Our paper centers on the study of k-Hessian equations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_{k}^{\frac{1}{k}}[\nu (D^{2}\varphi -A(|x|)I)]=f(-\varphi ,|\nabla \varphi |)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>k</mi> </mrow> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </msubsup> <mrow> <mo stretchy="false">[</mo> <mi>ν</mi> <mo stretchy="false">(</mo> </mrow> <msup> <mi>D</mi> <mn>2</mn> </msup> <mrow> <mi>φ</mi> <mo>-</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>φ</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the presence of a negative augmented term under the Robin boundary condition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varphi +\lambda \displaystyle \frac{\partial \varphi }{\partial n}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>φ</mi> <mo>+</mo> <mi>λ</mi> <mfrac> <mrow> <mi>∂</mi> <mi>φ</mi> </mrow> <mrow> <mi>∂</mi> <mi>n</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow> </mstyle> </math></EquationSource> </InlineEquation>. By using the fixed point theorem and mixed monotone iteration technique, we not only obtain the existence of the unique positive radial solution of the k-Hessian equation with a negative augmented term, but also construct the explicit monotone iterative sequences that converge uniformly to the unique positive radial solution. Subsequently, we give an example to support our existence conclusion.

The unique positive radial solution for a class of k-Hessian equation with a negative augmented term

期刊论文
发表日期：

2025年9月10日

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Huimin Li,
Guotao Wang

摘要

Our paper centers on the study of k-Hessian equations $S_{k}^{\frac{1}{k}}[\nu (D^{2}\varphi -A(|x|)I)]=f(-\varphi ,|\nabla \varphi |)$ $S_{k}^{\frac{1}{k}} [ν (D^{2} φ - A (| x |) I)] = f (- φ, | \nabla φ |)$ in the presence of a negative augmented term under the Robin boundary condition $\varphi +\lambda \displaystyle \frac{\partial \varphi }{\partial n}=0$ $φ + λ \frac{\partial φ}{\partial n} = 0$ . By using the fixed point theorem and mixed monotone iteration technique, we not only obtain the existence of the unique positive radial solution of the k-Hessian equation with a negative augmented term, but also construct the explicit monotone iterative sequences that converge uniformly to the unique positive radial solution. Subsequently, we give an example to support our existence conclusion.