1. Introduction

To contextualize our work, we will begin by discussing the scalar case (a single equation), which serves as motivation for the system addressed in our research. Let us consider the following problem:

(𝒫_λ)\begin{align} \left\{ \begin{array}{ll} -\Delta u=\lambda a(x)u^q+b(x)u^p& \text{in}~~\Omega\\ u=0&\text{on}~~\partial\Omega. \end{array} \right. \end{align}

Here, $\Omega \subset \mathbb{R}^N$ is a bounded domain, $a, b \in L^\infty(\Omega)$, λ > 0 and $0 \lt q \lt 1 \lt p$. When $a \equiv b \equiv 1$, Ambrosetti, Brézis and Cerami in [Reference Ambrosetti, Brézis and Cerami4] studied this problem. The authors showed that there exists $\Lambda \gt 0$ such that $(\mathcal{P}_\lambda)$ has a positive solution when $0 \lt \lambda \leq \Lambda$, and there is no positive solution when $\lambda \gt \Lambda$. Moreover, if $p + 1 \leq 2^* := 2N/(N - 2)$, then the solutions of $(\mathcal{P}_\lambda)$ correspond to critical points of the functional $I_\lambda \in C^1(H_0^1(\Omega), \mathbb{R})$ defined by

\begin{equation*} I_\lambda(u) = \int_\Omega |\nabla u|^2 - \frac{\lambda}{q + 1} \int_\Omega |u|^{q + 1} - \frac{1}{p + 1} \int_\Omega |u|^{p + 1}. \end{equation*}

Using variational methods and techniques introduced by Brézis and Nirenberg in [Reference Brézis and Nirenberg12], they showed the existence of a second positive solution for $0 \lt \lambda \lt \Lambda$ (it is necessary to demand some regularity for the domain in order to guarantee the existence of the unit exterior normal vector to $\partial\Omega$. See, for example, [Reference Ambrosetti, Brézis and Cerami4, Lemma 4.1]). From a purely mathematical perspective, problems with concave-convex type nonlinearities have received great interest since the seminal work [Reference Ambrosetti, Brézis and Cerami4]. The list of references is extensive, among which we highlight [Reference Barrios, Colorado, De Pablo and Sánchez6–Reference Bartsch and Willem8, Reference Brändle, Colorado, De Pablo and Sánchez10, Reference García-Azorero and Peral28–Reference García-Azorero, Peral and Rossi30, Reference Hirano, Saccon and Shioji34, Reference Marano and Papageorgiou39, Reference Tang44, Reference Wang45, Reference Wu47]. Recently, there has been considerable interest in problems with indefinite weights, that is, problems where the weight functions $a(\cdot)$ or $b(\cdot)$ may change sign. De Figueiredo, Gossez and Ubilla in the work [Reference De Figueiredo, Gossez and Ubilla25, Corollary 2.2 and Corollary 2.7] showed existence and non-existence results for $(\mathcal{P}_\lambda)$ when $a(\cdot)$ and $b(\cdot)$ satisfy certain conditions, including the possibility of $a(\cdot)$ and $b(\cdot)$ changing sign. More specifically, they showed that there exists $\lambda_0 \gt 0$ and $\overline{c} \gt 0$ such that $(\mathcal{P}_\lambda)$ admits two solutions when $p\leq 2^*-1$ and $0 \lt \lambda \lt \lambda_0$ and admits no solution if $p\leq 2^*-1$ and $\lambda \gt \overline{c}$. In a later work [Reference De Figueiredo, Gossez and Ubilla26, Theorem 4.1 and Theorem 4.2], assuming certain conditions, including: $a,\,b\in L^\infty(\Omega)$, $0\not\equiv a(x)\geq 0$ in Ω, and $\inf_{B_1}a(x) \gt 0$ for some ball $B_1\subset\Omega$, the same authors recovered the results obtained by Ambrosetti et al. [Reference Ambrosetti, Brézis and Cerami4], that is, they showed that there exists $0 \lt \Lambda\leq\infty$ such that: If $0\leq q \lt 1 \lt p$, the problem $(\mathcal{P}_\lambda)$ has at least one solution when $0 \lt \lambda\leq\Lambda$ and admits no solution when $\lambda \gt \Lambda\neq+\infty$. If $0\leq q \lt 1 \lt p\leq 2^*-1$, the problem $(\mathcal{P}_\lambda)$ admits at least two solutions when $0 \lt \lambda \lt \Lambda$.

The results provided by the works [Reference Ambrosetti, Brézis and Cerami4, Reference De Figueiredo, Gossez and Ubilla26] include nonlinearities with supercritical growth. In the context of the Laplacian operator, when Ω is bounded, the problem $-\Delta u=f(x,u)$ in Ω and u = 0 on $\partial\Omega$, is said to have supercritical growth when there is no C > 0 such that $|f(x,t)|\leq C(1+|t|^{2^{\ast}-1})$, a.e. $x\in\Omega$ and $t\in\mathbb{R}$, where $2^*=2N/(N-2)$ is the critical exponent of the Sobolev Embedding and $N\geq 3$. In this sense, the problem $(\mathcal{P}_\lambda)$ has supercritical growth when $p \gt 2^*-1$ and $N\geq 3$.

The literature concerning problems with concave-convex type nonlinearities, as can be seen in the works we mentioned earlier, is rich in problems whose nonlinearity can have supercritical growth. For problems with supercritical growth and nonlinearities that are not concave-convex type, see for example [Reference Albalawi, Mercuri and Moroz3, Reference Arcoya, Boccardo and Orsina5, Reference Candela, Palmieri and Salvatore15, Reference Candela, Perera and Sportelli16, Reference Clapp and Tiwari21]. To the best of our knowledge, very few existence results have been determined for elliptic systems with supercritical growth (see [Reference Candela and Sportelli17, Reference Da Silva22]).

Consider the following system:

(1.1)\begin{equation} -\Delta u=f_\lambda(x,u,v)\,\,\mbox{and}\,\,-\Delta v=g_\mu(x,u,v)\,\,\mbox{in}\,\,\Omega, \,\,u=v=0\,\,\mbox{on}\,\,\partial\Omega. \end{equation}

Although there is a substantial literature related to scalar problems involving concave-convex type nonlinearities, to this day the results of Ambrosetti et al. [Reference Ambrosetti, Brézis and Cerami4] have not been fully recovered for systems of type (1.1) in which:

(1.2)\begin{equation} \left\{ \begin{array}{lcl} f_\lambda(x,t,t)=\lambda a(x)t^{q_1}+c(x)t^{p_1}\,\,\mbox{and}\,\,g_\mu(x,t,t)=\mu b(x)t^{q_2}+d(x)t^{p_2}\\ a,b,c,d\in L^\infty(\Omega),\,\,0\leq q_i \lt 1 \lt p_i. \end{array} \right. \end{equation}

To date, issues such as non-existence of solution, as well as the existence of solutions for systems with nonlinearities exhibiting supercritical growth (i.e. $p_i \gt 2^*-1$ in Equation (1.2)), have not been addressed in the current literature. In general, research involving Equations (1.1)–(1.2) as well as its generalizations is almost entirely restricted to gradient-type systems, a concept we will explain next. Consider the following system:

(1.3)\begin{equation} \left\{ \begin{array}{ll} -\mathcal{L}_i u=\phi_i(x,u_1,u_2)& \mbox{in}~~\Omega \subset\mathbb{R}^N,\,\,i=1,2 \\ u_i=0&\mbox{on}~~\partial\Omega, \end{array} \right. \end{equation}

it will be called a gradient-type system if there exists $G(x,\cdot,\cdot)\in C^1(\mathbb{R}^2)$ a.e. in Ω, such that $\phi_i(x,t_1,t_2)=G_{t_i}(x,t_1,t_2)$. The importance of this type of system lies in the fact that it is possible to associate to them an Euler–Lagrange functional.

As far as we know, Wu in [Reference Wu46] was the first author to consider a system with nonlinearities of type (1.2) and Dirichlet boundary condition. The author studied the following gradient-type system: Equation (1.3) with $\mathcal{L}_i=\Delta$ and $\phi_i(x,u_1,u_2)=G_{u_i}(x,u_1,v_2)$ where

\begin{equation*} G(x,u_1,u_2)=\dfrac{\lambda a(x)}{q+1}|u_1|^{q+1}+\dfrac{\mu b(x)}{q+1}|u_2|^{q+1} +\dfrac{c(x)}{\alpha+\beta}|u_1|^\alpha|u_2|^\beta. \end{equation*}

When $0 \lt q \lt 1$, $2 \lt \alpha+\beta \lt 2^*$, $a^+(x):=\max\{a(x),0\}\not\equiv 0$, $b^+(x):=\max\{b(x),0\}\not\equiv 0$, and $c\in C^0(\overline{\Omega})$ with $0\not\equiv c(x)\geq 0$ in Ω, using variational methods and assuming certain conditions on the weight functions, the author provided results on existence and multiplicity of non-negative solutions, provided that λ > 0 and µ > 0 are sufficiently small.

There are many works dealing with the existence or multiplicity of non-negative solutions for systems with concave-convex nonlinearities of type (1.2), or even for systems involving the p-Laplacian operator and nonlinearities that generalize Equation (1.2). However, the approach we encounter in these works, in general, is the same as that used to address the problem $(\mathcal{P}_\lambda)$ from a variational perspective, so they are restricted to cases where the system is of gradient type and the nonlinearities exhibit subcritical or critical growth, i.e. $p_i\leq 2^*$ in Equation (1.2). Furthermore, the results are limited to local cases, in the sense that the existence of a solution is guaranteed only if the parameters λ > 0 and µ > 0 in Equation (1.1) are sufficiently small. The same applies to systems that generalize Equation (1.1) to more general operators. Regarding gradient-type systems (1.2)–(1.1), as well as their generalizations involving the operators: p-Laplacian (in this case $0\leq q_i \lt p-1 \lt p_i$), fractional Laplacian and p&q-Laplacian, we refer to the following works and the references contained therein [Reference Afrouzi and Rasouli2, 3.2, Reference Batkam and Colin9, Reference Chen and Deng18, Reference Cheng, Feng and Wei19, Reference He, Squassina and Zou32, Reference Li and Yang38, Reference Song, Chen and Yan42, Reference Yin48] (see also [Reference Brown and Wu14, Reference Rasouli and Afrouzi41] for nonlinearities on $\partial\Omega$).

As mentioned earlier, the current literature involving Equations (1.1)–(1.2) and their generalizations is almost entirely restricted to gradient-type systems. This implies that the exponent p_i appearing in Equation (1.2) is bounded by a power associated with the space in which the Euler–Lagrange functional is well-defined.

Some works involving systems require additional comments. In [Reference Da Silva22], the author of the present work studied the system (1.1)–(1.2) with:

(1.4)\begin{equation} \left\{ \begin{array}{lcl} f_\lambda(x,u,v)&=&\lambda a(x)u^{q_1}-\tau c(x)u^{\alpha-1}v^\beta \\ g_\mu(x,u,v)&=&\mu b(x)v^{q_2}-\delta d(x)u^\alpha v^{\beta-1}, \end{array} \right. \end{equation}

where $\lambda,\mu,\delta,\tau \gt 0$, $0 \lt q_i \lt 1$, $a^+(\cdot)\not\equiv0$, $b^+(\cdot)\not\equiv0$ in Ω and $c(x)\equiv d(x)\geq 0$ in Ω. In said work, we showed the existence of a solution (u, v) such that $0\not\equiv u\geq 0$ and $0\not\equiv v\geq 0$ in Ω, provided that $\alpha,\beta\geq 1$ and $\alpha+\beta \gt 2$ (in this case $p_i=\alpha+\beta-1 \lt +\infty$). This system, although only of gradient type for specific values of τ > 0 and δ > 0, can be converted into a gradient-type system for all $\tau,\delta \gt 0$ (see [Reference Da Silva22, Proof of Theorem 1.1 ]). However, the approach of this work cannot be applied to systems that do not have the specific form (1.4) or for cases where $c(\cdot)$ or $d(\cdot)$ in Equation (1.2) may change sign, or for cases where $c(\cdot)\not\equiv d(\cdot)$ in Ω.

The only work we are aware of that deals with a class of systems that are not of gradient type is [Reference Chhetri, Raynor and Robinson20]; in this work, Chhetri et al. showed the existence of a solution (u, v) for Equation (1.1) such that u and v are positive in Ω, without necessarily requiring the system to be of gradient type. However, f_λ and g_µ, by assumption, must have subcritical growth (see [Reference Chhetri, Raynor and Robinson20, Theorem 2.8 and Condition (H1)]). The conditions on f_λ and g_µ are more restrictive than those addressed in the present paper; for instance, it is required that there exists $R_0 \gt 0$ such that $f_\lambda(x,t,s) \gt \lambda_1(\Omega)s+1$ for every $t\geq0$ and $s \gt R_0$, and $g_\mu(x,t,s) \gt \lambda_1(\Omega)t+1$ for every $s\geq0$ and $t \gt R_0$ (see [Reference Chhetri, Raynor and Robinson20, p. 43]), where $\lambda_1(\Omega)$ is the first eigenvalue of $(-\Delta, H_0^1(\Omega))$. In the present work, f_λ and g_µ do not necessarily need to satisfy the latter condition nor Condition (H1) from [Reference Chhetri, Raynor and Robinson20]; the class of nonlinearities addressed in this paper is more general and allows for nonlinearities with supercritical growth.

Candela and Sportelli in [Reference Candela and Sportelli17] considered the following system:

(1.5)\begin{equation} \begin{cases} -\text{div}(a(x, u, \nabla u)) + A_t(x, u, \nabla u) = G_u(x, u, v) & \text{in } \Omega, \\ -\text{div}(b(x, v, \nabla v)) + B_t(x, v, \nabla v) = G_v (x, u, v) & \text{in } \Omega, \\ u = v = 0 & \text{on } \partial\Omega, \end{cases} \end{equation}

where $a(x,t,\xi)=(\partial_{\xi_1}A,\ldots,\partial_{\xi_N}A)$ and $b(x,t,\xi)=(\partial_{\xi_1}B,\ldots,\partial_{\xi_N}B)$, and the Euler–Lagrange functional associated with this system is defined on the Banach space $X:=(W^{1,p_1}_0(\Omega)\cap L^\infty(\Omega))\times(W^{1,p_2}_0(\Omega)\cap L^\infty(\Omega))$ (see [Reference Candela and Sportelli17, (1.6)]). Assuming certain conditions (see [Reference Candela and Sportelli17, $(h_0)-(h_7)$ and $(g_0)-(g_4)$]), among which $A(x,t,\xi)$ and $B(x,t,\xi)$ grow at least as fast as $(1 + |t|^{s_{1}p_{1}})|\xi|^{p_{1}}$ and $(1 + |t|^{s_{2}p_{2}})|\xi|^{p_{2}}$ respectively, where $p_i \gt 1$, $s_i\geq 0$, the authors showed that if

(1.6)\begin{equation} \limsup_{(u,v) \rightarrow (0,0)} \dfrac{G(x, u, v)}{|u|^{p_1} + |v|^{p_2}} \lt \alpha_2 \min\{\lambda_{1,1}, \lambda_{2,1}\} \quad \text{uniformly a.e. in } \Omega, \end{equation}

where $\lambda_{i,1}$ is the first eigenvalue of $(-\Delta_{p_i},W_0^{1,p_i})$ and $\alpha_2 \gt 0$ is a constant related to the growth of A and B (see [Reference Candela and Sportelli17, Theorem 4.1]), then Equation (1.5) admits a non-trivial solution (see [Reference Candela and Sportelli17, Theorems 1.1 and 4.1]), assuming additional conditions they showed the existence of infinitely many solutions for Equation (1.5). The notable characteristic of the work of these authors is that they provided an existence result for the system (1.5) with the possibility of $G(x,u,v)$ having a supercritical growth, in the sense of, the exponents q ₁ and q ₂ that appear in the growth of G_u and G_v respectively (see [Reference Candela and Sportelli17, $(g_1)$]), satisfy:

(1.7)\begin{equation} 1\leq q_i \lt p^*_i(s_i+1):=Np_i(s_i+1)/(N-p_i). \end{equation}

In this case, depending on the choice of s_i, it is possible to have $p_i^* \lt q_i \lt p^*_i(s_i+1)$ where $p^*_i$ is the critical Sobolev exponent associated with $W^{1,p_i}_0(\Omega)$; the multiplicity result provided by these authors also allows for supercritical growth for G. However, no information about the sign of the solutions was given. When $A=(1/p_1)|\xi|^{p_1}$ and $B=(1/p_2)|\xi|^{p_2}$, the system (1.5) becomes

\begin{equation*} -\Delta_{p_1}u=G_u(x,u,v)\,\,\mbox{and}\,\,-\Delta_{p_2}v=G_v(x,u,v)\,\,\mbox{in}\,\,\Omega, \,u=v=0\,\,\mbox{on}\,\partial\Omega. \end{equation*}

Thus $s_1=s_2=0$ and the nonlinearities G_u and G_v only have subcritical growth (see [Reference Candela and Sportelli17, Remark 3.1]). Although the techniques we employ in this work are not only directed at gradient-type systems, our results include nonlinearities that do not satisfy Equation (1.6). For example, if $\gamma_1,\gamma_2 \gt 0$ and $\gamma_1+\gamma_2 \lt p_i$ (or $\gamma_i \lt p_i$), condition (1.6) prevents G, near the origin, from having nonlinearities that grow like $|u|^{\gamma_1}|v|^{\gamma_2}$ (or $|u|^{\gamma_1}+|v|^{\gamma_2}$), which is a common type of nonlinearity in systems that generalize the results of [Reference Ambrosetti, Brézis and Cerami4].

The work of Adriouch and El Hamidi [Reference Adriouch and El Hamidi1] arose before [Reference Wu46]. In their work, the authors consider Equation (1.3) as a gradient-type system with subcritical growth, where $\mathcal{L}_1=-\Delta_{p}$ and $\mathcal{L}_2=-\Delta_{q}$. However, the nonlinearity the authors considered is concave-convex only in one of the equations. For example, when $p=q=2$, system (1.3) takes the form (1.1), the exponents of the nonlinearities in Equation (1.2) satisfy $0 \lt q_1 \lt 1 \lt p_1=p_2 \lt 2^*$ and $0 \lt q_2=1 \lt p_1=p_2 \lt 2^*$. That is, the second equation has a linear term instead of a concave one.

Another fact worth mentioning is that, as in Equation (1.1), in general, the nonlinearity $\phi_i(x,u,v)$ of Equation (1.3) also depends on a parameter. Typically, we have $\phi_1(x,u,v)=\phi_{1,\lambda}(x,u,v)$ and $\phi_2(x,u,v)=\phi_{2,\mu}(x,u,v)$. In all the works we are aware of, the authors ensure the existence of a solution only if the parameters λ > 0 and µ > 0 are sufficiently small. In other words, regarding the parameters, there are only local existence results. This raises an important question, for example: when $\phi_1=\phi_{1,\lambda}$ and $\phi_2=\phi_{2,\mu}$ are nonlinearities that generalize Equation (1.2), is the set $\mathcal{O}:=\{(\lambda,\mu)\in\mathbb{R}_+^2|$ (1.3) has a solution $(u,v)\,\,\mbox{with}\,\,u,v \gt 0\,\,\mbox{in}\,\Omega\,\}$ bounded or unbounded? We will show in this work that, surprisingly, this set can be unbounded. In relation to the non-existence of a positive solutions for Equations (1.1)–(1.2) (as well as for Equation (1.3) equipped with a condition that generalizes Equation (1.2)), while there is ample literature for the scalar case, there are no studies involving the non-existence issue for systems. None of the previously mentioned works addresses the non-existence of solutions for systems.

Although our work is inspired by that of Ambrosetti–Brézis–Cerami, our present contribution addresses interesting cases which, to the best of our knowledge, have not been considered before, such as non-gradient type systems. We will also consider nonlinearities with supercritical growth and present novel existence results even for critical and subcritical cases, as well as providing both existence and non-existence results.

In this work, we are dealing with the following systems:

\begin{equation*} (\mathcal{P}_1)\,\,\left\{ \begin{array}{ll} -\Delta u=\lambda a(x)u^q+c(x)u^\alpha v^\beta & \mbox{in}~~\Omega \\ -\Delta v=\mu b(x)u^p+d(x)u^\theta v^\gamma & \mbox{in}~~\Omega \\ 0\not\equiv u\geq 0,\,\,0\not\equiv v\geq 0& \mbox{in}~~\Omega \\ u=v=0&\mbox{on}~~\partial\Omega \end{array} \right. \end{equation*}

or

\begin{equation*} (\mathcal{P}_2)\,\,\left\{ \begin{array}{ll} -\Delta u=a(x)u^q+\lambda c(x)u^\alpha v^\beta& \mbox{in}~~\Omega \\ -\Delta v=b(x)v^p+\mu d(x)u^\theta v^\gamma & \mbox{in}~~\Omega \\ 0\not\equiv u\geq 0,\,\,0\not\equiv v\geq 0& \mbox{in}~~\Omega \\ u=v=0&\mbox{on}~~\partial\Omega \end{array} \right. \end{equation*}

where $p,\,q\in(0,1)$, concerning the exponents, throughout this work, we will always assume that $\max\{\alpha,\,\beta\} \gt 1$ and $\max\{\theta,\,\gamma\} \gt 1$. More specifically, we have the following hypotheses:

1. (PW₁) $\alpha,\,\gamma\geq1$ and $\beta,\,\theta\, \gt 0$;

2. (PW₂) $0 \lt \alpha \lt 1 \lt \beta$ and $0 \lt \gamma \lt 1 \lt \theta$;

3. (PW₃) $\alpha\geq 1,\,\,\beta \gt 0$ and $0 \lt \gamma \lt 1 \lt \theta$.

We will comment on the differences between $(\mathcal{P}_1)$ and $(\mathcal{P}_2)$ by comparing them with their scalar versions. Observe that u is a solution of $(\mathcal{P}_\lambda)$ if and only if $\overline{u}=\lambda^\frac{-1}{1-q}u$ is a solution to $(\mathcal{P}^{\overline{\lambda}})$ where $\overline{\lambda}=\lambda^\frac{p-1}{1-q}$, and

(𝒫^λ)\begin{align} \left\{ \begin{array}{ll} -\Delta u=a(x)u^q+\lambda b(x)u^p& \text{in}~~\Omega \\ u=v=0&\text{on}~~\partial\Omega. \end{array} \right. \end{align}

When $a(x)\geq 0$, the global results provided by [Reference De Figueiredo, Gossez and Ubilla26] imply global results for $(\mathcal{P}^\lambda)$. Here, global means that it is possible to establish for which parameters λ > 0 the problem has a solution and for which it does not, which contrasts with the local case where existence can only be established for small parameters. However, when $b(x)\geq 0$ and a(x) change sign, we cannot derive global results for $(\mathcal{P}^\lambda)$ from the results of [Reference De Figueiredo, Gossez and Ubilla26]. Global results for $(\mathcal{P}^\lambda)$ were obtained by De Paiva in [Reference De Paiva27] when $b(x)\geq 0$ and $1 \lt p\leq 2^*-1$.

Now, suppose that $(\alpha-1)(\gamma-1)\neq\theta\beta$. Then, for all $\overline{\lambda} \gt 0$ and $\overline{\mu} \gt 0$, the following system has a solution

(1.8)\begin{equation} \left\{ \begin{array}{lcl} (\alpha-1) x+\beta y&=&\ln\overline{\lambda} \\ \theta x+(\gamma-1) y&=&\ln{\overline{\mu}}. \end{array} \right. \end{equation}

Then, $\lambda:=e^{x(1-q)} \gt 0$ and $\mu:=e^{y(1-p)} \gt 0$ satisfy $\overline{\lambda}=\lambda^\frac{\alpha-1}{1-q}\mu^\frac{\beta}{1-p}$ and $\overline{\mu}=\lambda^\frac{\theta}{1-q}\mu^\frac{\gamma-1}{1-p}$. Setting $t=\lambda^\frac{-1}{1-q}$ and $s=\mu^\frac{-1}{1-p}$, we see that u and v are solutions for $(\mathcal{P}_1)$ with $\lambda,\mu \gt 0$ if and only if $\overline{u}=tu$ and $\overline{v}=sv$ are solutions for $(\mathcal{P}_2)$ with $\overline{\lambda},\overline{\mu} \gt 0$. Therefore, the global results for $(\mathcal{P}_1)$ with certain hypotheses on the weights yield global results for $(\mathcal{P}_2)$ with the same hypotheses.

But when the system (1.8) has no solution, this approach does not work. Therefore, even in the case that all weights are non-negative, we cannot obtain a solution $(u_2,v_2)$ to $(\mathcal{P}_2)$ with $u_2=tu_1$ and $v_2=sv_1$ for some $t,s \gt 0$ and $(u_1,v_1)$ solution to $(\mathcal{P}_1)$. This does not happen in the scalar case, since when the weights are non-negative, global results for $(\mathcal{P}_\lambda)$ guarantee global results for $(\mathcal{P}^\lambda)$.

We will see that the conditions $(PW_1)$–$(PW_3)$ affect the set of values $\lambda,\mu \gt 0$ for which the system has a solution (u, v). Before formalizing the concept of the solution we will employ, it is pertinent to note that since the weight functions lie in ${L^\infty(\Omega)}$, our optimal expectation is that solutions belong to ${C^1(\overline{\Omega})}$. Thus, we shall consider the following definition:

Concerning the existence of solutions, throughout this work, we will assume that ${\Omega \subset \mathbb{R}^N}$ is a bounded domain such that ${\partial \Omega \in C^{1,1}}$. Except for the solution of item (v) of Theorem 1.4, this regularity is sufficient to obtain solutions in ${W^{2,s}(\Omega) \cap W^{1,s}_0(\Omega) \cap C^1(\overline{\Omega})}$ for all s > 1. To this end, we will invoke the following classical results from elliptic regularity theory: Theorems 7.26 and 9.15 of [Reference Gilbarg and Trudinger31], which require that ${\Omega \subset \mathbb{R}^N}$ be a domain such that ${\partial \Omega\in C^{0,1}}$ and ${\partial \Omega\in C^{1,1}}$, respectively. Regarding the non-existence of solutions, we require the existence of ${\Omega_0 \subset \Omega}$ such that ${\partial \Omega_0 \in C^{1,1}}$, so that we can use Green’s identities (which require ${\partial \Omega_0 \in C^1}$) and the results of [Reference Gilbarg and Trudinger31, Theorems 9.15 and Lemma 9.17] in Proposition 2.7 (which require ${\partial \Omega_0 \in C^{1,1}})$.

Let $\Omega_0\subset\Omega$ be open and connected. We say that a function $h:\overline{\Omega}\rightarrow\mathbb{R}$ satisfies the condition $(P_{\Omega_0})$ if $h(x)\geq 0$ for all $x\in\Omega_0$, and the set $\{x\in\Omega_0\,\,|\,\,h(x) \gt 0\}$ has positive Lebesgue measure. The conditions on the weights are stated below.

1. (P₁) $a,b,c,d\in L^\infty(\Omega)\setminus\{0\}$;

2. (P₂) $a(x)\geq 0$ and $b(x)\geq 0$ in Ω;

Regarding condition $(P_1)$, we do not know if the results of the present work are valid for the case where the weight functions are in an appropriate L^p space. We have not delved into this issue.

Certain constants will appear in the form of powers associated with the questions of existence and non-existence. To better organize the presentation of our first theorem, we present these constants in the table below.

In our first result, we show that there exist $\tau,\sigma,\overline{\tau},\overline{\sigma}\geq0$ satisfying $\tau+\sigma \gt 0$, $\overline{\tau}+\overline{\sigma} \gt 0$, and $\Lambda^*_1,\Lambda^*_2 \gt 0$ such that problem $(\mathcal{P}_1)$ has no solution if $\lambda^\tau\mu^\sigma \gt \Lambda^*_1$ or $\lambda^{\overline{\tau}}\mu^{\overline{\sigma}} \gt \Lambda^*_2$. The same occurs with $(\mathcal{P}_2)$. Specifically, we prove the following:

In [Reference Ambrosetti, Brézis and Cerami4], the non-existence of positive solutions to $(\mathcal{P}_\lambda)$ is a consequence of the simple inequality: Given c > 0, there exists $\lambda=\lambda_c \gt 0$ such that $\lambda t^q+t^p \gt ct$, $\forall \,t \gt 0$ (see [Reference Ambrosetti, Brézis and Cerami4, (3.1) with $c=\lambda_1(\Omega)$]). However, even for the case $f_\lambda=\lambda t^q \,+\, t^\alpha s^\beta$ and $g_\mu=\mu s^p \,+\, t^\theta s^\gamma$, for all $\lambda,\mu \gt 0$, there is no c > 0 satisfying $f_\lambda(x,t,s) \gt ct$ and $g_\mu(x,t,s) \gt cs$ simultaneously for all $t,s \gt 0$. Hence, the approach for the non-existence of solutions is different in the context of systems. Lemma 2.1 plays a key role in our proof. It is a slight adaptation of [Reference Ambrosetti, Brézis and Cerami4, Lemma 3.3]. (Observe that in the work [Reference Ambrosetti, Brézis and Cerami4], this lemma is not relevant to show the result of non-existence.) Our proof is based on technical arguments, and in some cases, we use the Krein–Rutman theorem [Reference Krein and Rutman37].

The problem $(\mathcal{P}_2)$ is related to $(\mathcal{P}^\lambda)$. In [Reference De Paiva27, Theorem 1], the author showed a global non-existence result when $0 \lt q \lt 1 \lt p\leq 2^*-1$, $b(x)\geq 0$ in Ω, and $\overline{\{x\in\Omega\,\,|\,\,a(x) \gt 0\}}\cap\overline{\{x\in\Omega\,\,|\,\,a(x) \lt 0\}}=\emptyset$ with other conditions. Therefore, concerning the non-existence question, Theorem 1.2 complements this result since the hypotheses considered are weaker than [Reference De Paiva27, Theorem 1] and we are considering supercritical powers, that is, $\alpha+\beta \gt 2^*-1$ or $\theta+\gamma \gt 2^*-1$ in $(\mathcal{P}_2)$.

In order to present our next theorem, we define the following hypothesis:

\begin{equation*} \begin{array}{ll} (H_1) c(x)\geq 0 \mbox{ in } \Omega; & (H_4) d(x)\geq 0 \mbox{ in } \Omega; \\ (H_2) \alpha\geq 1; & (H_5) \gamma\geq 1; \\ (H_3) q \lt \alpha \lt 1 \mbox{ and } \inf\limits_\Omega a(x) \gt 0; & (H_6) p \lt \gamma \lt 1 \mbox{ and } \inf\limits_\Omega b(x) \gt 0. \\ \end{array} \end{equation*}

In our second result, in the main item, we show that there exist $\tau,\sigma,\overline{\tau},\overline{\sigma}\geq0$ satisfying $\tau+\sigma \gt 0$, $\overline{\tau}+\overline{\sigma} \gt 0$, and $\Lambda_1,\Lambda_2 \gt 0$ such that problem $(\mathcal{P}_1)$ has at least one solution if $\lambda^\tau\mu^\sigma\leq\Lambda_1$ and $\lambda^{\overline{\tau}}\mu^{\overline{\sigma}}\leq\Lambda_2$. The same occurs with $(\mathcal{P}_2)$. Specifically, we prove the following:

With the purpose of exemplifying the previous theorem, let us consider the following system:

(1.9)\begin{equation} \left\{ \begin{array}{ll} -\Delta u=\lambda u^q+u^\alpha v^\beta & \mbox{in}~~\Omega \\ -\Delta v=\mu u^p+u^\theta v^\gamma & \mbox{in}~~\Omega \\ u \gt 0,\,\,v \gt 0& \mbox{in}~~\Omega \\ u=v=0&\mbox{on}~~\partial\Omega. \end{array} \right. \end{equation}

Supposing that $p,q\in(0,1)$ and $\lambda,\mu \gt 0$. Regarding existence, Theorem 1.4-(i) ensures the existence of $\Lambda_i \gt 0$ such that:

1. (E₁) Assuming that $\alpha\geq1,\,\gamma\geq 1,\,\beta \gt 0,\,\theta \gt 0$, then Equation (1.9) has a solution if:

   \begin{equation*} \lambda^\frac{\alpha-1}{1-q}\mu^\frac{\beta}{1-p}\leq\Lambda_1 \,\,\,\mbox{and}\,\, \lambda^\frac{\theta}{1-q}\mu^\frac{\gamma-1}{1-p}\leq\Lambda_2 \end{equation*}

2. (E₂) Assuming that $0 \lt \alpha \lt 1 \lt \beta$ and $0 \lt \gamma \lt 1 \lt \theta$, then Equation (1.9) has a solution if:

   \begin{equation*} \lambda\leq \Lambda_1\,\,\,\mbox{and}\,\,\,\mu\leq\Lambda_2 \end{equation*}

3. (E₃) Assuming that $\alpha\geq 1$, β > 0, and $0 \lt \gamma \lt 1 \lt \theta$, then Equation (1.9) has a solution if:

   \begin{equation*} \lambda\leq\Lambda_1\,\,\,\mbox{and}\,\,\,\lambda^\frac{\alpha-1}{1-q}\mu^\frac{\beta}{1-p} \leq\Lambda_2. \end{equation*}

Note that if α > 1 or γ > 1, then the result of $(E_1)$ ensures that the set $\mathcal{E}:=$ $\{(\lambda,\mu)\in\mathbb{R}^2_+ |$ (1.9) $\mbox{has a solution}\,\}$ is unbounded. Furthermore, if α > 1, then the result of $(E_3)$ also implies that $\mathcal{E}$ is unbounded. These results contrast with the scalar case, since $\left\{\lambda\in\mathbb{R}_+\,\,|\,\,{(\mathcal{P}_\lambda)}\,\,\mbox{has a solution}\,\right\}=(0,\Lambda]$ and $\Lambda \lt +\infty$ for the case where the weight functions of $(\mathcal{P}_\lambda)$ are constants.

To prove Theorem 1.4, we use the sub-super solution method. The way we define the concept of sub-super solution for systems is somewhat different than in the scalar case. For example, a solution of $(\mathcal{P}_i)$ may not be a sub-solution or a super-solution for $(\mathcal{P}_i)$ (see Remarks 3.1 and 3.5).

The super-solutions are obtained by solving the systems Equation (3.1) and (3.2). The greatest difficulty lies in obtaining solutions when one of the parameters λ > 0 or µ > 0 can be arbitrarily large. The sub-solutions come in two types of functions. To obtain the first, we use the theory of principal eigenfunction for operators with indefinite weight functions. For that, we choose an appropriate weight function (see, for instance, Equation (3.4)). To obtain the second sub-solution, we solve an auxiliary problem employing variational techniques (see Proposition 3.6).

The novelty of this theorem lies in the fact that we solve the problem of the existence of a solution for a more general class of systems (not just gradient systems and not just with critical or subcritical growth powers), and we prove that the set of solutions $\mathcal{O}_i:=\left\{(\lambda,\mu)\in\mathbb{R}_+^2\,\,|\,\,(\mathcal{P}_i)\,\,\mbox{has solution}\,\right\}$ is unbounded in some cases. To prove (v), we use the same approach as in [Reference Ambrosetti, Brézis and Cerami4, p. 528]. However, the same idea does not work when $\alpha\leq 1$ or $\gamma\leq 1$, although we believe that this statement holds for other cases. Unfortunately, we cannot prove it. Despite this, in [Reference Da Silva23], we have a complete proof for when the system is of the gradient type.

Originally, this work contained a study involving the multiplicity of solutions for gradient-type systems, as well as analyzing cases where all weights change sign and the nonlinearities have supercritical growth terms. However, we decided to address this in another paper [Reference Da Silva23]. The issue of local minimization for functionals with two variables was also addressed in another paper [Reference Da Silva24]. However, the question of multiplicity for a system that is not of the gradient type remains open. A major difficulty is that if the system cannot be studied via variational methods, we do not know how to approach the issue of multiplicity. Unfortunately, we were unable to answer some other questions. For example, can we have the maps $\mu\mapsto\Lambda_\mu$ and $\lambda\mapsto\Lambda_\lambda$ continuous? Is it possible that $\Lambda_{1ij}=\Lambda^*_{1ij}$ and $\Lambda_{2ij}=\Lambda^*_{2ij}$? If $0 \lt \lambda=L^*_{ij} \lt +\infty$, can we have $\Lambda_\lambda \gt 0$? If $0 \lt \mu=M^*_{ij} \lt +\infty$, can we have $\Lambda_\mu \gt 0$?

After the introduction, this work is divided into three other sections. The second section is dedicated to the question of non-existence of solutions for $(\mathcal{P}_i)$. In the third section, we present the method of sub-super solution, and finally, in the fourth and last section, we present the proof of Theorem 1.4.