The existence of infinitely many boundary blow-up solutions to the <i>p</i>-<i>k</i>-Hessian equation

Abstract

Abstract The primary objective of this article is to analyze the existence of infinitely many radial <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p - <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -convex solutions to the boundary blow-up <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p - <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -Hessian problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msub> <m:mrow> <m:mi>σ</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>D</m:mi> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msub> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>H</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mspace width="0.33em"/> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> <m:mspace width="0.33em"/> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width="0.33em"/> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mo>+</m:mo> <m:mi>∞</m:mi> <m:mspace width="0.33em"/> <m:mspace width="0.1em"/> <m:mtext>on</m:mtext> <m:mspace width="0.1em"/> <m:mspace width="0.33em"/> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>.</m:mo> </m:math> {\sigma }_{k}\left(\lambda \left({D}_{i}\left({| Du| }^{p-2}{D}_{j}u)))=H\left(| x| )f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{0.33em}u=+\infty \hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega . Here, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> k\in \left\{1,2,\ldots ,N\right\} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>σ</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\sigma }_{k}\left(\lambda ) </jats:a

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