Existence of positive radial solutions of general quasilinear elliptic systems

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <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="0.33em"/> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> Ω ⊂ {{R}}ⁿ{0.33em}(n≥ 2) be either an open ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>R</m:mi> </m:mrow> </m:msub> </m:math> {B}R centred at the origin or the whole space. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true" columnspacing="0.33em"> <m:mtr> <m:mtd columnalign="right"> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> </m:mtd> <m:mtd columnalign="center"> <m:mo>=</m:mo> </m:mtd> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <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:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mspace width="1.0em"/> <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:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mi>v</m:mi> </m:mtd> <m:mtd columnalign="center"> <m:mo>=</m:mo> </m:mtd> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <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:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>h</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mspace width="1.0em"/> <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:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \{{array}{rcl}{Δ }ₚu& =& {f}₁(| x| ){g}₁(v){| ∇ u| }α{1.0em}{0.1em}in{0.1em}{0.33em}Ω ,\\ {Δ }ₚv& =& {f}₂(| x| ){g}₂(v)h(| ∇ u| ){1.0em}{0.1em}in{0.1em}{0.33em}Ω ,{array}. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:math> α ≥ 0 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {Δ }ₚ is the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -Laplace operator, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:math> p> 1 , and for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> i,j=1 , 2, we assume <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {f}ᵢ,{g}ⱼ , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>h</m:mi> </m:math> h are continuous, non-negative and non-decreasing functions. For functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {g}ⱼ which grow polynomially, we prove sharp conditions for the existence of positive radial solutions which blow up at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>∂</m:mo> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>R</m:mi> </m:mrow> </m:msub> </m:math> ∂ {B}R , and for the existence of global solutions.