<p>This paper is mainly concerned with the existence and asymptotic behavior of convex large classical solutions to the Monge-Ampère type equation with double weights <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{det}\ D^2 u(x)= a(x) f(u(x))+b(x)|\nabla u(x)|^q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>det</mtext> <mspace width="4pt" /> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x\in \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a strictly convex and bounded smooth domain in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$q>n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f(s)=s^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>s</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p>n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f(s)=\exp s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>exp</mo> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a, b\in C^\infty (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <msup> <mi>C</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$a_1d^\alpha (x)\le a(x)\le a_2 d^\alpha (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <msup> <mi>d</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <msup> <mi>d</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$ b_1d^\beta (x)\le b(x)\le b_2 d^\beta (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <msup> <mi>d</mi> <mi>β</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <msup> <mi>d</mi> <mi>β</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$x\in \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$a_i, b_i>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq15"> <EquationSource Format="TEX">$i=1, 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\alpha >-n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mo>-</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\beta \ge q-n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>≥</mo> <mi>q</mi> <mo>-</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We completely describe how <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$n, p, q, \alpha , \beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> affect the asymptotic behavior of solutions to such problem.</p>

Existence and asymptotic behavior of large solutions for a class of equations of Monge-Ampère type

期刊论文
发表日期：

2026年3月7日

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Zhijun Zhang

摘要

This paper is mainly concerned with the existence and asymptotic behavior of convex large classical solutions to the Monge-Ampère type equation with double weights $\textrm{det}\ D^2 u(x)= a(x) f(u(x))+b(x)|\nabla u(x)|^q$ $det D^{2} u (x) = a (x) f (u (x)) + b (x) {| \nabla u (x) |}^{q}$ , $x\in \Omega $ $x \in Ω$ , where $\Omega $ $Ω$ is a strictly convex and bounded smooth domain in $\mathbb {R}^n$ $R^{n}$ with $n\ge 2$ $n \geq 2$ , $q>n$ $q > n$ , $f(s)=s^p$ $f (s) = s^{p}$ with $p>n$ $p > n$ , or $f(s)=\exp s$ $f (s) = exp s$ , and $a, b\in C^\infty (\Omega )$ $a, b \in C^{\infty} (Ω)$ with $a_1d^\alpha (x)\le a(x)\le a_2 d^\alpha (x)$ $a_{1} d^{α} (x) \leq a (x) \leq a_{2} d^{α} (x)$ , $ b_1d^\beta (x)\le b(x)\le b_2 d^\beta (x)$ $b_{1} d^{β} (x) \leq b (x) \leq b_{2} d^{β} (x)$ , $x\in \Omega $ $x \in Ω$ for some $a_i, b_i>0$ $a_{i}, b_{i} > 0$ ( $i=1, 2$ $i = 1, 2$ ) and $\alpha >-n-1$ $α > - n - 1$ , $\beta \ge q-n-1$ $β \geq q - n - 1$ . We completely describe how $n, p, q, \alpha , \beta $ $n, p, q, α, β$ and $\partial \Omega $ $\partial Ω$ affect the asymptotic behavior of solutions to such problem.