The multiplicity of radial p-k-convex solutions for the p-k-Hessian equation

This paper focuses on radial p-k-convex solutions for the following p-k-Hessian equation \(\begin{aligned} \left\{ \begin{array}{ll} S_{k}(\xi (D_{i}(|Dv|^{p-2}D_{j}v)))=M(|z|){(|v|+1)}^{m}(ln (|v|+1))^{\mu }, z\in E,\\ v=+\infty , ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~z\in \partial E, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{S}_k(\xi (D_i{(|Dv|}^{p-2}{D}_{j}v)))=M(|z|){(|v|+1)}^m(ln(|v|+{1))}^{\mu },z\in E,\hfill \\ v=+\infty ,z\in \partial E,\hfill \end{array}\right)\end{array}$$ where \(p\ge 2\) $p\ge 2$ , \(k\in \{1,2,...,n\}\) $k\in \{1,2,...,n\}$ , \(E\subset \mathbb {R}^{n}(n\ge 2)\) $E\subset {\mathbb{R}}^n(n\ge 2)$ denotes a ball. For the case of \(0<m<(p-1)k\) $0<m<(p-1)k$ , \(\mu =0\) $\mu =0$ , the multiplicity of radial p-k-convex solutions of the above p-k-Hessian equation is established by the sub-supersolutions method. For the case of \(m=(p-1)k\) $m=(p-1)k$ , \(\mu >(p-1)k\) $\mu >(p-1)k$ , we construct a new supporting function to overcome the difficulty caused by logarithmic nonlinearity, which ensures that the above p-k-Hessian equation has infinitely many radial p-k-convex solutions.