Positive radial solutions of  p -Laplace equations with nonlinear gradient terms on annular domains

This paper discusses the existence of positive radial solutions for the boundary value problem of p -Laplace equation with nonlinear gradient term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mspace width="0.17em"/> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> <m:mo>,</m:mo> <m:mspace width="0.17em"/> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="0.17em"/> <m:mo stretchy="false">|</m:mo> <m:mi>∇</m:mi> <m:mi>u</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width="2em"/> <m:mspace width="2em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mfenced close="|" open=""> <m:mrow> <m:mfrac> <m:mrow> <m:mi>∂</m:mi> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>u</m:mi> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $${cases}-{{Δ}}ₚ u=f( x , u, ∇ u ), x∈ {Ω}, \\ {.∂ u/∂ n }_{∂ {B}₁}=0, u{ }_{∂ {B}₂}=0, {cases}$$ where Δ p u = div(|∇ u | p −2 ∇ u ) is the p -Laplace operator, p > 1, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>\</m:mo> <m:msub> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> Ω=B₂B̄₁ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mfenced close="}" open="{"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow>