Suppose <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n<5p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mn>5</mn> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p \in (1, 2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n<p+\frac{4p}{p-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mi>p</mi> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <mi>p</mi> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We establish an a priori Hölder estimate for regular stable solutions to nonlinear equations involving p-Laplacian <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-\Delta _p u = f(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f \in C^1(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>C</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f(t)\ge -A|t|^{\gamma }-K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mo>-</mo> <msup> <mrow> <mi>A</mi> <mo stretchy="false">|</mo> <mi>t</mi> <mo stretchy="false">|</mo> </mrow> <mi>γ</mi> </msup> <mo>-</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$t\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> with some nonnegative constants A, K and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>. Furthermore, when <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> we obtain the local Hölder regularity for stable energy solutions. Notice that the dimension range <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$n<p+\frac{4p}{p-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mi>p</mi> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <mi>p</mi> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is optimal when <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$p \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Boundedness of stable solutions to a class of nonlinear equations involving p-Laplacian

期刊论文
发表日期：

2026年1月28日

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Yaqing Peng,
Rongzhen Shi,
Wenming Zou

摘要

Suppose $n<5p$ $n < 5 p$ when $p \in (1, 2)$ $p \in (1, 2)$ and $n<p+\frac{4p}{p-1}$ $n < p + \frac{4 p}{p - 1}$ when $p \ge 2$ $p \geq 2$ . We establish an a priori Hölder estimate for regular stable solutions to nonlinear equations involving p-Laplacian $-\Delta _p u = f(u)$ $- Δ_{p} u = f (u)$ , where $f \in C^1(\mathbb {R})$ $f \in C^{1} (R)$ satisfies $f(t)\ge -A|t|^{\gamma }-K$ $f (t) \geq - {A | t |}^{γ} - K$ for all $t\in \mathbb {R}$ $t \in R$ with some nonnegative constants A, K and $\gamma $ $γ$ . Furthermore, when $p=2$ $p = 2$ we obtain the local Hölder regularity for stable energy solutions. Notice that the dimension range $n<p+\frac{4p}{p-1}$ $n < p + \frac{4 p}{p - 1}$ is optimal when $p \ge 2$ $p \geq 2$ .