Generalized convex-growth existence and uniqueness conjecture for regime-switching HJB systems

Let $k\geq 2$. For each $j\in\\{1,\dots,k\\}$, let $p_j>1$, set $q_j=p_j/(p_j-1)$, and suppose that there are a convex function $g_j\geq 0$ and constants $C_{\varepsilon,j}\geq 0$ for every $\varepsilon>0$ such that

$$(1-\varepsilon)g_j(x)-\varepsilon|x|^{q_j}-C_{\varepsilon,j}\leq f_j(x)\leq(1+\varepsilon)g_j(x)+\varepsilon|x|^{q_j}+C_{\varepsilon,j}.$$

Assume that the coupling matrix $(\alpha_{j\ell})$ satisfies $\alpha_{j\ell}\geq 0$ for $\ell\ne j$ and $\sum_{\ell=1}^k\alpha_{j\ell}=0$, and that $\delta_j>0$. Define

\mathcal{A}_g^k:=\left\\{u=(u_1,\dots,u_k):u_j\in W_{\mathrm{loc}}^{2,m}(\mathbb{R}^N)\text{ for all }m>N,\\ \liminf_{|x|\to\infty}u_j(x)|x|^{-q_j}\geq 0\right\\}.

Generalized convex-growth existence and uniqueness conjecture. Under these assumptions, the system admits a unique solution $u\in\mathcal{A}_g^k$. This conjecture proposes extending the paper's power-growth existence and uniqueness theorem to continuous running costs controlled by convex reference functions. The main unresolved issues are constructing compatible componentwise barriers and proving comparison when the exponents and reference functions differ across components.