ge k\}} |z|^{p}\nabla u \cdot \Psi . \end{aligned}$$

Therefore,

$$\begin{aligned} \displaystyle \left| \int _{\varOmega }[|z_n|^{p}\nabla u_n- |z|^{p}\nabla u]\cdot \psi \right|&\le \int \nolimits _{\{|z_n| \ge k\}} |z_n|^{p}|\nabla u_n| |\Psi |\\&\displaystyle + \int \nolimits _{\{|z_n| \ge k\}} |z|^{p}|\nabla u| |\Psi | + 2\varepsilon _{n}. \end{aligned}$$

Using that $|z_n|^{p}\nabla u_n$ is bounded in $(L^{2}(\varOmega ))^{N}$, and that $|z|^{p}\nabla u$ belongs to $(L^{2}(\varOmega ))^{N}$, we have

$$\begin{aligned} \displaystyle \int \nolimits _{\{|z_n| \ge k\}} |z_n|^{p}|\nabla u_n| |\Psi |&\le \displaystyle \left( \int _{\varOmega }|z_n|^{2p}|\nabla u_n|^{2}\right) ^{\frac{1}{2}} \left( \int \nolimits _{\{|z_n| \ge k\}} |\Psi |^{2} \right) ^{\frac{1}{2}} \\&\le \displaystyle C\,\left( \int \nolimits _{\{|z_n| \ge k\}} |\Psi |^{2} \right) ^{\frac{1}{2}}, \end{aligned}$$

and

$$\begin{aligned} \displaystyle \int \nolimits _{\{|z_n| \ge k\}} |z|^{p}|\nabla u| |\Psi |&\le \displaystyle \left( \int _{\varOmega }|z|^{2p}|\nabla u|^{2}\right) ^{\frac{1}{2}} \left( \int \nolimits _{\{|z_n| \ge k\}} |\Psi |^{2} \right) ^{\frac{1}{2}}\\&\le \displaystyle C\,\left( \int \nolimits _{\{|z_n| \ge k\}} |\Psi |^{2} \right) ^{\frac{1}{2}}, \end{aligned}$$

so that since $z_n$ is strongly convergent in $L^{1}(\varOmega )$, for every $\delta > 0$, there exists $k_{\delta } \ge 0$ such that

$$\begin{aligned} \left| \int \nolimits _{\{|z_n| \ge k\}} |z_n|^{p}\nabla u_n\cdot \Psi \right| \le \delta , \quad \left| \int \nolimits _{\{|z_n| \ge k\}} |z|^{p}\nabla u \cdot \Psi \right| \le \delta , \end{aligned}$$

for every $n$ in $\mathbb {N}$. Thus, choosing $n \ge n_{\delta }$, where $n_{\delta }$ is such that $\varepsilon _{n} \le \delta$, we have

$$\begin{aligned} \left| \int _{\varOmega }[|z_n|^{p}\nabla u_n- |z|^{p}\nabla u]\cdot \psi \right| \le 4\delta , \end{aligned}$$

for every $n \ge n_{\delta }$, which implies the desired weak convergence.

Choosing $v$ in $H_0^{1}(\varOmega )$ as test function in the first equation, we obtain

$$\begin{aligned} \int _{\varOmega }[a(x) + |z_n|^{p}]\nabla u_n\cdot \nabla v = \int _{\varOmega }f_{n}\,v, \end{aligned}$$

and, passing to the limit as $n$ tends to infinity, using the weak convergence just proved,

$$\begin{aligned} \int _{\varOmega }[a(x) + |z|^{p}]\nabla u \cdot \nabla v = \int _{\varOmega }f\,v, \end{aligned}$$

so that $u$ is a solution of the first equation of the system. If we take $v = u$, we then have

$$\begin{aligned} \int _{\varOmega }[a(x) + |z|^{p}]|\nabla u|^{2} = \int _{\varOmega }f\,u. \end{aligned}$$

(3.6)

On the other hand, choosing $u_n$ as test function in the first approximate equation, we obtain

$$\begin{aligned} \int _{\varOmega }[a(x) + |z_n|^{p}]|\nabla u_n|^{2} = \int _{\varOmega }f_{n}\,u_n. \end{aligned}$$

Passing to the limit as $n$ tends to infinity, and using (3.6), and the weak convergence of $u_n$ to $u$ in $H_0^{1}(\varOmega )$, we get

$$\begin{aligned} \displaystyle \lim _{n \rightarrow +\infty }\,\int _{\varOmega }[a(x) + |z_n|^{p}]|\nabla u_n|^{2}&= \displaystyle \lim _{n \rightarrow +\infty } \, \int _{\varOmega }f_{n}\,u_n\\&= \displaystyle \int _{\varOmega }f\,u = \int _{\varOmega }[a(x) + |z|^{p}]|\nabla u|^{2}. \end{aligned}$$

Therefore,

$$\begin{aligned} \lim _{n \rightarrow +\infty }\,\int _{\varOmega }[a(x) + |z_n|^{p}]|\nabla u_n|^{2} = \int _{\varOmega }[a(x) + |z|^{p}]|\nabla u|^{2}. \end{aligned}$$

(3.7)

Choosing $\sigma _{n} = a(x)$ and $\rho _{n} = \nabla u_n$ in Lemma 3.3, or $\sigma _{n} = |z_n|^{p}$ and $\rho _{n} = \nabla u_n$ in the same lemma, we have

$$\begin{aligned} \int _{\varOmega }a(x)|\nabla u|^{2} \le \liminf _{n \rightarrow +\infty }\,\int _{\varOmega }a(x)|\nabla u_n|^{2}, \end{aligned}$$

and

$$\begin{aligned} \int _{\varOmega }|z|^{p}|\nabla u|^{2} \le \liminf _{n \rightarrow +\infty }\,\int _{\varOmega }|z_n|^{p}|\nabla u_n|^{2} \le C, \end{aligned}$$

with the latter inequality being due to the fact that

$$\begin{aligned} \int _{\varOmega }|z_n|^{p}|\nabla u_n|^{2} \le \int \nolimits _{\{|z_n| \le 1\}} |\nabla u_n|^{2} + \int \nolimits _{\{|z_n| \ge 1\}} |z_n|^{2p}|\nabla u_n|^{2} \le C. \end{aligned}$$

Define now

$$\begin{aligned} A_{n} = \int _{\varOmega }a(x)|\nabla u_n|^{2}, \quad A = \int _{\varOmega }a(x)|\nabla u|^{2}, \end{aligned}$$

and

$$\begin{aligned} B_{n} = \int _{\varOmega }|z_n|^{p}|\nabla u_n|^{2}, \quad B = \int _{\varOmega }|z|^{p}|\nabla u|^{2} \end{aligned}$$

From (3.7), it follows that

$$\begin{aligned} \lim _{n \rightarrow +\infty } (A_{n} + B_{n}) = A + B, \end{aligned}$$

with

$$\begin{aligned} A \le \liminf _{n \rightarrow +\infty }\,A_{n}, \quad \text{ and } \quad B \le \liminf _{n \rightarrow +\infty }\,B_{n}. \end{aligned}$$

But then, since $A_{n} = (A_{n} + B_{n}) - B_{n}$, we have

$$\begin{aligned} \liminf _{n \rightarrow +\infty } \, A_{n} = A + B - \limsup _{n \rightarrow +\infty }\,B_{n}, \end{aligned}$$

and so

$$\begin{aligned} A + B = \liminf _{n \rightarrow +\infty } \, A_{n} + \limsup _{n \rightarrow +\infty }\,B_{n} \ge A + \limsup _{n \rightarrow +\infty }\,B_{n}, \end{aligned}$$

which then implies

$$\begin{aligned} \limsup _{n \rightarrow +\infty }\,B_{n} \le B. \end{aligned}$$

Therefore, $B_{n}$ converges to $B$, and so $A_{n}$ converges to $A$. Thus,

$$\begin{aligned} \lim _{n \rightarrow +\infty }\,\int _{\varOmega }a(x)|\nabla u_n|^{2} = \int _{\varOmega }\,a(x)|\nabla u|^{2}, \end{aligned}$$

which then implies that $u_n$ strongly converges to $u$ in $H_0^{1}(\varOmega )$. In particular, and always up to subsequences, $\nabla u_n$ almost everywhere converges to $\nabla u$.

We can now pass to the limit in the second equation; as in the proof of Theorem 2.2, the only difficult part is the proof of the strong convergence in $L^{1}(\varOmega )$ of the lower-order term. Due to the almost everywhere convergence of $z_n$ to $z$ and of $\nabla u_n$ to $\nabla u$, we have

$$\begin{aligned} z_n|z_n|^{2p-1}|\nabla u_n|^{2} \rightarrow z|z|^{2p-1}|\nabla u|^{2}, \quad \text{ almost } \text{ everywhere } \text{ in } \varOmega \text{, } \end{aligned}$$

so that we only have to prove the equiintegrability in order to apply Vitali theorem. If $E$ is a measurable subset of $\varOmega$, we have, recalling (2.11),

$$\begin{aligned} \displaystyle \int \nolimits _{E} |z_n|^{2p}|\nabla u_n|^{2}&\le \displaystyle k^{2p}\int \nolimits _{\{|z_n| \le k\} \cap E} |\nabla u_n|^{2} + \int \nolimits _{\{|z_n| \ge k\}} |z_n|^{2p}|\nabla u_n|^{2}\\&\le \displaystyle k^{2p}\int \nolimits _{E} |\nabla u_n|^{2} + \int \nolimits _{\{|z_n| \ge k-1\}}|g_{n}|. \end{aligned}$$

We now first choose $k$ such that the second term is small (which can be done since $g_{n}$ is strongly convergent in $L^{1}(\varOmega )$, as is $z_n$) and then choose $\mathrm{meas}(E)$ small enough so that the first term is small [and this can be done since $|\nabla u_n|$ is strongly convergent in $L^{1}(\varOmega )$]. Thus, the sequence $|z_n|^{2p}|\nabla u_n|^{2}$ is equiintegrable, and so, the lower-order term is strongly convergent in $L^{1}(\varOmega )$, as desired. $\square$

If we want to put ourselves in a “variational” setting, we have to make more restrictive assumptions on the data and on $p$.

Theorem 3.5

Let $p > 1$ be a real number, and let $a : \varOmega \rightarrow \mathbb {R}$ be a measurable function such that

$$\begin{aligned} 0 < \alpha \le a(x) \le \beta , \quad \text{ almost } \text{ everywhere } \text{ in } \varOmega . \end{aligned}$$

Let $f$ be a function in $L^{2_{*}}(\varOmega )$, where $2_{*} = \frac{2N}{N+2}$, let $g$ be a function in $L^{r}(\varOmega )$, with

$$\begin{aligned} r \ge \frac{N(p+2)}{N + 2 + 2p}, \end{aligned}$$

(3.8)

and let $B > 0$. Then, there exists a solution $(u,z)$ of the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}([a(x) + |z|^{p}] \nabla u) = f &{}\quad \text{ in } \varOmega , \\ -\mathrm{div}(b(x) \nabla z) + B\, z|z|^{p-2}\,|\nabla u|^2 = g &{}\quad \text{ in } \varOmega , \\ \, u = 0 = z &{}\quad \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}$$

with $u$ and $z$ in $H_0^{1}(\varOmega )$. Moreover, $z^{p}|\nabla u|^{2}$ belongs to $L^{1}(\varOmega )$.

Remark 3.6

We remark that for every choice of $p > 1$, we have

$$\begin{aligned} \frac{N(p+2)}{N + 2 + 2p} > 2_{*}, \end{aligned}$$

so that the datum $g$ always belongs to the dual space of $H_0^{1}(\varOmega )$. Furthermore, if $p$ tends to infinity, then $r$ has to be larger than $\frac{N}{2}$ (which is the assumption on $g$ in order to have bounded solutions $z$).

Remark 3.7

If we choose $B = \frac{p}{2}$, the system becomes variational, and the solutions can be found as minima of the functional

$$\begin{aligned} J(v,w) = \frac{1}{2} \int _{\varOmega }[a(x) + |w|^{p}]|\nabla v|^{2} + \frac{1}{2} \int _{\varOmega }b(x)|\nabla w|^{2} - \int _{\varOmega }f\,v - \int _{\varOmega }g\,w. \end{aligned}$$

Proof

The proof of the result follows the lines of the argument in Theorem 3.4. We begin by approximating the system, taking $f_{n}$ and $g_{n}$ sequences of $L^{\infty }(\varOmega )$ functions converging to $f$ and $g$ in $L^{2_{*}}(\varOmega )$ and $L^{r}(\varOmega )$, respectively, and such that (2.3) holds. We then consider a solution $(u_n,z_n)$, which exists by Theorem 3.2, of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}([a(x)+|z_n|^p] \nabla u_n) = f_{n} &{}\quad \text{ in } \varOmega , \\ -\mathrm{div}(b(x) \nabla z_n)+ B\,z_n|z_n|^{p-2}|\nabla u_n|^2 = g_{n} &{}\quad \text{ in } \varOmega , \\ \, u_n= 0 = z_n&{}\quad \text{ on } \partial \varOmega . \end{array}\right. } \end{aligned}$$

Since the lower-order term of the second equation has the same sign as the solution, standard elliptic techniques (see for example [4]) imply that $z_n$ is bounded in $H_0^{1}(\varOmega )$ (see Remark 3.6) and in $L^{r^{**}}(\varOmega )$. Since the assumption on $r$ implies

$$\begin{aligned} \frac{1}{r} + \frac{p+1}{r^{**}} \le 1, \end{aligned}$$

choosing $|z_n|^{p+1}T_{1}(G_{k}(z_n))$ as test function in the second equation, we obtain, dropping two nonnegative terms,

which is exactly (2.11). From now on, the proof is identical to that of Theorem 3.4, and therefore, we omit it. $\square$

4 Breaking of the maximum principle

When considering system (1.1), both with $a$ bounded or unbounded, one of the main properties of the solutions is that (no matter which assumptions we make on the data) $|h(x,z)||\nabla u|^{2}$ belongs to $L^{1}(\varOmega )$, as a consequence of the sign assumption on $h$. However, if the datum $f$ is in $L^{1}(\varOmega )$ [or, in general, if it does not belong to $H^{-1}(\varOmega )$], $u$ does not belong to $H_0^{1}(\varOmega )$, so that the function $|\nabla u|^{2}$ may not belong to $L^{1}(\varOmega )$. This fact, as we will prove in the next theorem, implies that the solution $z$ has to be zero inside $\varOmega$ even if the datum $g$ is nonnegative. In other words, the maximum principle fails for the second equation of system (1.1).

Theorem 4.1

Let $f \ge 0$ be a function in $L^{1}(\varOmega )$, and let $g \ge 0$ belong to $L^{1}(\varOmega )$. Let $a(x,s)$, $b(x)$ and $h(x,s)$ be such that (2.1) and (2.2) hold, and let $(u,z)$ be a solution of (1.1) given by Theorem 2.2. If, for every $\delta > 0$,

$$\begin{aligned} \inf _{\varOmega \times [\delta ,+\infty )} h(x,s) = \Gamma _{\delta } > 0, \end{aligned}$$

and if there exists $x_{0}$ in $\varOmega$ such that $f$ belongs to $L^{\infty }(\varOmega \!\setminus \!B_{r}(x_{0}))$, and $f$ does not belong to $H^{-1}(B_{r}(x_{0}))$ for every $r > 0$, then

$$\begin{aligned} \mathrm{ess\,inf}_{B_r(x_0)} z = 0, \quad \forall r > 0. \end{aligned}$$

(4.1)

Proof

First of all, observe that both $u$ and $z$ are nonnegative functions. If (4.1) is false, then there exist $R>0$ and $\delta >0$ such that

$$\begin{aligned} \mathrm{ess\,inf}_{B_R(x_0)} z = \delta . \end{aligned}$$

Taking $T_{\delta }(z_n)$ as test function in the second equation of (2.4), we obtain, dropping a nonnegative term,

$$\begin{aligned} \int \nolimits _{\varOmega } h(x,z_n)\,T_{\delta }(z_n)|\nabla u_n|^{2} \le \delta \int \nolimits _{\varOmega }\,|g|. \end{aligned}$$

Recalling (2.14), and the fact that $z \ge 0$, we then have

$$\begin{aligned} \delta \,\int \nolimits _{\{z \ge \delta \}} h(x,z)|\nabla u|^{2} \le \int \nolimits _{\varOmega } h(x,z)\,T_{\delta }(z)|\nabla u|^{2} \le \delta \int \nolimits _{\varOmega }\,|g|, \end{aligned}$$

so that

$$\begin{aligned} \int \nolimits _{B_{R}(x_{0})}\!\!\!\!\!\! h(x,z)|\nabla u|^{2} = \int \nolimits _{B_{R}(x_{0}) \cap \{z \ge \delta \}} \!\!\!\!\!\!\!\!\!\!\!\! h(x,z)|\nabla u|^{2} \le \int \nolimits _{\{z \ge \delta \}} \!\!\!\!\!\! h(x,z)|\nabla u|^{2} \le \int \nolimits _{\varOmega }\,|g|. \end{aligned}$$

Therefore, recalling the assumption on $h$, we have

$$\begin{aligned} \Gamma _\delta \int \nolimits _{B_R(x_0)}|\nabla u|^2 \le \int \nolimits _{B_R(x_0)}h(x,z)|\nabla u|^2 \le \int _{\varOmega }|g|. \end{aligned}$$

By the result of [5], we also have

so that $u$ belongs to $H_0^{1}(\varOmega )$. This, and the boundedness of $a(x,s)$, implies that $f$ belongs to $H^{-1}(\varOmega )$, which is false. $\square$

Note that, as a consequence of this result, if $f$ is an $L^{1}(\varOmega ) \setminus H^{-1}(\varOmega )$ function which has an isolated singularity in $x_{0}$, then the coupling between the equations of the system forces $z$ to be zero at $x_{0}$.

5 Measure data

Now, we turn to the case of measure data for system (1.1). In this case, as we will see, we can prove a negative (i.e., nonexistence) result also in the “simple” case of $a(x,s)$ bounded.

We recall that the main tool in the proof of the existence result of Theorem 2.2 is the fact that if $u_n$ and $z_n$ are sequences of approximating solutions, we can choose $k$ and $j$ large enough so that

$$\begin{aligned} \int \nolimits _{\{|z_n| \ge j\}}|g| \le \delta , \quad \int \nolimits _{\{|u_n| \ge k\}}|f| \le \delta , \quad \int \nolimits _{\{|u_n| \ge k\}}|g| \le \delta , \end{aligned}$$

uniformly with respect to $n$. To achieve that, it is of fundamental importance the fact that the sequences $f_{n}$ and $g_{n}$ strongly converge in $L^{1}(\varOmega )$, and not (for example) in the weak$*$ topology of measures. And indeed, if $f_{n}$ converges to (for example) a Dirac mass, and if $g_{n}$ converges to another (different) Dirac mass, there may not be existence of solutions for the system (1.1), as the following example shows.

Example 5.1

Let $\varOmega = B_{1}(0)$ be the unit ball of $\mathbb {R}^{N}$, and let us consider the sequences $u_n$ and $z_n$ of solutions of the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_n= f_{n} &{}\quad \text{ in } \varOmega , \\ -\Delta z_n+ z_n^{\gamma }|\nabla u_n|^2 = g_{n} &{}\quad \text{ in } \varOmega , \\ \, u_n= 0 = z_n&{}\quad \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}$$

(5.1)

where $f_{n}$ is the sequence of nonnegative $L^{\infty }(\varOmega )$ functions equal to $D_{N}\,n^{N}$ on $B_{\frac{1}{n}}(0)$, and zero elsewhere, with $D_{N}$ such that $f_{n}$ converges to $\delta _{0}$, the Dirac mass at the origin, and $g_{n}$ is a sequence of nonnegative $L^{\infty }(\varOmega )$ functions with support contained in $B_{\frac{1}{n}}(x_{0})$ and convergent to $\delta _{x_{0}}$, the Dirac mass at $x_{0} \ne 0$; on $\gamma$, we suppose that $\gamma > \frac{N}{N-2}$.

It is well known that $u_n$ converges to $u(x) = C_{N}(|x|^{2-N} - 1)$, the Green function of the Laplacian relative to the origin; calculating explicitly $u_n$, one can see that $u_n\equiv u$ outside of $B_{\frac{1}{n}}(0)$, so that

$$\begin{aligned} |\nabla u_n|^{2} = |\nabla u|^{2} = C'_{N}|x|^{2-2N} \ge C'_{N}, \quad \mathrm{for} \quad \frac{1}{n} \le |x| \le 1. \end{aligned}$$

(5.2)

Since from the second equation we have that

$$\begin{aligned} \int _{\varOmega }z_n^{\gamma }|\nabla u_n|^{2} \le \int _{\varOmega }g_{n} \le C, \end{aligned}$$

from (5.2), we obtain

$$\begin{aligned} C'_{N} k^{\gamma }\,\mathrm{meas}(\{z_n\ge k\} \cap (\varOmega \!\setminus \!B_{\frac{1}{n}}(0))) \le \int \nolimits _{\varOmega \setminus B_{\frac{1}{n}}(0)}z_n^{\gamma }|\nabla u_n|^{2} \le C, \end{aligned}$$

and so

$$\begin{aligned} \mathrm{meas}(\{z_n\ge k\} \cap \left( \varOmega \!\setminus \!B_{\frac{1}{n}}(0))\right) \le \frac{C}{k^{\gamma }}. \end{aligned}$$

(5.3)

On the other hand, since the support of $g_{n}$ is disjoint from the ball $B_{\frac{1}{n}}(0)$ if $n \ge n_{0}$, with $n_{0}$ large enough, the result of [5] implies that $z_n$ is bounded in $L^{\infty }(B_{\frac{1}{n_{0}}})$ by a constant $\overline{k}$. Therefore, if $k \ge \overline{k}$, we have

$$\begin{aligned} \mathrm{meas}(\{z_n\ge k\}) = \mathrm{meas}(\{z_n\ge k\} \cap \left( \varOmega \!\setminus \!B_{\frac{1}{n_{0}}}(0))\right) \le \frac{C}{k^{\gamma }}. \end{aligned}$$

On the other hand, choosing $T_{k}(z_{n})$ as test function in the second equation, and dropping the positive second term, we have

$$\begin{aligned} \int _{\varOmega }|\nabla T_{k}(z_n)|^{2} \le \int _{\varOmega }g_{n}\,T_{k}(z_n) \le C\,k. \end{aligned}$$

Reasoning as in [1], the latter two inequalities imply

$$\begin{aligned} \mathrm{meas}(\{|\nabla z_n| \ge \lambda \}) \le \frac{C}{\lambda ^{\frac{2\gamma }{\gamma +1}}}, \end{aligned}$$

so that $z_n$ is bounded in $W^{1,q}_{0}(\varOmega )$ for every $q < \frac{2\gamma }{\gamma + 1}$. Since $\gamma > \frac{N}{N-2}$, then $\frac{2\gamma }{\gamma + 1} > \frac{N}{N-1}$, which implies that $z_n$ is bounded in $W^{1,p}_{0}(\varOmega )$ for some $p > \frac{N}{N-1}$.

Since the set $\{x_{0}\}$ has zero $p'$-capacity (as every point in $\mathbb {R}^{N}$), for every $\delta > 0$ there exists a function $\psi _{\delta }$ in $C^{\infty }_{0}(\varOmega )$ (see [8]) such that

$$\begin{aligned} 0 \le \psi _{\delta } \le 1, \quad \int _{\varOmega }|\nabla \psi _{\delta }|^{p'} \le \delta , \quad \int _{\varOmega }g_{n}\,(1 - \psi _{\delta }) = 0, \end{aligned}$$

the latter being true for every $n$ large enough.

We choose now $T_{k}(z_n)\,(1-\psi _{\delta })$ as test function in the second equation, to obtain:

$$\begin{aligned}&\displaystyle \int _{\varOmega }|\nabla T_{k}(z_n)|^{2}(1 - \psi _{\delta }) + \int _{\varOmega }z_n^{\gamma }\,T_{k}(z_n)\,(1-\psi _{\delta })|\nabla u_n|^{2}\\&\displaystyle \quad = \int _{\varOmega }\nabla z_n\cdot \nabla \psi _{\delta }\,T_{k}(z_n) + \int _{\varOmega }g_{n}\,T_{k}(z_n)\,(1 - \psi _{\delta }). \end{aligned}$$

Dropping the nonnegative second term, observing that the last term is zero for $n$ large enough, and passing to the limit as $n$ tends to infinity, we obtain (by weak lower semicontinuity)

$$\begin{aligned} \int _{\varOmega }|\nabla T_{k}(z)|^{2}(1 - \psi _{\delta }) \le \int _{\varOmega }\nabla z \cdot \nabla \psi _{\delta }\,T_{k}(z). \end{aligned}$$

Letting $\delta$ tends to zero (and recalling that $|\nabla \psi _{\delta }|$ tends to zero in $L^{p'}(\varOmega )$, while $|\nabla z|$ belongs to $L^{p}(\varOmega )$), we obtain

$$\begin{aligned} 0 \le \int _{\varOmega }|\nabla T_{k}(z)|^{2} \le 0, \end{aligned}$$

which implies that $z \equiv 0$. However, $z \equiv 0$ is not a solution of the second equation of (1.1).

Remark 5.2

Remark that the conclusion of the previous example remains true every time that $g_{n}$ converges to a Dirac mass concentrated in a point $x_{0} \ne 0$, and the sequence $u_n$ of solutions of the first equation is such that $|\nabla u_n| \ge C > 0$ outside a neighborhood of $0$.