DOI: 10.1515/dema-2025-0314 ISSN: 2391-4661

Existence and asymptotic behavior of radial solutions for a k -Hessian equation with the gradient

Zedong Yang
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Abstract

In this work, we focus on the existence and asymptotic behavior of radial solutions for the k -Hessian equation with the nonlinear gradient terms

S k σ D 2 u = λ b ( | x | ) φ ( − u ) | ∇ u | m , i n Ω , u = 0 , o n ∂ Ω , $$\begin{cases}_{k}\left(\sigma \left({D}^{2}u\right)\right)=\lambda b\left(\vert x\vert \right)\varphi \left(-u\right)\vert \nabla u{\vert }^{m}, \mathrm{i}\mathrm{n} {\Omega},\hfill \\ u=0, \mathrm{o}\mathrm{n} \partial {\Omega},\hfill \end{cases}$$

where λ > 0, 0 ≤ m < k and |∇ u | denotes the gradient of u . The main methods of proof rely on the eigenvalue theory established by Guo and Lakshmikantham, as well as Guo’s fixed point theorem, combined with techniques such that Jensen inequality.

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