This paper studies the existence and multiplicity of radial p-k-convex boundary blow-up solutions for the following augmented p-k-Hessian equation \(\left\{ \begin{array}{lc} \mathcal {S}_{k}\left( \lambda \left( D_{i}\left( |D u|^{p-2} D_{j} u\right) -\varepsilon I\right) \right) =\beta (|z|)(|u|+l)^{m}(\ln (|u|+l))^{\mu },z \in E,\\ u=+\infty ,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad z \in \partial E, \end{array}\right. \) where \(k \in \{1,2, \ldots , n\}\) , \(p \geqslant 2\) , \(m > 0\) , \({\mu } \ge 0\) , \(l>1 \) , \(\varepsilon \) is a constant \((\varepsilon \ne 0 )\) , I is identity matrix and \( E \subset \mathbb {R}^n (n \geqslant 2)\) denotes a ball. In the case where \(0< m < (p-1)k\) and \({\mu }=0\) , the multiplicity of radial p-k-convex boundary blow-up solutions for the aforementioned equation is established by applying the sub-super solution method. In the case where \(m = (p-1)k\) and \(0< {\mu } < (p-1)k\) , a novel auxiliary function is constructed to overcome the challenges posed by the logarithmic nonlinearity, thereby guaranteeing the existence of infinitely many radial p-k-convex solutions for the aforementioned augmented p-k-Hessian equation.
