Nonlinear equations and weighted norm inequalities

作者：N. J. Kalton, Igor E. Verbitsky · 发表于：Transactions of the American Mathematical Society · 年份：1999 · DOI：10.1090/s0002-9947-99-02215-1 · 被引用次数：128 · 研究领域：Advanced Harmonic Analysis Research、Nonlinear Partial Differential Equations、Numerical methods in inverse problems

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column minus normal upper Delta u 2nd Column a m p semicolon equals v u Superscript q Baseline plus w comma u greater-than-or-equal-to 0 3rd Column a m p semicolon 4th Column a m p semicolon on normal upper Omega comma 2nd Row 1st Column u 2nd Column a m p semicolon equals 0 3rd Column a m p semicolon on partial-differential normal upper Omega comma EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>u</mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:mi>v</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>u</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:...