The existence and asymptotic behavior of large solutions to the k -Hessian equations with nonlinear gradient terms

In this paper, we establish the first- and second-order asymptotic behaviors (expansions) of boundary blow-up solutions to the k -Hessian problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>b</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>∇</m:mi> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mtext> in </m:mtext> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width="0.28em"/> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mo>+</m:mo> <m:mi>∞</m:mi> <m:mtext> on </m:mtext> <m:mi>∂</m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>.</m:mo> </m:math> Sₖ(D²u)=b(x)f(u)(C₀+ ∇ u ²)q/2\,in\,Ω, u=+∞ \,on\,∂ Ω. Here, Ω is a smooth, bounded, and strictly convex domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mspace width="0.3333em"/> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> RN (N≥ 2) , q ∈ [0, k + 1), C 0 > 0 is a constant, b ∈ C ∞ (Ω) is positive in Ω and may be (critically) singular or vanish on ∂ Ω, and f ∈ C ∞ (0, + ∞ ) ∩ C [0, + ∞ ) (or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> f∈ C∞(R) ) is positive and increasing on [0, + ∞ ) (or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">R</m:mi> </m:math> R ). Under various conditions, we reveal the influences of principal curvatures of ∂ Ω and the gradient term on the coefficients of the expansions of large solutions to this problem. Particularly, when b becomes critically singular on