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- W4378802589 abstract "Abstract In this paper, we mainly consider the singular k -Hessian equations <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msup> <m:mi>D</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mi>h</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>f</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo stretchy=false>|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo mathvariant=italic separator=true> </m:mo> <m:mrow> <m:mtext>in </m:mtext> <m:mo></m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> S_{k}(lambda(D^{2}u))=h(x)f(-u)+g(|Du|)quadtext{in }Omega and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msup> <m:mi>D</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>h</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>f</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo stretchy=false>|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo mathvariant=italic separator=true> </m:mo> <m:mrow> <m:mtext>in </m:mtext> <m:mo></m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> S_{k}(lambda(D^{2}u))=h(x)f(-u)(1+g(|Du|))quadtext{in }Omega with the Dirichlet boundary condition <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {u=0} on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>∂</m:mo> <m:mo></m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:mrow> </m:math> {partialOmega} , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi mathvariant=normal>Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:math> {Omegasubsetmathbb{R}^{N}} ( <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {Ngeq 2} ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h , f and g . This problem is more difficult to solve than the k -Hessian problem without gradient terms, and requires additional new conditions in the proof process." @default.
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- W4378802589 date "2023-06-01" @default.
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- W4378802589 title "Existence and asymptotic behavior of strictly convex solutions for singular <i>k</i>-Hessian equations with nonlinear gradient terms" @default.
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- W4378802589 doi "https://doi.org/10.1515/gmj-2023-2033" @default.
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