Let L be a second order uniformly elliptic differential operator in a domain D of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\psi :\mathbb {R}_+\rightarrow \mathbb {R}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>:</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo stretchy="false">→</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation> be a nondecreasing continuous function and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\xi ,g:D\rightarrow \mathbb {R}_+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>D</mi> <mo stretchy="false">→</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation> be locally bounded Borel measurable functions. Under appropriate conditions, we determine a function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> with values in ]0, 1] such that for every nonnegative solution to inequality <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-Lu+\xi \psi (u) \ge g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>L</mi> <mi>u</mi> <mo>+</mo> <mi>ξ</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation> in D and for every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$x\in D$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>D</mi> </mrow> </math></EquationSource> </InlineEquation>, <Equation ID="Equ64"> <EquationSource Format="TEX">$\begin{aligned} u(x)\ge p(x)\,\varphi \left( \frac{G_D(\xi \psi (p))(x)}{p(x)}\right) , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mi>φ</mi> <mfenced close=")" open="("> <mfrac> <mrow> <msub> <mi>G</mi> <mi>D</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p=G_Dg$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <msub> <mi>G</mi> <mi>D</mi> </msub> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation> is the Green function of g. The function <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is completely determined by <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> and does not depend on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L,D,\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>,</mo> <mi>D</mi> <mo>,</mo> <mi>ξ</mi> </mrow> </math></EquationSource> </InlineEquation> or g.

Lower Bound Estimates of Nonnegative Solutions to Semilinear Elliptic Inequalities

期刊论文
发表日期：

2025年12月17日

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Basma Nayli,
Khalifa El Mabrouk

摘要

Let L be a second order uniformly elliptic differential operator in a domain D of $\mathbb {R}^{d}$ $R^{d}$ , $\psi :\mathbb {R}_+\rightarrow \mathbb {R}_+$ $ψ : R_{+} \to R_{+}$ be a nondecreasing continuous function and let $\xi ,g:D\rightarrow \mathbb {R}_+$ $ξ, g : D \to R_{+}$ be locally bounded Borel measurable functions. Under appropriate conditions, we determine a function $\varphi $ $φ$ with values in ]0, 1] such that for every nonnegative solution to inequality $-Lu+\xi \psi (u) \ge g$ $- L u + ξ ψ (u) \geq g$ in D and for every $x\in D$ $x \in D$ , $\begin{aligned} u(x)\ge p(x)\,\varphi \left( \frac{G_D(\xi \psi (p))(x)}{p(x)}\right) , \end{aligned}$ $\begin{matrix} u (x) \geq p (x) φ (\frac{G_{D} (ξ ψ (p)) (x)}{p (x)}), \end{matrix}$ where $p=G_Dg$ $p = G_{D} g$ is the Green function of g. The function $\varphi $ $φ$ is completely determined by $\psi $ $ψ$ and does not depend on $L,D,\xi $ $L, D, ξ$ or g.