<p>In this article, we will prove existence results for the equations of the type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$-\Delta _{N}u=H_{l}(u)+\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>N</mi> </msub> <mi>u</mi> <mo>=</mo> <msub> <mi>H</mi> <mi>l</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$F_{\frac{N}{2}}[-u]=H_{l}(u)+\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </msub> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mi>u</mi> <mo stretchy="false">]</mo> </mrow> <mo>=</mo> <msub> <mi>H</mi> <mi>l</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> in a bounded domain <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, with Dirichlet boundary condition, where the source term <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H_{l}(r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>l</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> takes the form <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$e^{r}-\sum _{j=0}^{l-1}\frac{r^{j}}{j!}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>e</mi> <mi>r</mi> </msup> <mo>-</mo> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>l</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mfrac> <msup> <mi>r</mi> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is a nonnegative Radon measure.</p>

N-Laplacian and N/2-Hessian Type Equations with Exponential Reaction Terms and Measure Data

期刊论文
发表日期：

2025年12月19日

查看全文

Shiguang Ma,
Zijian Wang

摘要

In this article, we will prove existence results for the equations of the type $-\Delta _{N}u=H_{l}(u)+\mu $ $- Δ_{N} u = H_{l} (u) + μ$ and $F_{\frac{N}{2}}[-u]=H_{l}(u)+\mu $ $F_{\frac{N}{2}} [- u] = H_{l} (u) + μ$ in a bounded domain $\Omega $ $Ω$ , with Dirichlet boundary condition, where the source term $H_{l}(r)$ $H_{l} (r)$ takes the form $e^{r}-\sum _{j=0}^{l-1}\frac{r^{j}}{j!}$ $e^{r} - \sum_{j = 0}^{l - 1} \frac{r^{j}}{j!}$ and $\mu $ $μ$ is a nonnegative Radon measure.