Existence conjecture for weak solutions of the coupled p-Laplacian system

Let $\Omega\subset\mathbb{R}^N$ be the domain, let $p^*$ denote the Sobolev critical exponent, and let $(p^*)'$ be its conjugate exponent. Let $A>0$, let $r>1$, and let $0\leq\theta<p-1$. The system referred to as

has unknowns $(u,\varphi)$ and seeks a weak solution in $W^{1,p}_{0}(\Omega)\times W^{1,p}_{0}(\Omega)$. **Existence conjecture.** There \exists $1<\underline{m}<(p^*)'$ such that, if $f\in L^m(\Omega)$ with $m\geq\underline{m}$, then system

admits a weak solution $(u,\varphi)\in W^{1,p}_{0}(\Omega)\times W^{1,p}_{0}(\Omega)$.

The conjecture proposes a regularizing effect for the first equation in the case $\theta>0$, where the authors state that their method does not establish existence of a finite-energy solution. The proposed threshold $\underline{m}$ depends on the parameters and is required to lie strictly between $1$ and $(p^*)'$; no resolution is supplied here.